diff --git "a/ferrimagnetic resonance/1.json" "b/ferrimagnetic resonance/1.json" new file mode 100644--- /dev/null +++ "b/ferrimagnetic resonance/1.json" @@ -0,0 +1 @@ +[ { "title": "1510.03545v1.Optomagnonic_whispering_gallery_microresonators.pdf", "content": "Optomagnonic whispering gallery microresonators\nXufeng Zhang,1Na Zhu,1Chang-Ling Zou,1and Hong X. Tang1,\u0003\n1Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA\n(Dated: October 14, 2015)\nMagnons in ferrimagnetic insulators such as yttrium iron garnet (YIG) have recently emerged as\npromising candidates for coherent information processing in microwave circuits. Here we demon-\nstrate optical whispering gallery modes of a YIG sphere interrogated by a silicon nitride photonic\nwaveguide, with quality factors approaching 106in the telecom c-band after surface treatments.\nMoreover, in contrast to conventional Faraday setup, this implementation allows input photon po-\nlarized colinearly to the magnetization to be scattered to a sideband mode of orthogonal polarization.\nThis Brillouin scattering process is enhanced through triply resonant magnon, pump and signal pho-\nton modes - all of whispering gallery nature - within an \\optomagnonic cavity\". Our results show\nthe potential use of magnons for mediating microwave-to-optical carrier conversion.\nHybrid magnonic systems have been emerging recently\nas an important approach towards coherent informa-\ntion processing1{9. The building block of such systems,\nmagnon, is the quantized magnetization excitation in\nmagnetic materials10,11. Its great tunability and long life-\ntime make magnon an ideal information carrier. Partic-\nularly, in magnetic insulator yttrium iron garnet (YIG),\nmagnons interact with microwave photons through mag-\nnetic dipole interaction, which can reach the strong and\neven ultrastrong coupling regime thanks to the large spin\ndensity in YIG4{6. Besides, the magnon can also couple\nwith the elastic wave12,13and optical light14,15, it is of\ngreat potential as an information transducer that medi-\nates inter-conversion among microwave photon, optical\nphoton and acoustic phonon. Long desired functions,\nsuch as microwave-to-optical conversion, can be realized\non such a versatile platform.\nMagneto-optical (MO) e\u000bects such as Faraday e\u000bect\nhave been long discovered and utilized in discrete op-\ntical device applications16{18. Based on such e\u000bects,\nmagnons can coherently interact with optical photons.\nOn the one hand, magnon can be generated by optical\npumps19{22. On the other hand, optical photons can be\nused to probe magnon through Brillouin light scattering\n(BLS)15,23. However, in previous studies the typical ge-\nometries are all thin \flm or bulk samples inside which\nthe optical photon interacts with magnon very weakly,\nusually only through a single pass. For high e\u000ecient\nmagnon-photon interaction, it is desirable to obtain triple\nresonance condition of high quality ( Q) factor modes, i.e.,\nthe magnon, the input and the output optical photons are\nsimultaneously on resonance.\nIn this Letter, we demonstrate the magnon-photon in-\nteraction in a high Qoptomagnonic cavity which simul-\ntaneously supports whispering gallery modes (WGMs) of\noptical and magnon resonances. With high-precision fab-\nrication and careful surface treatment, the widely used\nYIG sphere structure, which is inherently an excellent\nmagnonic resonator, exhibits high optical Qfactors in\nour measurements. YIG has a high refractive index (2.2\nin the telecom c-band), which poses a challenge for e\u000e-\ncient light coupling with silica \fber tapers. By employingan integrated silicon nitride optical waveguide with an ef-\nfective index (around 2.0) matching that of YIG, we can\ne\u000eciently couple to both the TM and TE optical reso-\nnances in the YIG sphere. By driving the system with\nan optical pump, the magnons excited by an input mi-\ncrowave signal can be converted into optical signal of a\ndi\u000berent color. Our demonstration shows the great po-\ntential of YIG sphere as a platform to bridge the gap\nbetween magnon and optical photons, paving the way\ntowards using magnon as a transducer for coherent in-\nformation processing between distinct carriers.\nIn magnetized YIG material, magnons are the collec-\ntive excitations of spin states of Fe3+ions24. The cre-\nation or annihilation of a single magnon corresponds to\nthe ground spin \rip. At the microwave frequencies, the\nmagnon state can be manipulated straightforwardly by\nmagnetic dipolar transition using the oscillating mag-\nnetic \felds of microwave photons. While at the optical\nfrequencies, the magnon manipulation becomes di\u000ecult\nbecause the magnetic transition is negligible whilst di-\nrect spin-\rip by electric dipole transition is forbidden25.\nAlternatively, the optical photons can modify the ground\nstate spin through a two-photon transition by means of\nan orbital transition and spin-orbit interaction (the MO\ne\u000bect)14. Such a process has been previously studied us-\ning conventional Faraday setups26, in which light prop-\nagates parallel to the magnetic \feld and interacts with\nmagnon in a single pass. It is natural to consider shaping\nthe YIG into an optical cavity to boost up the magnon-\nphoton interaction as light passes the magnetic material\nmultiple times. Therefore, we propose to use a whisper-\ning gallery resonator made by YIG to provide enhanced\nmagnon-photon coupling in a triply resonant con\fgura-\ntion, and the mechanism is explained in the following\ndiscussions.\nThe optomagnonic cavity we used in our experiments\nis a single crystal YIG sphere. Due to the spherical\nsymmetry, lights are con\fned in the sphere by total in-\nternal re\rection and form WGMs. Each optical WGM\nis characterized by three mode numbers ( q;l;n ), which\ncorrespond to the radial, angular and azimuthal order\n(n=\u0000l;:::;l ), respectively27,28. Moreover, the WGMsarXiv:1510.03545v1 [cond-mat.mes-hall] 13 Oct 20152\nTM\nTE\nωωmSignal Pumpx\nyzσ+\nπ\nσ+\nnm−1nmπ σ+ π(b)(a)\n(c)\nFIG. 1. (a) Schematic illustration of the magnon-photon in-\nteraction. The YIG sphere is biased by a magnetic \feld along\nzdirection, while the WGMs propagate along the perimeter\nin thex-yplane. The TM input light excites the \u0019WGM\nin the YIG sphere, which is scattered by magnon into \u001b+\npolarized photon and then converts to the TE output in the\nwaveguide. (b) Energy level diagram of the magnon-photon\ninteraction. (c) Triple resonance condition for the enhanced\nmagnon-photon interaction process in the optomagnonic res-\nonator.\nare also characterized by their polarization, i.e., the di-\nrection of their electric \feld distribution. Conventional\nFaraday setups require the bias magnetic \feld to be par-\nallel to the direction of light propagation. However, for\nWGMs light propagates along the circumference of the\nsphere [Fig. 1(a)], therefore the bias magnetic \feld should\nbe inx-yplane. Due to the geometry symmetry, the MO\ne\u000bect vanishes for such a Faraday con\fguration. At a\n\frst glance, the MO e\u000bect also vanishes for bias mag-\nnetic \feld along zdirection, since it requires circular\npolarization in respect to\u0000 !Hwhile WGMs are linearly\npolarized, either parallel (TE) or perpendicular (TM) to\nthezdirection. However, thanks to the \feld gradient\nat the dielectric interface29{31, there are non-zero optical\nelectric \felds along the propagation direction for the TE\npolarized WGMs. As a result, the electric \feld rotates\nwithin thex-yplane and forms a cycloid trajectory, sim-\nilar to the elliptically polarized light propagating in free\nspace. Therefore, the TE WGMs possess partial circular\npolarization ( \u001b+) and can have magnetic response via\nFaraday e\u000bect, as schematically illustrated in Fig. 1(a).\nNote that the pump light can propagate either clockwise\n(CW) or counterclockwise (CCW), with di\u000berent conser-\nvation conditions accordingly, as will be shown below.\nSimilar to the optical WGMs, the magnon modes in\nYIG sphere can also be characterized by three mode\nnumbers (qm;lm;nm)32. For the uniform magnon mode\nH\nz\nxy\nOutput fiberIntput fiber\nBefore A/g332er\nFibersWaveguideMagnetChip\nAntenna\nMagnet\n(c) (e)(a)\n(d)(b)FIG. 2. (a) and (b) Schematic and optical image of the ex-\nperimental assembly of our optomagnonic device, respectively.\n(c) Scanning electron microscope image of the polished YIG\nsphere. The scale bar is 100 \u0016m. (d) The surface of the YIG\nsphere before and after our surface treatment process. Scale\nbars are 1\u0016m. The sub-micrometer particles vanish after the\nsurface treatment. (e) Optical image of the silicon nitride\ncoupling waveguide chip with glued \fbers on the two sides.\nThe chip and the \fbers are attached to a piece of glass holder\nfor mechanical support and reducing long-term drift.\nwith all the spins precessing in phase, the corresponding\nmode numbers are (1 ;1;1). The microscopic mechanism\nof the magnon-photon interaction is intrinsically a three-\nwave process, as schematically illustrated by Fig. 1(b).\nDue to the spin angular momentum conservation, every\ntime when the magnon number increases by 1 it indi-\ncates that the electron spin increases by 1, which corre-\nsponds to a two-photon transition in the form of \u001b+!\u0019\n(CCW) or \u0019!\u001b\u0000(CW). As a result, there would be\nonly one optical sideband generated for a given pump-\ning light direction. The mesoscopic model of the MO\ne\u000bect is represented by the permittivity tensor \"ij=\n\"0(\"r\u000eij\u0000if\u000fijkMk)24, where\"0is the vacuum permittiv-\nity,\"ris the relative permittivity of YIG, \u000eijand\u000fijkare\nKronecker and Levi-Civita symbols, fis the Faraday co-\ne\u000ecient,Mkis the magnetization, and i,j,kcorrespond\ntox,y,zdirection, respectively. When the energy is con-\nserved for the two-photon and magnon transitions that\n!1\u0000!2=!m, the coupling strength between two opti-\ncal modes is g=\u0001\n\u0001\"ij(\u0000 !x)E\u0003\n1;i(\u0000 !x)E2;j(\u0000 !x)d\u0000 !x3, where\nEp;i(\u0000 !x) (p= 1;2) is the normalized \feld of optical WGM\u0001\n\"ii(\u0000 !x)jEp;i(\u0000 !x)j2d\u0000 !x3=!p, and magnon induced per-\nmittivity change is \u0001 \"ij(\u0000 !x) =\u0000if\"0\u000fijkMk(\u0000 !x). As the\n\feld distributions are in the form of ein\u001ein the spherical\ncoordinate along the azimuthal direction, gis non-zero\nonly for the conservation of orbit angular momentum\nn1\u0000n2=nm. Therefore, when the energy, spin and\norbit angular momentum conservation relations, i.e., the\ntriple resonance condition [Fig. 1(c)] and selection rule\nfor our optomagnonic resonator, are simultaneously sat-\nis\fed, the coupling strength gcan be greatly enhanced.\nThe schematic and optical images of the experiment\nassembly of out optomagnonic cavity integrated with3\nphotonic and microwave circuits are shown in Figs. 2(a)\nand (b), respectively. A 300- \u0016m-diameter single crys-\ntal YIG sphere [Fig. 2(c)] is glued to a 125- \u0016m-diameter\nsupporting silica \fber. Although YIG spheres have been\nwidely used as magnon resonators, their potential as op-\ntical high-QWGM microresonators has been overlooked.\nIn fact, the low absorption loss of YIG in the infrared\nwavelengths (0.13 dB/cm)33can lead to Qfactors as\nhigh as 3\u0002106. Nonetheless, the surface defects and\ncontamination of commercial YIG sphere products in-\nduce strong scattering losses, limiting the highest achiev-\nableQfactor in our experiment. A major contribution\nof the surface contamination is the residual of the sub-\nmicrometer aluminum oxide polishing grit used in the\nYIG sphere production process, which is very di\u000ecult to\nremove using conventional cleaning procedures. By com-\nbining a mechanical polishing procedure (using silicon ox-\nide slurry) and a follow-up chemical cleaning procedure\n(using bu\u000bered oxide etch), we e\u000eciently removed these\ncontamination and obtained very clean sphere surface\n[Fig. 2(d)]. To excite the high- QWGMs, conventional ta-\npered silica \fber (refractive index 1.44) approach cannot\nachieve high e\u000eciency because of the index mismatch34.\nTherefore, we integrate the YIG sphere with a silicon\nnitride optical circuit [Fig. 2(e)], whose waveguide mode\nindex matches that of YIG. The chip is glued to silica\noptical \fbers using UV curable epoxy after careful align-\nment, which provides high e\u000eciency and stable transmis-\nsion. Another coplanar loop antenna circuit is placed in\nvicinity of the YIG sphere to convert microwave signal to\nmagnon. In our experiments, the YIG sphere is always\nbiased by an external magnetic \feld along the support-\ning \fber (z) direction according to the spin conservation\ncondition discussed above.\nBefore studying the magnon-photon interaction, we\n\frst characterize the optical and magnon modes. The\nre\rection microwave spectrum at H= 1840 Oe is plot-\nted in Fig. 3(a), showing multiple dips that correspond\nto magnon modes (to observe high order modes, the YIG\nsphere is placed at the non-uniform \felds of the antenna).\nIn the zoomed-in spectrum of Fig. 3(b), the loaded Qfac-\ntor of the fundamental magnon mode (1 ;1;1) is 1230. In\nthe following magnon-photon interaction measurement,\nthe YIG sphere is placed at the uniform microwave \felds\nof the antenna output such that only the (1 ;1;1) mode\nis excited. The optical transmission spectra are plotted\nin Fig. 3(c), where TE/TM polarized light in the waveg-\nuide are used to probe \u001b+/\u0019polarized WGMs in the YIG\nsphere, receptively. Groups of optical resonances show up\nin the spectra, exhibiting large extinction ratio (beyond\n10 dB) for both polarizations, which con\frms the e\u000e-\ncient coupling between the silicon nitride waveguide and\nthe WGMs. The measured free spectral ranges for both\nthe\u001b+(1.0765 nm) and \u0019(1.1068 nm) polarization agree\nwith the prediction (1.1580 nm) for WGMs. Thanks to\nour surface treatments, very high optical Qfactors are\nachieved:Q\u001b+= 0:593\u0002106andQ\u0019= 0:763\u0002106\n[Figs. 3(d) and (e)].\nMicrowave refl. (dB)\nFrequency (GHz)\nMicrowave refl. (dB)\n5.16 5.17-25-20-15\nf (GHz)Q = 1230\n1546 1548 1550 1552 1554-45-40-35-30-25-20-15Op/g415cal transmission (dBm)\nWavelength (nm)σ+\nπ\nTrans. (dBm) Trans. (dBm)σ+\nQ = 593000\n1542.355 1542.357-31-30-29\nWavelength (nm)\nπ\nQ = 763000\n1540.828 1540.830-33.5-33.0\nWavelength (nm)5.1 5.2 5.3 5.4 5.5-30-25-20-15(a) (b)\n(c) (d)\n(e)FIG. 3. (a) Magnon resonances measured on a 300- \u0016m-\ndiameter YIG sphere biased at 1840 Oe. (b) The zoomed-\nin spectrum of the fundamental magnon mode. (c) Optical\nWGMs for both polarizations ( \u001b+and\u0019) measured on the\nsame YIG sphere using the silicon nitride coupling waveguide.\nLarge extinction ratio and the periodic mode distribution is\nevident. (d) and (e) are the zoomed-in spectrum for the two\npolarizations, respectively.\nTo measure the interaction between optical photon\nand magnon, the YIG sphere is biased at H= 2410\nOe, corresponding to a magnon resonance frequency of\n!m=2\u0019= 6:75 GHz. The optomagnonic resonator is\npumped by a TM polarized laser beam with 1 mW power,\nand the magnons are excited by an on-resonance mi-\ncrowave signal. The laser wavelength is scanned to search\nfor the optical modes that satisfy the energy, spin and\nangular momenta conservation conditions. During the\nsearching process, lock-in technique is adopted to im-\nprove the converted light signal to noise ratio. When the\nconservation conditions are satis\fed, the output light is\nsent to a high resolution spectrometer for further analy-\nsis. It is worth noting that the density of optical WGMs\nis very large, as there are mode degeneracy in the polar\ndirection and high order modes in the radial direction.\nAs a result, the conservation conditions can be satis\fed\naccidentally, similar to the Brillouin scattering in micro-\nsphere optomechanical cavities35,36. A typical spectrum\nof converted photons as a function of the sweeping pump\nlaser wavelength is shown in Fig. 4(a), where the passive\ntransmission spectrum of the pump light is also shown ac-\ncordingly. The correspondence between the resonances\nfor pump light and the peaks of magnon-photon con-\nversion implies the triple resonance enhancement in our\noptomagnonic cavity. The dependence of the converted\nphotons on the microwave resonance [Fig. 4(b)] also con-\n\frms the participation of magnon in the inelastic light4\n1539.0 1539.2 1539.40.000.050.101E-41E-30.01Opt. signal (a.u.)\nWavelength (nm)Pump trans. (mW)\n-6 -4 -2 0 2 4 61E-91E-81E-71E-61E-81E-71E-61E-51E-41E-3Op/g415cal power (mW)\nFrequency (GHz)\n6.67 6.68 6.690.00.51.01.5-20-18-16Opt. signal (a.u.)\nFrequency (GHz)MW refl. (dB)\n-6.9 -6.8 -6.7 -6.6 -6.5 -6.4 -6.351015 Power (nW)\nFrequency (GHz) 245 mW\n 255 mW 270 mW 280 mW 305 mW 320 mW\n250 300 350101520Power (nW)\nP (mW)MWTM\nTE(a) (c)\n(b) (d)\nFIG. 4. (a) Optical pump transmission and the generated op-\ntical signal as a function of pump laser wavelength. The corre-\nspondence of the generated optical signal peak and the optical\npump resonance dip indicates the satisfaction of the conser-\nvation conditions. (b) Microwave re\rection and the generated\noptical signal as a function of the microwave frequency. (c)\nOptical spectrum of the device output when the triple reso-\nnance condition is satis\fed. The TM and TE components of\nthe output light are separated by a polarization beam splitter.\nThe TM component corresponds to the direct transmission of\nthe pump light, while the TE component contains the scat-\ntered sideband. (d) Power dependence of the sideband on\nthe input microwave power PMW. Inset: extracted sideband\npower as a function of the input microwave power.\nscattering process.\nThe detailed spectrum for one selected tripe-resonance\ncondition is plotted in Fig. 4(c), where the optomagnonic\nresonator is pumped by a TM light at 1534.599 nm. A\npolarization beam splitter is used to separate the two\npolarizations. The TM component of the output light\nshows a single peak as it only contains the transmitted\npump light. On the contrary, the TE component shows\ntwo peaks: a strong peak which corresponds to the trans-mitted pump light that is not completely \fltered out,\nand a weak sideband which corresponds to the gener-\nated photons. Therefore we do have orthogonal polariza-\ntions for the generated signal and the pump light, which\nagrees well with our theory model. The linewidths of the\nmeasured pump and sideband signal are not the physical\nlinewidth of the light but instead only represent the \f-\nnite resolution (67 MHz) of the \flter in the spectrometer.\nThe centers of the pump peak and sideband di\u000ber from\neach other by 6 :75 GHz, matching the input magnon fre-\nquency. The sideband appears only on one side of the\npump as a result of the conservation conditions, as ex-\nplained in above analysis. We measured the converted\nlight at various microwave input powers, which clearly\nshows a linear power dependence [Fig. 4(d)], indicating\na linear magnon to photon conversion. The \ftted raw\npower (system) conversion e\u000eciency is 5 \u000210\u00008. Consid-\nering the imperfect resonance coupling and in-line inser-\ntion losses for both the optical and microwave circuits,\nthe internal power conversion e\u000eciency is about 5 \u000210\u00003.\nThe conversion e\u000eciency can be improved by using YIG\nspheres of smaller size and smoother surface. Further ge-\nometry optimization, such as using YIG microdisk whose\nmodal volume is orders of magnitude smaller, in combina-\ntion with Faraday e\u000bect enhancement via doping, could\nlead to much improved conversion e\u000eciencies.\nIn conclusion, we have demonstrated an excellent op-\ntomagnonic resonator that is made by a highly polished\nYIG sphere. Utilizing an integrated optical chip for\nhigh e\u000eciency optical coupling, high- Qoptical WGMs\nare observed in addition to magnon resonances in the\nYIG sphere after our careful surface treatment. When\nthe triple resonance condition and angular momentum\nconservation condition are satis\fed, the magnon is con-\nverted to optical photon with internal power e\u000eciency\nof about 0:5%. This e\u000eciency can be further improved\nby doping or geometry optimization. Our \fndings show\nthat YIG sphere is a promising platform for designing\ncomplex hybrid systems, which holds great potential to\nrealize information inter-conversion among magnon, mi-\ncrowave photon, and optical photons.\nThe authors thank Liang Jiang and Michael Flatt\u0013 e\nfor fruitful discussions, and funding support from\nDARPA/MTO MESO program (N66001-11-1-4114), a\nUS Army Research O\u000ece grant (W911NF-14-1-0563),\nan AFOSR MURI grant (FA9550-15-1-0029), and the\nPackard Foundation.\n\u0003corresponding email: hong.tang@yale.edu\n1A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nature Phys. 11, 453 (2015).\n2Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. 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Bahl,\nNature Phys. 11, 275 (2015)." }, { "title": "2103.10711v1.Domain_wall_dynamics_of_ferrimagnets_induced_by_spin_current_near_the_angular_momentum_compensation_temperature.pdf", "content": "Domain wall dynamics of ferrimagnets induced by spin-current near the angular\nmomentum compensation temperature\nV.V. Yurlov,1K.A. Zvezdin,2, 3,\u0003P.N. Skirdkov,2, 3and A.K. Zvezdin2\n1Moscow Institute of Physics and Technology, Institutskiy per. 9, 141700 Dolgoprudny, Russia\n2Prokhorov General Physics Institute of the Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia\n3New Spintronic Technologies, Russian Quantum Center,\nBolshoy Bulvar 30, bld. 1, 121205 Moscow, Russia\n(Dated: March 22, 2021)\nWe report on a theoretical study of the spin-current excited dynamics of domain walls (DWs) in\nferrimagnets in the vicinity of the angular momentum compensation point. E\u000bective Lagrangian\nand nonlinear dynamic equations are derived for a two-sublattice ferrimagnet taking into account\nboth spin-torques and external magnetic \feld. The dynamics of the DW before and after the Walker\nbreakdown is calculated for any direction of the spin current polarization. It is shown that for the\nin-plane polarization of the spin current, the DW mobility reaches a maximum near the temperature\nof the angular momentum compensation. For the out-of-plane spin polarization, in contrast, a spin\ncurrent with the densities below the Walker breakdown does not excite the dynamics of the DW.\nAfter overcoming the Walker breakdown, the domain wall velocity increases linearly with increasing\nthe current density. In this spin-current polarization con\fguration the possibility of a gigahertz\noscillation dynamics of the quasi-antiferromagnetic vector under the action of a damping-like torque\nin the angular momentum compensation point is demonstrated. Possible structures for experimental\ndemonstration of the considered e\u000bects are discussed.\nI. INTRODUCTION\nSpintronics, which is a rapidly developing branch of\nnanoelectronics, is based on the concept that the prin-\ncipal role in information processing belongs to spins of\nelectrons instead of charges[1, 2]. The mainstream of\nspintronics is attaining the winning combination of the\nspin transport e\u000eciency and nanoscale size of spintronic\ndevices. In this regard, magnetic DWs attract increas-\ning attention[3{5]: they can be used to store and trans-\nmit information in race track or magnetic random access\nmemories (MRAM)[6{10].\nConventional spintronic devices use ferromagnetic\n(FM) materials owing to their property to create and\nsubsequently to use spin-polarization of the conducting\nelectrons. The nanural restrictions of ferromagnetic spin-\ntronic devices relate to the limited operation frequen-\ncies and general energy e\u000eciency. Recent advances in\nspin current injection into insulating antiferromagnets\n(AFMs) have revealed the prospects of AFM spintron-\nics, whose advantages is an extremely high frequency in\ncomparison to the operation frequency of ferromagetic\ndevices[11, 12]. At the same time the AFM spintron-\nics has its own shortcomings associated with the di\u000ecul-\nties to detect the magnetization states and magnetiza-\ntion dynamics. These motivate a boosting development\nof ferrimagnetic (FiM) spintronics, which combines the\nultra-high operation frequences close to those of AFM de-\nvices, with much more reliable ways to detect its magne-\ntization states. Very rich and interesting magnetization\ndynamics[13, 14], in terms of fundamental and applied\n\u0003zvezdin.ka@phystech.eduphysics, is observed in these materials near the points of\ncompensation of magnetization and angular momentum.\nMoreover, by manipulating the temperature of the ferri-\nmagnet near the compensation points, outstanding mag-\nnetization switching characteristics can be obtained[15{\n17]. It has been shown that the electrical current can\nbe an e\u000ecient approach to magnetization switching[18{\n21]. GdFeCo FiM layer demonstrates ultrafast magneti-\nzation reversal in\ruenced by femtosecond laser pulses in\nvarious experiments[22]. These results suggests that the\nFiMs based structures can form a promising technologi-\ncal platform for ultrafast spintronic memory devices.\nAngular momentum compensation point TA, where\nM1=\r1=M2=\r2,\riis the gyromagnetic ratio of the i-\nsublattice (i = 1, 2), represents a very promising line\nof research of FiMs magnetization dynamics[13, 23, 24].\nRecent \feld-driven experiments demonstrated high ve-\nlocity and great mobility of a domain wall in FiM near\ntheTA[25]. The next natural step in this direction is to\nuse the spin-currents to manipulate DWs position and\ndynamics[26]. While the spin-current induced phenom-\nena in FMs seems to be well comprehensible, the mech-\nanisms of spin transfer in AFMs and compensated FiMs\nare still not \fgured out properly.\nIn the present research we develop a model to de-\nscribe DW motion in ferrimagnets near the angular mo-\nmentum compensation point in case of arbitrary spin\ncurrent polarization and torque type. DW dynamics\nin\ruenced by a spin-current in FiMs is studied gener-\nally by using collective coordinates model and Landau-\nLifshitz-Gilbert equation with addition of spin-transfer\ntorque components[27{29]. Here instead we employ\nthe Lagrangian formalism for two subblatice ferrimag-\nnet. This approach allows us to strictly de\fne ferrimag-\nnetic parameters such as width of the DW, velocity ofarXiv:2103.10711v1 [physics.app-ph] 19 Mar 20212\nthe magnons, transverse magnetic susceptibility, e\u000bec-\ntive Gilbert damping parameter and gyromagnetic ratio\nby using perturbation theory.\nWe derive non-linear dynamic equations based on the\nLagrangian formalism, which is similar to Slonszewski\nequations[30]. Using this model, we calculate the dynam-\nics of the domain wall in FiM, depending on the direction\nof the polarizer, electric current density and temperature,\nbefore and after the Walker breakdown. In our model-\ning we observe that for the in-plane polarization of the\nspin current, the DW mobility reaches a maximum near\nthe temperature of the angular momentum compensa-\ntion, and vanishes after bypassing the Walker breakdown.\nFor the out-of-plane spin polarization, in contrast, a spin-\ncurrent with the densities below the Walker breakdown\ndoes not excite the stationary dynamics of the DW. After\novercoming the Walker breakdown, the domain wall ve-\nlocity increases linearly with increasing the electric cur-\nrent density. In this con\fguration of the spin current,\nnear the compensation point TAwe observe gigahertz os-\ncillations of the quasi-antiferromagnetic vector.\nII. MODEL AND BASIC EQUATIONS\nz\nxyL\nθ\nφ\nFiMs layerDW\nspin-currentσVℓ\nFIG. 1. Schematic of considered FiMs with single domain\nwall,\u001bis polarization vector of the spin-current, l- is thick-\nness of the sample; \u0012and'are the polar and azimuthal angels\nof quasi-antiferromagnetic vector L.\nWe develop a model based on the Lagrange formalism\nfor describing DW dynamics due to spin-current. For two\nsublatticed FiMs ordering parameters related to magne-\ntizations M1,M2of these sublattices can be introduced\nin the vicinity of compensation temperatures as quasi-\nantiferromagnetic vector L=M1\u0000M2and ferromag-\nnetic M=M1+M2order parameters. We consider a\nspin-current with a polarization \u001b\rowing through theFiM \flm (see Fig. 1). By analogy with the approach\nused for ferromagnets, we use the adiabatic approxima-\ntion, assuming that the AFM order parameter Lchanges\nslowly in comparison with the spins of the injected elec-\ntrons. The spin-current excites a spin transfer torque\nacting on local magnetization of the i-sublatice respec-\ntively and in general case composed to the in{plane and\nthe out{of plane components Ti\nST=Ti\nFL+Ti\nDL. The\nTi\nFLcomponent due to its symmetry is usually refer-\need as a \feld{like torque and has the following form:\nTi\nDL\u0018[Mi\u0002\u001b]. The Ti\nDLtorque component has a sym-\nmetry similar to the damping torque and usually refereed\nas an damping-like torque (or anti-damping-like torque):\nTi\nDL\u0018[Mi\u0002[Mi\u0002\u001b]]. Typically the magnitude of\nthe anti-damping-like torque component is signi\fcantly\nlager than the \feld-like one for magnetic tunnel junc-\ntions, however in case of spin-orbit torques can be of a\nsimilar magnitude.\nThe magnetization dynamics is described by a system\nof Euler-Lagrange equations:\n8\n>><\n>>:d\ndt\u0010@L\n@_\u0012i\u0011\n\u0000\u000eL\n\u000e\u0012i=\u0000@Ri\n@_\u0012i\u0000@W\n@_\u0012i\nd\ndt\u0010@L\n@_'i\u0011\n\u0000\u000eL\n\u000e'i=\u0000@Ri\n@_'i\u0000@W\n@_'i; (1)\nwhereLandRiare the Lagrangian and Rayleigh func-\ntions,Wis spin transfer torque power density; \u0012iand\n'iare the polar and azimuthal angles characterizing the\norientation of the i-th sublattice magnetization (i = 1,\n2). Note that \u000eWrepresents external spin-current ef-\nfect on magnetic structure and consists of the damping-\nlike and the \feld-like components. Due to its symme-\ntry the \feld-like component can be included in the La-\ngrangian by using the quasi-antiferromagnetic approxi-\nmation. Thus, \u000eWin the Euler-Langrange equations con-\nsist of only damping-like spin-current component. Here-\ninafter we turn to the e\u000bective Lagrangian Le\u000b, e\u000bective\nRayleigh function Re\u000band power density of a spin cur-\nrent\u000eWin quasi-antiferromagnetic approximation appli-\ncable in the vicinity of compensation temperatures (see\nin Supplementary)[31]\nLe\u000b=\u001f?\n2\u0010_\u0012\n\re\u000b\u00112\n+m\u0010\nH\u0000_'\n\re\u000b\u0011\ncos\u0012+\n\u001f?\n2\u0010\nH\u0000_'\n\re\u000b\u00112\nsin2\u0012\u0000Kusin2\u0012\u0000\n\u0000K?sin2\u0012sin2'\u0000A\u0010\u0010d\u0012\ndx\u00112\n+ sin2\u0012\u0010d'\ndx\u00112\u0011\n\u0000\n\u0000\u001f?\n2\u0010B\nM\u00112\u0010\nsin2\u0012n?+\n+ cos2\u0012cos2('\u0000 )nk+ sin2('\u0000 )nk\u0011\n;\nRe\u000b=\u000be\u000bM\n\re\u000b\u0010\n_\u00122+ sin2\u0012\u0001_'2\u0011\n;\n\u000eW=\u0000Asin('\u0000 )nk\u0001\u000e_\u0012+\n+(\u0000An?sin2\u0012+Ankcos\u0012cos('\u0000\u0012))\u0001\u000e_';(2)3\nwherem=M2\u0000M1,M=M1+M2;\u001f?=M=Hex\nis transverse magnetic susceptibility, Hexis an exchange\nmagnetic \feld acting between sublattices; KuandK?are\nconstants of uniaxial and in-plane magnetic anisotropies\nrespectively; A is an exchange sti\u000bness constant, \u0012and\n'are the polar and azimuthal angles of an quasi-\nantiferromagnetic vector L,H= (0;0;Hz) is a mag-\nnetic \feld applied along the \\easy magnetization axis\";\n\u000be\u000b=\u000bm=(m\u0000m0),\re\u000b=\rm=(m\u0000m0),\re\u000b=\n\r\u0001(1\u0000m\u0001m0=M2)\u00001,\u000b= (\u000b1\r2+\u000b2\r1)=2(\r1+\r2),\n1=\r= (1=\r1+ 1=\r2)=2, where\u000biand\riare a damping\nconstant and a gyromagnetic ratio for the i-sublatice re-\nspectively,m0=M(\r1\u0000\r2)=(\r1+\r2);A=~JPDL=(2el)\nandB=~JPFL=(2el) are the \feld-like and the damp-\ning (or anti-damping) spin transfer torque coe\u000ecients,\nwhereJis electrical current density, lis the thickness of\nthe magnetic \flm, e>0 is the electron charge; n?andnk\nare the out-of- and the in- plane components of unit vec-\ntorn= (nx;ny;nz) along the polarization of spin-current\n\u001b, is an angle between the projection of polarization\nvector of the spin-current \u001bon the x-y plane and the\nx-axis;PDLandPFLare the \feld-like and the damp-\ning (or anti-damping) polarizations of the spin current,\nrespectively.\nImplementing the procedure which is described in the\nSupplementary, we derive the system of dynamic equa-\ntions for the 180\u000eDW without external magnetic \feld:\n8\n><\n>:2\u000bM\n\r\u00010_q+m_'\n\re\u000b=fT\u0012\n\u0000\u001f?\n\r2\ne\u000b'+m\n\re\u000b_q\n\u00010\u0000K?sin 2'\u00002\u000bM\n\r_'=fT';\n(3)\nwhere q is a coordinate of the DW centre, \u0001 0=p\nA=Ku\nis a width of the DW. The spin transfer torque compo-\nnents are written as\nfT\u0012=\u0000\u0019\n2Asin('\u0000 )nk\nfT'=\u0000An?+\u001f?\n2\u0010B\nM\u00112\nsin 2('\u0000 )nk: (4)\nNote that in general case the width of the DW is deter-\nmined as \u0001 = \u0001 0p\n1\u0000( _q=c)2, wherec=\re\u000bp\n2A=\u001f?is\na magnons velocity (see in Supplementary). For our set\nof parameters it can be estimated as c\u00188 km/s. As a\nresult, the variation of the DW width for the considered\nvelocities is of the order of one percent (\u0001 =\u00010\u00180:01).\nThus we can assume that _ q\u001ccand DW width \u0001 \u0019\u00010.\nIII. DYNAMIC EQUATION ANALYSIS\nTo understand peculiar features of the current induced\nDW dynamics in compensated FiMs following from eqs.\n(3) and (4) we analyze several particular cases. To calcu-\nlate the DW dynamics we use typical GdFeCo param-\neters: [25]: Ku\u00181\u0001105erg/cc,M \u0019 900 emu/cc,\n\u000b\u00180:02,\r\u00182\u0001107,A\u00181\u000110\u00006erg/cm,gd= 2:2,gf= 2,TM= 220 K,TA= 310 K,l= 10 nm, where\ngdandgfare Lande g-factors for d- and f-sublatices re-\nspectively. The constant of in-plane magnetic anisotropy\nin case of in\fnite \flm is K?= 2\u0019m2, however in case\nof a narrow FiMs nanowire it has a di\u000berent form due\nto magnetostatic interaction. Note, that all dynamic pa-\nrameters (such as velocity, DW displacement and others)\nare functions of \u0017=m=M, which can be rewritten in\nterm of temperature Tby using the following expression\n:\n\u0017=m\nM=T\u0000TM\nT\u0003; (5)\nwhereT\u0003= 1891 K is obtained from the GdFeCo\nparameters[25]. For all further mentioned modelling re-\nsultsPDL= 0:3 andPFL= 0:03.\nT=TA\nT=290 K\nT=280 K\nT=270 KT=350 K\nJT=TA\nT=290 K\nT=280 K\nT=270 KT=350 K\nJIn-plane spin-current polarization (n =1)║\n(a) (b)\nFIG. 2. a) Average DW velocity in Walker and post Walker\nregimes as a function of the electrical current density J; b)\nAbsolute value of the azimuthal angle 'in Walker and post\nWalker regimes as a function of the electrical current density\nJ; blue, green, yellow, red and black curves correspond to the\ntemperatures T= 270 K,T= 280 K,T= 290 K,T=TA\nandT= 350 K respectively; the black arrow indicate the\ntransition in the post Walker regime. All curves are plotted\nfor the in-plane spin current polarization.\nFirst, let us analyze DW dynamics for the in-plane\nspin-current when K?6= 0 and the spin polarization\nalong the y axis n= (0;1;0) ( =\u0019=2 andnk= 1). In\nthis geometry stationary DW motion ( _ '= 0 and a con-\nstant DW) is observed below Walker breakdown. In the\nTAthe azimuthal angle 'tends to zero (see red curve in\nFig. 2(b)) and the stationary DW motion is observed with\nthe velocity _ q=\u00010=\u0019\rA=4\u000bM, which follows from the\n\frst equation in (3). As follows from (3) and (4) in this\ncase the damping-like (or anti-damping-like) spin trans-\nfer torque component with magnitude Ainitiates DW\ndynamics, while the \feld-like one only modi\fes the mag-\nnetostatic term. The magnitude of the azimuthal angle '\nincreases with increasing in the electric current density\nand tends to \u0019=2 which is demonstrated in Fig. 2(b).\nNote that in the case of the in-plane polarizer after\nreaching the critical current density corrisponding to the\nWalker breakthrough J\u0003= 16elj\u000be\u000bjK?=\u0019\u0017~PAD, there\nis no domain wall motion observed - it is indicated by the4\nblack arrows in the Fig. 2(a) and Fig. 2(b). This range\ncorresponds to the constant azimuthal angle '\u0019\u0019=2.\nSpin-current cannot push the domain-wall when angle '\nexceeds\u0019=2 (see Fig. 2(b)) for the case of in-plane polar-\nization. This means that the steady precessional motion\nof DW is impossible for in-plane polarized spin current\nand DW velocity eventually drops to zero for all temper-\natures except for TA.\nNow let us discuss a more di\u000ecult situation when the\n\u001bis parallel to the z-axis n= (0;0;1) andn?= 1. Actu-\nally, if we assume that K?= 0,\u001f?\u001c1 and consider the\ntemperatures in the vicinity of angular momentum com-\npensation point TA, the system (4) describes the steady\nmotion of the DW with velocity _ qand precession rate _ ':\n_q\n\u00010=\u0000\r\n2M\u000b\u0001A\u0017=2\u000be\u000b\n1 + (\u0017=2\u000be\u000b)2; (6)\n_'=\r\n2M\u000b\u0001A\n1 + (\u0017=2\u000be\u000b)2: (7)\nBy using the equation (5) we can rewrite the (6) and\n(7) in term of temperature T and study the dependence\nof the DW velocity and precession rate on temperature\nand current density. Fig. 3(a) demonstrates that the DW\nvelocity has two maximum values near the angular mo-\nmentum compensation point and these values increase\nwith growth in current density. These curves are asym-\nmetric with respect to TA. Thus, the velocity of the DW\nchanges its sign passing through the angular momentum\ncompensation point. This situation is also realized in\nFig. 3(b), where the dependence of the DW velocity on\nelectric current density at di\u000berent temperatures is given.\nAs it is seen from the equation (6) DW velosity linearly\ndepends on the electrical current density. The blue and\ngreen line (see Fig. 3(b)) lie below the TAand the slope\nof this curves (DW mobility _ q=J) decreases. The DW ve-\nlocity changes its direction above the angular momentum\ncompensation point (red curve in Fig. 3(b)).\nNote, that the DW velocity reaches 260 m/s at cur-\nrent densities by about 3 \u0002107A/cm2. Precession rate is\nnot zero _'6= 0 in the vicinity of the angular momentum\ncompensation temperature compared with \feld-driving\nDW motion[32] (where magnetic \feld is applied along\nthe easy magnetization axis). The equation (7) shows\nthat _'reaches its maximum by about 17 GHz at low\ncurrent density (\u00183\u0002107A/cm2) near theTA(see in\nFig. 3(c)). As it follows from the equations (6) and (7)\nin case of considered polarization direction both oscilla-\ntion of the 'angel and DW motion is triggered by the\ndamping (or anti-damping) spin transfer torque compo-\nnent with magnitude A; hence DW dynamic and oscilla-\ntion freezes without spin-current. Field-like spin transfer\ntorque is neglected at the out-of-plane spin-current po-\nlarization case due to decomposition of Lagrangian of the\ntwo-sublattice ferrimagnet as a next order small param-\neter (see in Supplementary).\n(a) (c)\n(b)Out-of-plane spin-current polarization (n⟂=1) \nT=290 K\nT=300 K\nT=320 KTA TMJ=1×107 A/cm2\nJ=2×107 A/cm2\nJ=3×107 A/cm2J=1×107 A/cm2\nJ=2×107 A/cm2\nJ=3×107 A/cm2\nTA TM\nJFIG. 3. a) Dependence of the DW velocity on temperature\nat the di\u000berent current densities; b) Dependence of the DW\nvelocity on electrical current density at the di\u000berent temper-\natures; c) Precession rate as a function of temperature at the\ndi\u000berent current densities. All curves are plotted for the out-\nof-plane spin current polarization.\nA\nBStationary mode\n JJ*Out-of-plane spin-current polarization (n⟂=1) J\nFIG. 4.J\u0000Tdiagram which describes the ranges of a steady\nand non-steady motion of the DW, green curve shows the\ntemperature dependence of the critical current J\u0003; the ranges\nabove (J > J\u0003) and below ( J < J\u0003) of the green curve\ncorrespond to non-stationary (post Walker) and stationary\n(Walker) mode of the DW, respectively. Insets show the time\ndependence of DW displacement in non-stationary range for\npoint A (T= 320 K and J= 2\u0001107A/cm2) and stationary\nrange for point B ( T= 280 K and J= 0:3\u0001107A/cm2) of\nthe diagram. All curves are plotted for the out-of-plane spin\ncurrent polarization.\nLet us analyse the DW dynamic in the presence of in-\nplane magnetic anisotropy K?6= 0 and n= (0;0;1). We\n\fnd out that there are two di\u000berent regimes of the DW\nmotion: steady ( _ '= 0) and non-steady ( _ '6= 0). Let us\ndiscuss the non-steady one. An analytical solution to the5\nsystem of di\u000berential equations (4) can be written as:\ntan'=J\u0003\nJ+r\n1\u0000\u0010J\u0003\nJ\u00112\ntan(!0t\u0000'0);(8)\nwhereJ\u0003=4\u0019M2\u00172el\n~PDLis the critical current density,\n!0=\r\n\u000b\u0019\u00172Mp\n1\u0000(J\u0003=J)2\n1+(\u0017=2\u000beff)2,'0= arctanJ\u0003=Jp\n1\u0000(J\u0003=J)2. The\nnon-stationary regime realises when the current density J\nhigher than critical current J\u0003(J >J\u0003). This situation\nis represented on the J\u0000Tdiagram in Fig. 4, where the\ngreen curve is the temperature dependence of the critical\ncurrent density J\u0003. The equation (8) describes the oscil-\nlation of the angle 'in the non-stationary range of the\nJ\u0000Tdiagram (J >J\u0003) and inset for the point A in Fig. 4\nshows the time dependence of the DW displacement in\nthis range at \fxed temperature T= 320 K and current\ndencityJ= 2\u0001107A/cm2. Stationary regime of the DW\nrealises when the current density Jis lower than critical\ncurrentJ\u0003. Inset in Fig. 4 for the point B ( T= 280 K\nandJ= 0:3\u0001107A/cm2) shows that after a small period\nof time\u00180:15 ns the DW displacement stops changing\nin time. Therefore, the precession rate _ '= 0, velocity\nof the DW tends to zero and the magnetization freezes\nin the stable state, which corresponds to the equation\nsin 2'=J=J\u0003as follows from the (3). Hence stationary\n(Walker) mode in considered case corresponds to absence\nof DW motion, while non-stationary mode is responsible\nfor DW motion.\nJT=290 K\nT=TA\nT=330 K\nJ*\n1 J*\n2Out-of-plane spin-current polarization (n⟂=1) \nFIG. 5. Average DW velocity in stationary and non-\nstationary mode as function of the electrical current den-\nsity J; blue, green and red curves correspond to temperatures\nT= 290 K,T=TAandT= 330 K, respectively; J\u0003\n1;2are\ncritical current densities which corresponds to T= 290 K\nandT= 330 K, respectively. All curves are plotted for the\nout-of-plane spin current polarization.\nThe dependence of the average DW velocity on theelectrical current density in the stationary and non-\nstationary regimes is shown in Fig. 5. In the stationary\nmode the DW velocity is zero. Near the critical current\nJ\u0003an increase in the value of velocity occurs. However,\nin the non-stationary mode ( J > J\u0003) the average DW\nvelocity linearly increases. Note that in the angular mo-\nmentum compensation point the average velocity is equal\nto zero (green curve in Fig. 5). Besides Fig. 5 shows that\nvelocity changes its sign passing through the T A, which\nis demonstrated by blue ( T= 280 K< TA) and red\n(T= 330 K< TA) curves in Fig. 5. These results are\nconsistent to the J\u0000Tdiagram in Fig. 4. It's impor-\ntant to note that in the non-stationary mode nonlinear\nspin waves can be excited and a\u000bect the dynamics of\nthe DW. However, frequencies of the spin-wave in ferri-\nmagnetic or antiferromagnetic materials lies in terahertz\nrenge[27, 33, 34]. In contrast precession rate of quasi-\nantiferromagnetic vector lies in gigahertz range and we\nsuppose that nonlinear spin waves have weak e\u000bect on\nthe DW dynamics. Moreover, our model itself has a lim-\nitation (see Supplementary) in precession rate, which co-\nincides with the frequencies of spin waves.\nNow, let us discuss the directions of the spin-current\npolarisation \u001band type of torques, which can lead to\ne\u000bects mentioned above, and possibility of their experi-\nmental realization. As follows from the reported results,\nthe damping (or anti-damping) spin transfer torque is\nresponsible for considered motion and oscillation regimes\nfor both planar and perpendicular spin-current polarisa-\ntion\u001b.\nThe \frst possible way to create damping (or anti-\ndamping) torque is to use magnetic tunnel junction\n(MTJ) structure. It is consist of free magnetic layer\nand polariser, which are separated by thin insulating ma-\nterial (usually MgO). In such a structure electric cur-\nrent \rows perpendicularly to the plane and creates Slon-\nczewski torque in the free layer, while spin-current polar-\nisation \u001bdirection is determined by magnetization direc-\ntion of the polariser. Example of MTJ structure is pre-\nsented in Fig. 6(a). The typical polarization value PDL\nin MTJ with ferromagnets is about 0.2-0.4. Hence one\ncan add thin FM layer between MgO and FIMs, which is\nusually done even in classic MTJ to improve TMR and\npolarization values [10], to achieve the level of PDL= 0:3\nused in our simulations.\nAnother way to create damping (or anti-damping)\ntorque is to use heavy metal / FIMs heterostructure. In\nsuch structure electric current \rows through heavy metal\n(like Ta, W, Pt, Au etc.) in plane of the \flm and due\nto the spin Hall e\u000bect creates perpendicular spin current\nwith polarization \u001b, which is perpendicular to the both\nelectric and spin currents. This spin current can cre-\nate anti-damping torque in FIMs. The examples of spin\nHall based geometry in case of perpendicular and pla-\nnar polarization \u001bis presented in Fig. 6(b) and Fig. 6(c)\nrespectively. The value of polarisation in these cases is\nequal to spin Hall angle. This angle can be up to 0.3\n[35, 36], which again makes our PDL= 0:3 is reasonable.6\nz\nx y\nFiMs layerDW\nJV\nmrefMgOFM layer(a)\n(b)\nV\nFiMs layer\nJHMσ\nFiMs layerV\nHM\nJ(c)\nσ\nFIG. 6. Examples of a) MTJ base, b)-c) spin Hall based struc-\ntures, which can be used to observe reported DW motion and\noscillation regimes in FIMs \flm or nanostripe. b) corresponds\nto perpendicular and c) - to planar direction of spin-current\npolarisation \u001b.\nMoreover, it is possible to use topological insulator in-\nstead of heavy metal to achieve spin Hall angles more\nthan 1 [37], which signi\fcantly decrease required current\ndensities.IV. DISCUSSION AND CONCLUSION\nThe theoretical study of the DW dynamics caused by\nthe spin-current near the angular momentum compensa-\ntion point is performed by using the Lagragian formal-\nism. The non-linear dynamic equations describing the\nDW motion are derived from the e\u000bective Lagrangian\nof the two sublattice ferrimagnets. We analyse the DW\nmotion at di\u000berent directions of the spin-current polar-\nizations and show the di\u000berent types of magnetic het-\nerostructures where this spin-current polarization can\nbe realised. In the case of the out-of-plane polarizer\n(n= (0;0;1)) we analyse dependence of DW velocity\nand precession rate on temperature and current den-\nsity. We foresee the possibility to generate oscillations of\nthe quasi-antiferromagnetic vector Lwith the frequen-\ncies by about 17 GHz at low current densities in the\nvicinity of the angular momentum compensation tem-\nperature. This oscillations are initiated by the damping\n(or anti-damping) component of the spin-transfer torque.\nThis precession movement can be associated with a re-\ncent micromagnetic modelling of THz oscillation caused\nby a spin current in antiferromagnetic materials[38] at\nhigh current densities. Furthermore, the DW velocity\nchanges the direction passing trough this temperature\nand this e\u000bect is observed experimentally in GdFeCo fer-\nrimagnet due to spin-current[39]. We explore the DW\nmotion in the stationary (Walker) and non-stationary\n(post Walker) modes and construct the diagram that pro-\nvides the values of current densities and temperatures for\nwhich these modes are realised. The model shows that\nin the Walker regime no DW motion occurs, while in the\npost Walker range DW velocity linearly increases with\nthe current. Note, that the similar dependence of the\nDW velocity was observed due to the spin Hall e\u000bect[26]\nin the TbCo ferrimagnet sample in presence of external\nmagnetic \feld. We also analyze the DW dynamics for\nthe in-plane spin-current polarization and obtain the de-\npendence of the DW velocity as a function of current in\nthe Walker and post Walker regimes. Finally, we deter-\nmine the directions of the spin-current polarisation \u001band\ntype of torques, which lead to e\u000bects mentioned above,\nand possibility of their experimental realization. These\nresults may be useful for experimental studying of do-\nmain wall dynamics in ferrimagnets.\nThis research has been supported by RSF grant No.\n19-12-00432.\n[1] M. Tsoi, R. E. Fontana, and S. S. P. Parkin,\nApplied Physics Letters 83, 2617 (2003),\nhttps://doi.org/10.1063/1.1578165.\n[2] J. Grollier, P. Boulenc, V. Cros, A. Hamzi\u0013 c, A. Vaur\u0012 es,\nA. Fert, and G. 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Ono, Nature Electronics 2,\n389393 (2019)." }, { "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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Buhrman, Spin transfer torque devices utilizing the giant \nspin Hall effect of tungsten, Appl. Phys. Lett. 101, 122404 \n(2012). \n[72] L. Zhu, L. Zhu, R. A. Buhrman, Fully Spin -Transparent \nMagnetic Interfaces Enabled by the Insertion of a Thin \nParamagnetic NiO Layer, Phys. Rev. Lett. 126, 107204 \n(2021). \n[73] A. R. Mellnik, J. S. Lee, A. Richardel la, J. L. Grab, P. J. \nMintun, M. H. Fischer, A. Vaezi, A. Manchon, E. -A. Kim, N. \nSamarth , D. C. Ralph, Spin -transfer torque generated by a \ntopological insulator, Nature 511, 449 –451 (2014).\n \n " }, { "title": "2012.02790v3.Nutation_in_antiferromagnetic_resonance.pdf", "content": "Nutation in antiferromagnetic resonance\nRitwik Mondal,\u0003Sebastian Großenbach, Levente Rózsa, and Ulrich Nowak\nFachbereich Physik, Universität Konstanz, DE-78457 Konstanz, Germany\n(Dated: April 16, 2021)\nThe effect of inertial spin dynamics is compared between ferromagnetic, antiferromagnetic and\nferrimagnetic systems. The linear response to an oscillating external magnetic field is calculated\nwithin the framework of the inertial Landau–Lifshitz–Gilbert equation using analytical theory and\ncomputersimulations. Precessionandnutationresonancepeaksareidentified,anditisdemonstrated\nthat the precession frequencies are reduced by the spin inertia, while the lifetime of the excitations\nis enhanced. The interplay between precession and nutation is found to be the most prominent in\nantiferromagnets, where the timescale of the exchange-driven sublattice dynamics is comparable to\ninertial relaxation times. Consequently, antiferromagnetic resonance techniques should be better\nsuited for the search for intrinsic inertial spin dynamics on ultrafast timescales than ferromagnetic\nresonance.\nI. INTRODUCTION\nDeterministic spin switching at ultrashort timescales\nbuilds the fundament for future spin-based memory tech-\nnology [1–5]. At femtosecond timescales inertial switch-\ning becomes particularly relevant, where the reversal is\nachieved with a linear momentum gained by the interac-\ntion of an ultrashort pulse and spin inertia [6, 7]. The\nunderstanding of magnetic inertia has been pursued along\ntwo different directions so far.\nOn the one hand, spin dynamics in antiferromagnets\n(AFMs) and ferrimagnets (FiMs) has successfully been de-\nscribedbytheLandau–Lifshitz–Gilbert(LLG)equation[8–\n10] for two sublattices coupled by the exchange interaction.\nThe exchange energy created by tilting the sublattice mag-\nnetization directions away from the antiferromagnetic ori-\nentation is dynamically transformed into anisotropy energy\nby collectively rotating the sublattices away from the easy\nmagnetic direction [11], analogously to the transition be-\ntween kinetic and potential energy terms in a harmonic\noscillator. While the LLG equation for the two sublat-\ntices is of first order in time, this effect gives rise to an ef-\nfectively inertial second-order differential equation for the\norder parameter in AFMs [12, 13]. The interaction be-\ntween exchange and anisotropy degrees of freedom causes\nan exchange enhancement of AFM resonance frequencies\nand linewidths [14].\nOn the other hand, an intrinsic inertia also arises in mag-\nnetic systems, if it is assumed that the directions of spin\nangular and magnetic moments become separated in the\nultrafast dynamical regime [15, 16]. The inertia gives rise\nto spin nutation, a rotation of the magnetization around\nthe angular momentum direction [17], caused by the en-\nergytransferbetweenmagnetickineticandpotentialenergy\nterms. The emergence of spin inertia has been explained\nbased on an extension of the breathing Fermi surface model\n[18, 19], calculated from a s\u0000dlike interaction between\nthe magnetization density and electron spin [20] and de-\nrived from a fundamental relativistic Dirac theory [21, 22].\nMagnetic inertia can be associated with a torque term\ncontaining a second-order time derivative of the magnetic\nmoment appearing in the inertial LLG (ILLG) dynamical\n\u0003ritwik.mondal@uni-konstanz.deequation. The characteristic inertial relaxation time, using\nits definition in Eq. (1) below, is expected to range from\n1 fs [15, 20, 23, 24] to a few hundred fs [25].\nLinear-response theory predicted the emergence of a nu-\ntation resonance besides the conventional precession reso-\nnance in ferromagnets (FMs) [26–28], providing a possible\nway of detecting inertial dynamics by applying oscillating\nexternal fields. An indirect evidence of the inertial dynam-\nics was found in NiFe and Co samples [23] by following\nthe field dependence of the ferromagnetic precession reso-\nnance (FMR) peaks. The experimental observation of the\nnutation resonance has only been achieved very recently\nin NiFe and CoFeB using intense terahertz magnetic field\ntransients [25].\nWhile the notion of inertial dynamics has been applied\nboth in the context of the LLG equation for AFMs as well\nas in the ILLG equation for FMs, the linear response of\nthese two examples is fundamentally different. While in\nboth cases a pair of resonances is found in contrast to the\nsingle FMR peak, the excitation frequencies in an AFM are\ndegenerate in the absence of a static external field, while\ntheydifferbyseveralordersofmagnitudeintheILLGequa-\ntion. The effective damping parameter of the precession,\ndefined as the half-width of the peak at half-maximum, is\nconsiderably higher in AFMs than in FMs, where it corre-\nsponds to the Gilbert damping. In contrast, it was demon-\nstrated that the effective damping decreases in the ILLG\nequation applied to FMs [27], particularly at the nutation\nresonance [29]. However, the ILLG has not been applied\nto AFMs so far.\nHere, we explore the effects of the ILLG equation in\ntwo-sublattice AFMs and FiMs using linear-response the-\nory and computer simulations. It is shown that a pair of\nnutation resonance peaks emerges, and that the inertial re-\nlaxation time influences the precessional resonance signifi-\ncantly stronger in AFMs than in FMs due to the exchange\ncoupling between the sublattices. The effective damping\nparameter is found to decrease in AFMs, reaching consid-\nerably lower values than the Gilbert damping at the nuta-\ntion peak, thereby enhancing the lifetime of these excita-\ntions. The inertial effects in FiMs are found to interpolate\nbetween those in AFMs and FMs.\nII. METHODS\nAsderivedinearlierworks[15,21,22],theILLGequationarXiv:2012.02790v3 [cond-mat.mtrl-sci] 15 Apr 20212\nreads\n_Mi=\u0000\riMi\u0002Hi+\u000bi\nMi0Mi\u0002_Mi+\u0011i\nMi0Mi\u0002Mi;\n(1)\ngeneralized here to multiple sublattices indexed by i. The\nfirst, second and third terms in Eq. (1) describe spin pre-\ncession with gyromagnetic ratio \ri, transverse relaxation\nwith Gilbert damping \u000bi, and inertial dynamics with re-\nlaxation time \u0011i. Note that an alternative notation for\nthe inertial term with \u0011i=\u000bi\u001ciis also used in the liter-\nature [15, 23, 25]; where comparison with earlier works is\nmentioned in the following, the relaxation time is converted\nto the formulation of Eq. (1). The equation of motion was\ntreated analytically as described in the following sections,\nand also solved numerically using an algorithm presented\nin detail in Appendix A.\nIII. INERTIAL EFFECTS IN FERROMAGNETS\nFirst, we summarize the effects of the inertial term on\nFM resonance. The FM is described by the free energy\nF(M) =\u0000H0Mz\u0000KM2\nz=M2\n0, modeling a single sublat-\ntice where spatial modulations of the magnetization are ne-\nglected.M0is the magnitude of the magnetic moment, H0\nistheappliedexternalfieldand Kistheuniaxialanisotropy\nenergy, also considered to include demagnetization effects\nin the form of a shape anisotropy. The effective field can\nbe written as H=\u0000@F=@M= (H0+ 2KMz=M2\n0)^ez, and\nthe magnetic moment is oriented along the zdirection in\nequilibrium.\nThe linear response to a small transversal external field\ncomponent h(t)is calculated considering M=M0^ez+\nm(t)and expanding Eq. (1) up to first order in h(t)and\nm(t). The exciting field is assumed to be circularly polar-\nized,h\u0006=hx\u0006ihy=he\u0006i!t, with a similar time depen-\ndence for the response, m\u0006=mx\u0006imy=me\u0006i!t. The\ncalculated susceptibility reads (see Appendix B for details)\nm\u0006=\u001f\u0006h\u0006=\rM0\n\n0\u0000!\u0000\u0011!2\u0006i\u000b!h\u0006;(2)\nwith \n0=\r(H0M0+ 2K)=M0. It is found that the\nGilbert damping is associated with the imaginary part of\nthe susceptibility, while the inertial term contributes to the\nreal part of the susceptibility, which is consistent with the\nprevious calculation in Ref. [21]. The dissipated power is\ncalculated as P=_m\u0001h=!Im(\u001f+)jhj2. We note that a\nlinearly polarized exciting field can be described as a linear\ncombination of circularly polarized fields with !and\u0000!\nfrequencies.\nThe dissipated power with and without the inertial term\nis shown in Fig. 1. The data points denoted by symbols\nin Fig. 1 denote the results of the atomistic spin simula-\ntions (see Appendix A for details). The relaxation time is\nchosen to range from \u0011= 10\u000015s to\u0011= 10\u000012s. This\ncovers the fs timescales described in Refs. [20, 23, 24] andthe values of around 300 fs in Ref. [25]. It can be observed\nthat the inertial dynamics reduces the precession resonance\nfrequency. The resonance peak position is well approxi-\nmated as!p=\u0000p1 + 4\fFM\u00001\u0001\n=(2\u0011)\u0019\n0(1\u0000\fFM),\nwith\fFM=\u0011\n0. The associated shift in the resonance\nfieldHpwas investigated in Ref. [23]. However, note that\nthe relative value of this shift is very low since \fFM\u001c1,\nmeaning that it can only be observed if \n0is shifted to high\nvalues, for example by a strong external field H0.\nThe most profound effect of the inertial dynamics is the\nemergence of a second resonance peak, associated with\nthe spin nutation. Its frequency is approximately !n=\n\u0000\u0000p1 + 4\fFM+ 1\u0001\n=(2\u0011)\u0019\u00001=\u0011\u0000\n0(1\u0000\fFM). Simi-\nlarly to the precession frequency, the subleading corrections\n\fFM\n0are small. The negative sign of the frequency im-\nplies an opposite rotational sense [30]: while the precession\nis excited by a circularly polarized field rotating counter-\nclockwise, the nutation resonance reveals an opposite po-\nlarization.\nThe effective damping parameter is defined as the ra-\ntio of the imaginary and the real parts of the frequency\nwhere Eq. (2) has a node, and is approximately expressed\nas\u000beff,p=\u000beff,n\u0019\u000b(1\u00002\fFM), see Appendix B for the\nderivation. Since the imaginary part characterizes the half-\nwidth of the resonance peak at half maximum, the latter\nsuggests that the linewidth of FMR decreases due to the\ninertia, in agreement with the numerical results in Ref. [27].\nThe relative value of the reduction is once again governed\nby the factor \fFM.\nIV. INERTIAL EFFECTS IN\nANTIFERROMAGNETS AND FERRIMAGNETS\nNext, we consider AFMs and FiMs with two sublattices\nAandB. Assuming once again homogeneous sublattice\nmagnetizations, the free energy is expressed as\nF(MA;MB) =\u0000H0(MAz+MBz)\n\u0000KA\nM2\nA0M2\nAz\u0000KB\nM2\nB0M2\nBz+J\nMA0MB0MA\u0001MB;(3)\nwith the external field applied along the zdirec-\ntion, H0=H0^ez, uniaxial easy-axis anisotropy con-\nstantsKA;KBand intersublattice exchange coupling J.\nFrom the free energy, the associated fields entering\nthe sublattice ILLG equations (1) can be determined\nusing HA=B =\u0000@F(MA;MB)=@MA=B =H0^ez+\n2KA=BMA=Bz=M2\nA=B 0^ez\u0000JMB=A=(MA0MB0). Inequilib-\nrium, the sublattice magnetizations are aligned antiparallel\nalong thezdirection. Linear response to the transverse ho-\nmogeneous external field hA(t) =hB(t)may be calculated\nsimilarly to the FM case, using the expansions MA(r;t) =\nMA0^ez+mA(t)andMB(r;t) =\u0000MB0^ez+mB(t).\nThe two-sublattice susceptibility tensor is expressed as\nfollows (see Appendix C for details):\n\u0012\nmA\u0006\nmB\u0006\u0013\n=\u001fAB\n\u0006\u0012\nhA\u0006\nhB\u0006\u0013\n=1\n\u0001\u0006\u00121\n\rBMB0\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u00001\nMA0MB0J\n\u00001\nMA0MB0J1\n\rAMA0\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0013\u0012\nhA\u0006\nhB\u0006\u0013\n;(4)3\n-200-150-1000.00.51.0Dissipated power£Æ∞M0|h|2(a)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°15ssimulations-20-100.00.51.0Dissipated power£Æ∞M0|h|2(b)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°14ssimulations\n-2-10.00.51.0Dissipated power£Æ∞M0|h|2(c)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°13ssimulations-0.3-0.2-0.10.00.51.0Dissipated power£Æ∞M0|h|2(d)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°12ssimulations\nFigure 1. The rate of energy dissipation in the ferromagnet as a function of frequency for several values of the inertial relaxation\ntime, (a)\u0011= 1fs, (b)\u0011= 10fs, (c)\u0011= 100fs, and (d) \u0011= 1ps. The lines denote the results of the analytical calculations\nand the symbols of the atomistic simulations for a single macrospin. All curves are compared to the analytical expression obtained\nwithout the inertial term. The other parameters are \r= 1:76\u00021011T\u00001s\u00001,M0= 2\u0016B,H0= 1T,K= 10\u000023J,\u000b= 0:05, and\njhj= 0:001T.\nHereweusethedefinitions \u0001\u0006= (\rAMA0\rBMB0)\u00001\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u0000J2=\u0000\nM2\nA0M2\nB0\u0001\nas well as \nA=\rA=MA0(J+ 2KA+H0MA0)and\nB=\rB=MB0(J+ 2KB\u0000H0MB0).\nTo compare with FMR, we compute the dissipated power\nfor AFMR, P=_mA\u0001hA+_mB\u0001hB, with the explicit for-\nmula given in Appendix C. The result is shown in Fig. 2,\nusing the same parameters for both sublattices as for the\nFM in Fig. 1. The insets of Fig. 2 show that without the\ninertial term the AFM precession resonance peaks are sup-\npressed with respect to the FM one by a factor of about\nJ=(2K) = 50. This is caused by the fact that the magne-\ntization in the two sublattices rotates around the equilib-\nrium direction with a phase shift of \u0019, meaning that the\nhomogeneous exciting field only couples to the difference of\nthe sublattice precession amplitudes [14] in the dissipated\npower. Also, the inertial term shifts the precession reso-\nnancepeakstolowerfrequenciesconsiderablystrongerthan\nin the FM, and further reduces their magnitude. At higher\nfrequency, two additional nutation resonance peaks can be\nobserved. Remarkably, their height is significantly larger\nthan that of the precession resonances, even exceeding the\nintensity of the FMR peaks (cf. Fig. 1 where the same\nnormalization was used). The latter suggests that probing\nthe AFM nutation resonance peak is experimentally moresuitable than in the FM case. Most of these effects can be\nexplained by the fact that the precession and nutation res-\nonance frequencies lie much closer in AFMs than in FMs,\nas will be discussed in detail below.\nTo obtain the AFM resonance frequencies, we calculate\nthe nodes of the susceptibility tensor in Eq. (4), obtaining\n\u0001\u0006=a\u0006!4+b\u0006!3+c\u0006!2+d\u0006!+e\u0006= 0:(5)\nwith the following definitions:\na\u0006=\u0011A\u0011B; (6)\nb\u0006=\u0007i(\u000bA\u0011B+\u000bB\u0011A)\u0000(\u0011A\u0000\u0011B); (7)\nc\u0006=\u00001\u0006i(\u000bA\u0000\u000bB)\u0000(\nA\u0011B+ \nB\u0011A)\n\u0000\u000bA\u000bB; (8)\nd\u0006= (\nA\u0000\nB)\u0006i(\u000bB\nA+\u000bA\nB); (9)\ne\u0006=\u0000\rA\nMA0\rB\nMB0J2+ \nA\nB: (10)\nNote that inertial effects enter via a;b, andc, terms which\nare of higher order in frequency. Setting the inertial re-4\n-200-100010020000.51.0Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°15ssimulations\n-0.6-0.4-0.200.20.40.600.010.02\n-20-100102000.51.0Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°14ssimulations\n-0.6-0.4-0.200.20.40.600.010.02\n-0.8-0.6-0.4-0.200.20.40.60.800.51.01.52.0Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°12ssimulations\n-0.6-0.4-0.200.20.40.600.010.02-3-2-1012300.51.01.5Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°13ssimulations\n-0.6-0.4-0.200.20.40.600.010.02\nFigure2. Therateofenergydissipationfor theantiferromagnetasafunctionof frequencyforseveralvaluesofthe inertialrelaxation\ntime\u0011A=\u0011B=\u0011, (a)\u0011= 1fs, (b)\u0011= 10fs, (c)\u0011= 100fs and (d)\u0011= 1ps. The lines denote the results of the analytical\ncalculations and the symbols of the atomistic spin simulations for two coupled macrospins. All curves are compared to the analytical\nexpression obtained without the inertial term. The other parameters are MA0=MB0= 2\u0016B,\rA=\rB= 1:76\u00021011T\u00001s\u00001,\n\u000bA=\u000bB= 0:05,KA=KB= 10\u000023J,J= 10\u000021J,H0= 1T, andjhAj=jhBj= 0:001T. The insets show the precession\nresonances on a smaller frequency and power scale.\nlaxation times to zero, we obtain a second-order equa-\ntion that results in well-known antiferromagnetic reso-\nnance frequencies [31–33]. For equivalent sublattices and\nassuming\u000b\u001c1andK\u0019H0M0\u001cJ, these read\n!p\u0006\u0019\u0010\n1\u0006i\u000bp\nJ=(4K)\u0011\u0010\n\rH0\u0006\r=Mp\n4KJ\u0011\n. Com-\npared to the FM case, two resonance frequencies are found,\nand they are exchange enhanced by about a factor ofp\nJ=K. However, the lifetime of the excitations is reduced\nsince the effective damping is also higher by a factor ofp\nJ=(4K).\nIn the presence of the inertial term, the resonance fre-\nquencies are found as a solution of a fourth-order equation.\nThe real and imaginary parts of the calculated frequencies\nare denoted by Re(!p;n\u0006)andIm(!p;n\u0006)for precession and\nnutation resonances, respectively. These have been calcu-\nlated for an AFM and a FiM as a function of the relaxation\ntime\u0011A=\u0011B=\u0011in Fig. 3. In the absence of external field\nand damping, Eq. (5) simplifies to a second-order equa-\ntion in!2. The precession resonance frequencies are given\nby!p\u0006\u0019\u0006\r=Mp\n4KJ(1 + 2\fAFM)\u00001\n2forK\u001cJ. It is\nimportant to note here that the relative strength of the in-\nertial corrections is defined by the dimensionless parameter\n\fAFM =(\u0011\r=M 0)J, which is enhanced by a factor of J=Kas\ncompared to \fFM. The characteristic time scale of the ex-change interactions typically falls into the fs range in AFMs\nwhich are ordered at room temperature ( \rJ=M\u00191013s\u00001\nwiththeparametersusedhere), whichissimilartothetypi-\ncal values of the inverse inertial relaxation time [20, 23, 25].\nThis explains the considerable decrease of the AFMR pre-\ncession frequencies in Fig. 2, while Fig. 3(a) demonstrates\nthat deviations from the non-inertial case already become\nobservable for \u0011\u00191fs. This more pronounced inertial ef-\nfect should also be observable if the resonance is measured\nby sweeping the external field, as in Ref. [23]. The strongly\nasymmetric ( MA0= 5MB0) FiM in Fig. 3(b) is charac-\nterized by a high-frequency exchange mode, strongly influ-\nencedbyinertialeffectsasintheAFM,andalow-frequency\nmode which is less affected like in the FM.\nThe nutation resonance frequencies in the AFM can be\nexpressed as !n\u0006\u0019\u0006p1 + 2\fAFM=\u0011. Just as for the pre-\ncession resonance, the correction factor arising due to the\ninterplaybetweeninertiaandmagneticinteractionsisgiven\nby\fAFM, which is exchange enhanced compared to the FM\ncase. This gives rise to an increase of the nutation frequen-\ncies, as demonstrated in Fig. 2. For the FiM in Fig. 3(b),\nthe nutation frequency Re(!n+)belonging to the exchange\nmode Re(!p\u0000)starts deviating from the low-inertia \u0011\u00001\nasymptote at considerably lower frequencies than the FM-\nlike nutation Re(!n\u0000).5\n10−1610−1510−1410−1310−12\nη(s)10−1100101102103ωAFM\n±/2π(THz)\n(a)\n1/η\nRe(ωp+)\nRe(ωp−)\nRe(ωn+)\nRe(ωn−)\n10−1610−1510−1410−1310−12\nη(s)10−1100101102103ωFiM\n±/2π(THz)\n(b)\n1/η\nRe(ωp+)\nRe(ωp−)\nRe(ωn+)\nRe(ωn−)\nFigure 3. (Color Online) Real part of the precession resonance frequencies as a function of inertial relaxation time \u0011, (a) for AFMs\nwithMA0=MB0= 2\u0016Band (b) for FiMs with MA0= 5MB0= 10\u0016B. The other parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,\n\u000bA=\u000bB= 0:05,KA=KB= 10\u000023J,J= 10\u000021J,H0= 1T.\nThe effective damping parameters of the excitation\nmodes, defined as the ratio of the imaginary to the real part\nof the frequencies, are shown in Fig. 4. They no longer co-\nincide between precession and nutation as in the FM case,\nsince the exchange enhancement discussed above does not\naffect the nutation resonance. A reduction of the effec-\ntive damping is observed with increasing inertial relaxation\ntimes, which becomes noticeable for \fAFM =O\u0000\n10\u00002\u0001\n, just\nas in the case of the resonance frequencies. The consid-\nerable reduction of the effective damping compared to the\nGilbert damping leads to sharper nutation resonance peaks\nas demonstrated in Fig. 2, with higher intensities than for\nthe FM. In the FiM, the exchange modes !n+and!p\u0000\nstarttobecomeinfluencedatlowerinertialrelaxationtimes\nthan the FM modes !n\u0000and!p+[34]. The difference be-\ntween the effective damping parameters vanishes between\nexchange and FM modes for higher \u0011, but it remains to be\nobservable between precession and nutation modes.\nV. CONCLUSIONS\nTo conclude, we applied the ILLG equation to FMs and\ntotwo-sublatticeAFMsandFiMs, andinvestigatedtheres-\nonance frequencies using linear-response theory and com-\nputer simulations. The precession frequencies are found to\ndecrease with increasing inertial relaxation time and addi-\ntional high-frequency nutation peaks become observable.\nFurthermore, the calculation of the resonance linewidth\nshows that the effect of inertia reduces the effective damp-\ning parameter. While in FMs these corrections scale with\n\fFM=\u0011\n0, in AFMs the dimensionless coupling between\nprecession and nutation is given by \fAFM = (\u0011\r=M 0)J,\nwhich is typically several orders of magnitude higher.\nTherefore, an antiferromagnetic system with higher ex-\nchange to anisotropy energy ratio and higher \u0011will be suit-\nable to observe inertial effects. Such antiferromagnetic sys-\ntems include NiO [35] and CrPt [36, 37], even though the\ncharacteristic inertial relaxation time \u0011is unknown. The\nFiM is observed to interpolate between the FM and AFM\nlimits. The reduced effective damping gives rise to particu-\nlarly sharp and high-intensity nutation resonance peaks in\nAFMs, with frequencies comparable to the values alreadyobserved in FMs [23, 25]. These findings are expected to\nmotivatethesearchforthesignsofintrinsicallyinertialspin\ndynamics on ultrafast timescales using AFMR techniques.\nACKNOWLEDGMENTS\nWe acknowledge financial support from the Alexander\nvon Humboldt-Stiftung, the Deutsche Forschungsgemein-\nschaft via Project No. NO 290/5-1, and the National Re-\nsearch, Development, and Innovation Office of Hungary via\nProject No. K131938.\nAppendix A: Atomistic simulations of the ILLG\nequation\nThe inertial Landau-Lifshitz-Gilbert (ILLG) equation of\nmotion, given in Eq. (1) in the main text, can be rewritten\nfor the normalized spin si(t) =Mi(t)=Mi0as [21]\n@tsi=\u0000\risi\u0002Hi+\u000bisi\u0002@tsi+\u0011isi\u0002@ttsi:(A1)\nThe first term denotes precession of the spins around an\neffective field Hi, the second term corresponds to a trans-\nverse relaxation of the spins, and the last term defines the\ninertial dynamics [15]. The ILLG equation can be rewrit-\nten from the implicit form of Eq. (A1) to an explicit dif-\nferential equation which can be solved numerically without\niterations. By taking a scalar product of Eq. (A1) with siit\nis easy to see that the length of the spin remains conserved\nin the ILLG equation, i.e., @tjsij2= 0andsi\u0001@tsi= 0.\nFurthermore, we use\nsi\u0002(si\u0002@ttsi) =si(si\u0001@ttsi)\u0000@ttsi;(A2)\n@t(si\u0001@tsi)|{z}\n=0= (@tsi)2+si\u0001@ttsi: (A3)\nBy multiplying Eq. (A1) by si\u0002and using the conditions\nEqs. (A2) and (A3), we obtain the explicit equation of mo-\ntion (cf. Ref. [30])\n@ttsi=\u0000\ri\n\u0011isi\u0002(si\u0002Hi)\u0000\u000bi\n\u0011i@tsi\u00001\n\u0011isi\u0002@tsi\n\u0000si(@tsi)2=Fi(s;@ts;t): (A4)6\n10−1610−1510−1410−1310−12\nη(s)0.000.050.100.150.200.25Im(ω±)/Re(ω±)|AFM\n(a)\nη= 0\nIm(ωp+)/Re(ωp+)\nIm(ωp−)/Re(ωp−)\nIm(ωn+)/Re(ωn+)\nIm(ωn−)/Re(ωn−)\n10−1610−1510−1410−1310−12\nη(s)0.000.010.020.030.040.050.060.07Im(ω±)/Re(ω±)|FiM\n(b)\nη= 0\nIm(ωp+)/Re(ωp+)\nIm(ωp−)/Re(ωp−)\nIm(ωn+)/Re(ωn+)\nIm(ωn−)/Re(ωn−)\nFigure 4. (Color Online) Effective damping parameters of the resonance modes as a function of inertial relaxation time \u0011, for (a)\nAFMs with MA0=MB0= 2\u0016Band (b) FiMs with MA0= 5MB0= 10\u0016B. The other parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,\n\u000bA=\u000bB= 0:05,KA=KB= 10\u000023J,J= 10\u000021J,H0= 1T.\nNote that a second-order explicit differential equation is\nobtained because of the inertial term, while the LLG equa-\ntion is of first order. With the definition pi=@tsi, we can\nconvert the second-order differential equation into a system\nof first-order differential equations as follows:\n@ttsi=@tpi=Fi(s;p;t); (A5)\n@tsi=pi=Gi(s;p;t): (A6)\nIt is obvious that one has to solve six coupled differential\nequations of first order per lattice site i. We numerically\nsolve these equations with Heun’s method [38], where the\npredictor steps are\n\u0016si=si(t) + \u0001tGi(s;p;t); (A7)\n\u0016pi=pi(t) + \u0001tFi(s;p;t); (A8)\nand the corrector steps are implemented as\nsi(t+ \u0001t) =si(t) +\u0001t\n2[Gi(s;p;t) +Gi(\u0016s;\u0016p;t+ \u0001t)];\n(A9)\npi(t+ \u0001t) =pi(t) +\u0001t\n2[Fi(s;p;t) +Fi(\u0016s;\u0016p;t+ \u0001t)]:\n(A10)\nIn order to calculate the resonance curves, we employed a\ncircularly polarized field h(t)\u0018ei!tin thexyplane in ad-\ndition to the static magnetic field H0along thezdirection,and solved the equations of motion for one and two spins by\nstarting from the equilibrium state along the zdirection.\nBy multiplying Eq. (A4) by Mi0\u0011i@tsi=\ri, summing over\nthe sublattices, and rearranging the terms, one arrives at\n@t X\niMi0\u0011i\n2\ri(@tsi)2+F!\n=X\ni@tMi0@tsihi\n\u0000X\ni\u000biMi0\n\ri(@tsi)2: (A11)\nThe left-hand side of Eq. (A11) describes the change of rate\nof the energy of the system, consisting of a kinetic part and\na potential partF. The former sheds light on the meaning\nof\u0011ias an inertial parameter. The right-hand side con-\nsists of the power loss due to damping processes, which\nis compensated by the external driving force in a steady\nstate. Accordingly, we computed the dissipated power us-\ningP=P\niMi0@tsi\u0001hi.\nAppendix B: Calculation of the linear response in\nferromagnets\nIn ferromagnets, we consider that the initial magnetiza-\ntion points towards the zdirection, such that the magne-\ntization is expanded as M=M0^ez+m(t)in linear order.\nThe considered dynamical field is denoted by h(t). Using\nthe effective field in the main text, the linearized ILLG\nequation can be written in the following way:\n@tm=\u0000\r2\n664M0^ez\u0002H0^ez|{z}\n= 0+M0^ez\u00022K\nM0^ez\n|{z}\n= 0+M0^ez\u0002h(t) +m(t)\u0002H0^ez+m(t)\u00022K\nM0^ez+m(t)\u0002h(t)|{z}\nnegligible3\n775\n+\u000b\nM02\n664M0^ez\u0002@m\n@t+m\u0002@m\n@t|{z}\nnegligible3\n775+\u0011\nM02\n664M0^ez\u0002@2m\n@t2+m\u0002@2m\n@t2|{z}\nnegligible3\n775: (B1)7\nThus, we obtain the following two equations for the\ntransversal components:\n@tmx=\rM0hy\u0000\rH0my\u00002\rK\nM0my\u0000\u000b@tmy\u0000\u0011@ttmy;\n(B2)\n@tmy=\u0000\rM0hx+\rH0mx+2\rK\nM0mx+\u000b@tmx+\u0011@ttmx:\n(B3)\nWe define \n0=\r=M 0(H0M0+ 2K)as in the main text.\nTherefore, Eqs. (B2) and (B3) can be recast as\nhx=1\n\rM0[\n0mx+\u000b@tmx+\u0011@ttmx\u0000@tmy];(B4)\nhy=1\n\rM0[\n0my+\u000b@tmy+\u0011@ttmy+@tmx]:(B5)\nIn matrix form we write\n\u0012\nhx\nhy\u0013\n=1\n\rM0\u0012\n\n0+\u000b@t+\u0011@tt\u0000@t\n@t \n0+\u000b@t+\u0011@tt\u0013\u0012\nmx\nmy\u0013\n:\n(B6)\nWe switch to the circularly polarized basis, m\u0006=mx\u0006imy\nandh\u0006=hx\u0006ihy, where the equations decouple,\n\rM0\u0012\nh+\nh\u0000\u0013\n=\n\u0012\n\n0+\u000b@t+\u0011@tt+i@t 0\n0 \n 0+\u000b@t+\u0011@tt\u0000i@t\u0013\u0012\nm+\nm\u0000\u0013\n:\n(B7)\nFor the time dependence we consider h\u0006=he\u0006i!t, describ-\ningtwotypesofpolarizationwithoppositehandedness. We\nassumem\u0006=me\u0006i!t. Thus, we have\nhei!t=1\n\rM0\u0000\n\n0+i\u000b!\u0000\u0011!2\u0000!\u0001\nmei!t\n)m+=\rM0\n\n0+i\u000b!\u0000\u0011!2\u0000!hei!t; (B8)\nhe\u0000i!t=1\n\rM0\u0000\n\n0\u0000i\u000b!\u0000\u0011!2\u0000!\u0001\nme\u0000i!t\n)m\u0000=\rM0\n\n0\u0000i\u000b!\u0000\u0011!2\u0000!he\u0000i!t: (B9)\nThis leads to the susceptibility given in Eq. (2). Its real\nand imaginary parts are derived as\nRe(\u001f\u0006) =\rM0\n0\u0000!\u0000\u0011!2\n(\n0\u0000!\u0000\u0011!2)2+\u000b2!2;(B10)\nIm(\u001f\u0006) =\u0006\rM0\u000b!\n(\n0\u0000!\u0000\u0011!2)2+\u000b2!2:(B11)The dissipated power can be calculated according to its\ndefinition based on Eq. (A11),\nP=@tm\u0001h\n= (@tmxhx+@tmyhy)\n=1\n2(@tm+h\u0000+@tm\u0000h+)\n=i!\n2(\u001f+\u0000\u001f\u0000)jhj2\n=i!\n2\u0012\u00002i\u000b!\rM 0\n(\n0\u0000!\u0000\u0011!2)2+\u000b2!2\u0013\njhj2\n=!Im(\u001f+)jhj2: (B12)\nThe positions and the linewidths of the resonance peaks\nmay be analyzed by finding the poles of the susceptibility\nin Eq. (B8),\n!=1\n2\u0011\u0014\n\u0000(1\u0000i\u000b)\u0006q\n(1\u0000i\u000b)2+ 4\fFM\u0015\n=1\n2\u0011\u0002\n\u00001\u0006a+i\u000b\u0000\n1\u0007a\u00001\u0001\u0003\n; (B13)\nwhere\fFM=\u0011\n0andais the single positive real solution\nof the fourth-order equation\na4\u0000\u0000\n1\u0000\u000b2+ 4\fFM\u0001\na2\u0000\u000b2= 0:(B14)\nFor\fFM\u001c1, one hasa= 1 + 2\fFM+O\u0000\n\f2\nFM\u0001\n. For\nthe real parts of the frequencies, corresponding to the\npeak positions, one obtains !p\u0019\n0(1\u0000\fFM)and!n\u0019\n\u00001=\u0011\u0000\n0(1\u0000\fFM), as described in the main text. Note\nthat the latter expression agrees with Eq. (14) in Ref. [28],\nbut the correction terms are different from Ref. [27], where\n!n=\u0000p1 +\fFM=\u0011\u0019\u00001=\u0011\u0000\n0=2\u0000\n1\u0000\fFM=4\u0001\nwas sug-\ngested. It is apparent from Eq. (B13) that effective damp-\ning parameter, i.e. the ratio of the imaginary and the real\nparts of the frequency, is \u000ba\u00001\u0019\u000b(1\u00002\fFM)both for\nthe precession and the nutation peaks. The full width of\nthe resonance peaks at half maximum can be expressed as\n\u0001!=!1\u0000!2, which frequencies satisfy\n\n0\u0000!1\u0000\u0011!2\n1=\u0000\u000b!1; (B15)\n\n0\u0000!2\u0000\u0011!2\n2=\u000b!2: (B16)8\nThe ratio of the linewidth and the peak position is given by\n\u0001!\n!p=(\n0+\u000b\n0(1\u0000\fFM))\u0000\u0011(\n0+\u000b\n0)2\u0000(\n0\u0000\u000b\n0(1\u0000\fFM)) +\u0011(\n0\u0000\u000b\n0)2\n\n0(1\u0000\fFM)\n=2\u000b\n0\u00006\u000b\fFM\n0\n\n0(1\u0000\fFM)= 2\u000b1\u00003\fFM\n1\u0000\fFM\u00192\u000b(1\u00002\fFM) (B17)\nfor the precession resonance, confirming that dividing the\nhalf-width at half maximum by the resonance frequency\nis approximately equal to the effective damping parameter\ndescribed above.Appendix C: Calculation of the linear response in\ntwo-sublattice antiferromagnets and ferrimagnets\nWe expand the magnetization around the equilibrium\ndirection in small deviations, MA=MA0^ez+mAand\nMB=\u0000MB0^ez+mB, which are induced by the trans-\nverse external field hA=B(t). The linearized ILLG equation\nfor the two sublattices reads\n@tmA=\u0000\rA\nMA0[\u0000(H0MA0+ 2KA+J)mAx^ey+ (H0MA0+ 2KA+J)mAy^ex] +\rA\nMB0[JmBx^ey\u0000JmBy^ex]\n\u0000\rAMA0(hAx^ey\u0000hAy^ex) +\u000bA(@tmAx^ey\u0000@tmAy^ex) +\u0011A(@ttmAx^ey\u0000@ttmAy^ex); (C1)\n@tmB=\u0000\rB\nMB0[\u0000(H0MB0\u00002KB\u0000J)mBx^ey+ (H0MB0\u00002KB\u0000J)mBy^ex]\u0000\rB\nMA0[JmAx^ey\u0000JmAy^ex]\n+\rBMB0(hBx^ey\u0000hBy^ex)\u0000\u000bB(@tmBx^ey\u0000@tmBy^ex)\u0000\u0011B(@ttmBx^ey\u0000@ttmBy^ex): (C2)\nFor thexandycomponents we obtain\n\rAMA0hAy=\rA\nMA0(H0MA0+ 2KA+J)mAy+\rA\nMB0JmBy+\u000bA@tmAy+\u0011A@ttmAy+@tmAx; (C3)\n\rAMA0hAx=\rA\nMA0(H0MA0+ 2KA+J)mAx+\rA\nMB0JmBx+\u000bA@tmAx+\u0011A@ttmAx\u0000@tmAy; (C4)\n\rBMB0hBy=\rB\nMB0(\u0000H0MB0+ 2KB+J)mBy+\rB\nMA0JmAy+\u000bB@tmBy+\u0011B@ttmBy\u0000@tmBx;(C5)\n\rBMB0hBx=\rB\nMB0(\u0000H0MB0+ 2KB+J)mBx+\rB\nMA0JmAx+\u000bB@tmBx+\u0011B@ttmBx+@tmBy:(C6)\nIn the circularly polarized basis with mA=B\u0006=mA=Bx\u0006imA=By;hA=B\u0006=hA=Bx\u0006ihA=Byand defining \nA=\n\rA=MA0(H0MA0+ 2KA+J);\nB=\rB=MB0(J+ 2KB\u0000H0MB0), we obtain\n\rAMA0hA\u0006= (\nA+\u000bA@t+\u0011A@tt\u0006i@t)mA\u0006+\rA\nMB0JmB\u0006; (C7)\n\rBMB0hB\u0006= (\nB+\u000bB@t+\u0011B@tt\u0007i@t)mB\u0006+\rB\nMA0JmA\u0006: (C8)\nThe four equations of motion are separated into two pairs of coupled equations for the +and\u0000components. In matrix\nformalism we have\n\u0012\nhA\u0006\nhB\u0006\u0013\n=0\nB@1\n\rAMA0(\nA+\u000bA@t+\u0011A@tt\u0006i@t)1\nMA0MB0J\n1\nMA0MB0J1\n\rBMB0(\nB+\u000bB@t+\u0011B@tt\u0007i@t)1\nCA\u0012\nmA\u0006\nmB\u0006\u0013\n: (C9)\nBy substituting the time dependence hA=B\u0006;mA=B\u0006/e\u0006i!twe have\n\u0012\nhA\u0006\nhB\u0006\u0013\n=0\nB@1\n\rAMA0\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001 1\nMA0MB0J\n1\nMA0MB0J1\n\rBMB0\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u00011\nCA\u0012\nmA\u0006\nmB\u0006\u0013\n:(C10)9\nWe introduce the definition \u0001\u0006 = (\rAMA0\rBMB0)\u00001\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u0000\nJ2=\u0000\nM2\nA0M2\nB0\u0001\nfor the determinant of the matrix above. The susceptibility tensor is obtained by matrix inversion,\n\u0012\nmA\u0006\nmB\u0006\u0013\n=1\n\u0001\u00060\nB@1\n\rBMB0\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u00001\nMA0MB0J\n\u00001\nMA0MB0J1\n\rAMA0\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u00011\nCA\u0012\nhA\u0006\nhB\u0006\u0013\n=\u001fAB\n\u0006\u0012\nhA\u0006\nhB\u0006\u0013\n;\n(C11)\nas also given in Eq. (4).\nSimilarly to the ferromagnet, we calculate the dissipated power from Eq. (A11) as\nPAB=@tmA\u0001hA+@tmB\u0001hB\n=1\n2h\n@tmA+hA\u0000+@tmA\u0000hA++@tmB+hB\u0000+@tmB\u0000hB+i\n=i!\n2h1\n\u0001+\u00121\n\rBMB0\u0000\n\nB+i!\u000bB\u0000\u0011B!2+!\u0001\nhA+\u00001\nMA0MB0JhB+\u0013\nhA\u0000\n\u00001\n\u0001\u0000\u00121\n\rBMB0\u0000\n\nB\u0000i!\u000bB\u0000\u0011B!2+!\u0001\nhA\u0000\u00001\nMA0MB0JhB\u0000\u0013\nhA+\n+1\n\u0001+\u0012\n\u00001\nMA0MB0JhA++1\n\rAMA0\u0000\n\nA+i!\u000bA\u0000\u0011A!2\u0000!\u0001\nhB+\u0013\nhB\u0000\n\u00001\n\u0001\u0000\u0012\n\u00001\nMA0MB0JhA\u0000+1\n\rAMA0\u0000\n\nA\u0000i!\u000bA\u0000\u0011A!2\u0000!\u0001\nhB\u0000\u0013\nhB+i\n=!2jhAj2\n\rBMB02\n4(\rAMA0\rBMB0)\u00001\u000bAh\u0000\n\nB\u0000\u0011B!2+!\u00012+!2\u000b2\nBi\n+J2=\u0000\nM2\nA0M2\nB0\u0001\n\u000bB\n\u0001+\u0001\u00003\n5\n+!2jhBj2\n\rAMA02\n4(\rAMA0\rBMB0)\u00001\u000bBh\u0000\n\nA\u0000\u0011A!2\u0000!\u00012+!2\u000b2\nAi\n+J2=\u0000\nM2\nA0MB0\u00012\u000bA\n\u0001+\u0001\u00003\n5\n\u00002!2JjhAhBj\n\rAM2\nA0\rBM2\nB0\u0014(\nA\u000bB+ \nB\u000bA) + (\u000bA\u0000\u000bB)!\u0000(\u0011A\u000bB+\u0011B\u000bA)!2\n\u0001+\u0001\u0000\u0015\n: (C12)\nAs discussed in the main text, the peak positions and the linewidths may be understood by finding the nodes of the\ndeterminant \u0001\u0006,\n\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u0000\rA\rB\nMA0MB0J2= 0\n)\u0011A\u0011B|{z}\n=a\u0006!4+ [\u0007i(\u000bA\u0011B+\u000bB\u0011A)\u0000(\u0011A\u0000\u0011B)]| {z }\n=b\u0006!3\n+ [\u00001\u0006i(\u000bA\u0000\u000bB)\u0000(\nA\u0011B+ \nB\u0011A)\u0000\u000bA\u000bB]| {z }\n=c\u0006!2\n+ [(\nA\u0000\nB)\u0006i(\u000bB\nA+\u000bA\nB)]| {z }\n=d\u0006!+ \nA\nB\u0000\rA\rB\nMA0MB0J2\n|{z}\n=e\u0006= 0: (C13)\nThe fourth-order equation (C13) may be solved in a\nclosed form. However, in order to arrive at solutions which\nhave a simpler form, we consider the antiferromagnet with\nidentical sublattices, MA0=MB0=M0,\u000bA=\u000bB=\u000b,\n\u0011A=\u0011B=\u0011, andKA=KB=K. Furthermore, we\nassume\u000b\u001c1andM0H0;K\u001cJ, as is typical in most\nsystems. Consequently, we will treat the terms propor-\ntional to the damping and the external field in first-orderperturbation theory, leading to\n\u00112!4\u0000\u0012\n1 + 2\u0011\r\nM0(J+ 2K)\u0013\n!2\u0000i2\u000b\u0011!3\n(0)+ 2\rH0!(0)\n+i2\u000b\r\nM0(J+ 2K)!(0)+\r2\nM2\n0(J+ 2K)2\u0000\r2(H0)2\n\u0000\r2\nM2\n0J2= 0; (C14)\nwhere!(0)is the solution for \u000b= 0andH0= 0, and we10\nonly treat \u0001+for simplicity since \u0001\u0000may be obtained\nby complex conjugation. Equation (C14) is a second-order\nequation in !2, the solutions of which are simple to express.Expanding them up to first order in \u000bandH0for consis-\ntency with the order of the perturbation, and also in first\norder inK=J\u001c1, one obtains\n!p\u0006\u0019\u0006\r\nM0p\n4K(J+K)r\n1 + 2\u0011\r\nM0(J+ 2K)+1r\n1 + 2\u0011\r\nM0(J+ 2K)\n\u0002\f\f!(0)\f\f\n\r\nM0p\n4K(J+K)\u0014\n\rH0+i\u000b\u0012\r\nM0(J+ 2K)\u0000\u0011!2\n(0)\u0013\u0015\n; (C15)\n!n\u0006\u0019\u00061\n\u0011r\n1 + 2\u0011\r\nM0(J+ 2K)0\nBBB@1\u0000\u00112\r2\nM2\n04K(J+K)\n2\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u001521\nCCCA\n\u0000\u0011\f\f!(0)\f\f\n\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u00153\n2\u0014\n\rH0+i\u000b\u0012\r\nM0(J+ 2K)\u0000\u0011!2\n(0)\u0013\u0015\n; (C16)\nfor the precession and the nutation frequencies, respectively. Substituting in\f\f!(0)\f\ffrom the leading term in the expression\ninto the perturbative terms, one arrives at\n!p\u0006\u0019\u0006\r\nM0p\n4K(J+K)r\n1 + 2\u0011\r\nM0(J+ 2K)\n+1\n1 + 2\u0011\r\nM0(J+ 2K)\u0014\n\rH0+i\u000b\r\nM0(J+ 2K)\u0015\n; (C17)\n!n\u0006\u0019\u00061\n\u0011r\n1 + 2\u0011\r\nM0(J+ 2K)0\nBBB@1\u0000\u00112\r2\nM2\n04K(J+K)\n2\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u001521\nCCCA\n\u00001\n1 + 2\u0011\r\nM0(J+ 2K)\u0014\n\rH0\u0000i\u000b\u00121\n\u0011+\r\nM0(J+ 2K)\u0013\u0015\n: (C18)\nThe leading-order terms for H0;\u000b= 0and usingJ+K\u0019\nJare also reported in the main text. As discussed there,\nin the antiferromagnet the corrections caused by the in-\nertial dynamics surpass in magnitude those in the ferro-\nmagnet, since the characteristic dimensionless parameter\n\fFM=\u0011\n0is replaced by \fAFM =\u0011\r=M 0(J+ 2K)\u0019\n\u0011\r=M 0J+2K. Thisdifferenceisalsomanifestinthedepen-\ndence of the excitation frequencies on the static magnetic\nfieldH0: while in the ferromagnet the Larmor frequency\nis renormalized as (1\u0000\fFM)\rH0, in the antiferromagnet\nthecorrespondingfactoris (1 + 2\fAFM)\u00001\rH0forboththe\nprecession and the nutation frequencies, causing an appar-\nent decrease in the gyromagnetic factor.\nFrom Eqs. (C17) and (C18), the effective damping pa-rameters in the antiferromagnet may be expressed as\nIm(!p)\nRe(!p)\u0019\u000bs\n(J+ 2K)2\n4K(J+K)1r\n1 + 2\u0011\r\nM0(J+ 2K);\n(C19)\nIm(!n)\nRe(!n)\u0019\u000b1 +\u0011\r\nM0(J+ 2K)\n\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u00153\n2: (C20)\nWhile the inertial dynamics decrease the resonance\nlinewidth of the antiferromagnet by a larger factor\n(1 + 2\fAFM)\u00001=2compared to the ferromagnet (1\u00002\fFM),\nthis is compensated by the exchange enhancement ex-\npressed in the factorp\nJ=4K. 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Suga5,9 \n \n \n1 Institute for Materials Research, Tohoku University, 2- 1-1 Katahira, Sendai 980- 8577, \nJapan \n2 Center for Spintronics Research Network, Tohoku University, 2 -1-1 Katahira, Sendai \n980-8577, Japan \n3 Division of Materials Physics , Graduate School of Engineering Science, Osaka \nUniversity, 1- 3 Machikaneyama, Toyonaka, Osaka 560- 8531, Japan \n4 Center for Spintronics Research Network, Osaka University, 1-3 Machikaneyama, \nToyonaka, Osaka 560- 8531, Japan \n5 Institute of Scientific and Industrial Research, Osaka University, 8- 1 Mihogaoka, Ibaraki \n567-0047, Japan \n6 CMI2-MaDIS, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki \n305-0047, Japa n \n7 Institute for Solid State Physics, The University of Tokyo, Kashiwanoha, Chiba \n277-8581, Japan \n8 Synchrotron Radiation Research Organization, The University of Tokyo, Sayo -cho, \nHyogo 679 -5198, Japan \n9 Forschungszentrum Jü lich, PGI -6, 52425 Jü lich, Germany \n \n \n 2 \n Abstract \nDetailed information on the electronic states of both V and Mn 3d electrons in the \nferrimagnet Mn 2V Al is obtained by the bulk sensitive resonant inelastic soft x -ray scattering \n(SX-RIXS) excited with the circularly polarized light under an external magnetic field for the \nfirst time . The results under the V L -edge excitation have revealed the negligible partial \ndensity of states (PDOS) of the V 3 d states around the Fermi energy as well as their rather \nlocalized character. Under the M n L-edge excitation, on the other hand, the spectra are \ndominated by fluorescence with clear magnetic circular dichroism with noticeable excitation \nphoton energy dependence. Compared with the theoretical prediction of the RIXS spectra \nbased on the density -functional -theory band structure calculation, an itinerant, spin- dependent \ncharacter of the Mn 3 d states and decays of the Mn 2p core states are confirmed in \nconsistence with t he half -metallicity of the Mn 3 d states . \n \nKeywords: SX-RIXS, RIXS magnetic circular dichroism , bulk electronic states, partial \ndensity of states, half -metal, Heusler alloy \n Corresponding author: * rieume@imr.tohoku.ac.jp \n \n \n 3 \n I. INTRODUCTION \nSince the hal f-metal lic electronic structure was predicted in the half-Heusler alloys of \nNiMnSb and PtMnSb [1] and in full -Heusler alloys such as Co -based Heusler alloys [2,3] , a \nlarge number of investigations have been carried out from the interest in the field of \nspintronics . When electrons around the Fermi energy (EF) are completely spin -polarized, a \nsystem must be very useful as a ferromagnetic electrode for the spin injection and tunnel \nmagnetoresistance as well as various spin utilizable devices. For example, magnetoresistance \nof the magnetic tunneling junction with using complete half -metal ferromagnets would ideally \nbe infinity. Very recently, other types of Heulser alloys such as Mn -based Heusler alloys and \ntheir quasi -ternary alloy s have also b een pointed out to show the half -metallic electronic \nstates [4-10]. Among them Mn 2V Al is one of the most attractive materials for device \napplication. Its magnetic properties and theoretically predicted electronic structures were \nreported in early 1980’s [11-13]. \nIf the Mn 2VA l orders completely, atoms of Mn, V and Al occupy the Wyckoff positions \n8c, 4b, and 4a, respectively , with the space group Fm3m as schematically shown in Fig. S 1 in \nSupplemental Material . The spontaneous magnetization per formula unit of 1.9 µ B/f.u. at 4.2 \nK is close to 2 µ B/f.u.[ 11] predicted by the g eneralized Slater -Pauling rule [14] and is much \nsmaller than those in Co- based Heuser alloys. The magnetic critical Curie temperature TC of \nthis ferrimagnet Mn 2VA l is quite high to be about 760 K with antiferromagnetically coupled V \nand Mn spins [11,12]. This material is thought to be very promising for spintronic devices at \nroom temperature because the expected current to switch its spin would be rather low. \nInvestigations of Mn 2V Al have recently been extensively carried out , for example, by the \nx-ray absorption magnetic circular dichroism (XAS -MCD ) for bulk and film specimens \n[15-18]. \nFor the fundamental investigations of half -metallic materials, researcher s should pay \nattention how to provide convincing evidence of the half -metallic electronic states. To date \nfundamental magnetic properties reflecting the specific character of the half -metallic \nelectronic states have been investigated. For example, rather small high -field magnetic \nsusceptibility [19] and the negligibly small pressure dependence of the magnetization in some \nCo-based Heusler alloys were reported. These results are understood as reflect ing the fact that \nthe electron ic state near EF is insensitive to applied external fields in the case of half -metallic \nferromagnets [20]. Recently, anisotropic magne toresistance is also predicted to serve as one of \nthe practical screening tests of the half -metallic ferromagnets from both theore tical and \nexperimental aspects [21,22] . \nWe report here for the first time the direct evidence of the spin -polarized electronic 4 \n structures of t he half-metallic ferrimagnet Mn 2V Al Heusler alloy with highly ordered L 21-type \nstructure by means of resonant inelastic soft x -ray scattering (SX -RIXS) measurements of the \nV and Mn 2p core excitation with use of right and left helicity circularly polarized l ight and \nwith external magnetic field. RIXS is a bulk sensitive photon- in and photon- out spectroscopy, \nand very powerful to investigate such as d- d excitations for open shell 3d orbitals and \nmagneti c excitations for spin systems [23-26] as well as 2 p-3d transitions in element - and \nsymmetry -specific ways. These excitations and decays are sensitive to spin, electron \ncorrelation, crystalline symmetry, and the strength of hybridization with the ligand band. \nFurthermore , RIXS is insensitive to the surface conditions because of its long probing depth \n(> 100 nm) in contrast to any kind of photoelectron spectroscopy (PES). Since the emitted \nlight is probed, RIXS is not affected by such an external perturbation as magnetic field in contrast to PES for electrons . In the present study, the photon energy ( hv\nin) dependence of the \nmagnetic circular dichroism (MCD) of RIXS was measured in detail in order to obtain the \nspin- dependent information. The results obtained in the present experiments and theoretical \nanalyses confirmed the half -metallicity of Mn 2V Al, demonstrating that the RIXS and \nRIXS -MCD are extremely powerful for the study of the electronic structures of the \nhalf-metallic ferro magnetic or ferrimagnetic materials. \n \n \nII. EXPERIMENTAL \nA. Sample preparation \nMother ingot of polycrystalline Mn 2VAl was fabricated by induction melting in an argon \natmosphere. Since the vapor pressure of Mn is high during the melting, excess Mn elements are contained in the mother ingot. Single crystal was grown by the Bridgeman me thod with a \nsize of 12 mm in diameter and about 30 mm in length. The obtained ingot was annealed at \n1473 K to grow the crystal grains . Furthermore, a two -step annealing process at 1123 K and \nthen 873 K was employed in order to control the microstructure s and to heighten the degree of \norder. These sample preparation processes result ed in rather high degree of order as S = 0.84 \nin our sample compared with S = 0.5 by Kubota et al. [15] and S = 0.4 by Meinert et al . [17] \nin their samples. Crystal orientation was checked by the back Laue method and the specimen \nwas cut in a strip form in the direction parallel to <100>. The sample composition was confirmed to be Mn: 50.5, V: 26.9, Al: 22.6 (atomic %) with an electron probe microanalyzer. \nSample magnetization wa s measured with a superconducting quantum interference devices \n(SQUID) magnetometer. Magnetization ( M-H) curve measured at 5 K for Mn\n2VA l showed \nthat more that 90% of saturation magnetization is realized already at 0.2 T of the external 5 \n magnetic field. The expected value of the magnetic moment for Mn 2V Al by Slater -Pauling \nrule, which is predicted by Galanakis et al. [14] is 2 µB/f.u. Slight deviation of the saturation \nmagnetic moment 1.82 µB/f.u. of our sample from 2 µ B/f.u. might be caused by a small \namount of the off -stoichiometric composition. Although atomic content of Mn is well \ncontrolled in the present specimen, atomic content of V is slightly increased compared to that \nof Al. \n \nB. Measurement s \n Resonant inelastic soft x -ray scattering (SX -RIXS) for V and Mn 2p core excitation was \nmeasured at room temperature at the HORNET end station installed in the long undulator \nbeam line BL07LSU of SPring -8, Japan [27,28] . Measurement was performed with an \nexternal magnetic field of 0.25 T, which was applied by a permanent magnet with two poles \nfor passing the excitation light [29,30] . The direction of the magnetic field was repeatedly \nreversed by the rotatable feed through supporting the magnet (The schematic view of the \nexperimental geometry is given in Fig. S 2 in Supplemental Material). Right and left helicity \ncircularly polarized lights (RCP and LCP) parallel to the magnetic field were incident at 45° onto the (100) plane for excitation. Light emitted at 45 ° from this surface was dispersed by a \ngrating and det ected by a two dimensional detector. The <100> axis was contained in the \nscattering plane. The direction of the magnetic field was reversed to confirm the genuine magnetic circular dichroism in this RIXS experiment. The total energy resolution was set to \n~140 (170) meV at the V (Mn) 2p\n3/2 edge. RIXS measurements were performed on a surface \nof the sample obtained by fracturing in an Ar globe box in advance and transferred to a RIXS \nchamber with a vacuum of 1×10-5 Pa without any exposure to atmosphere. \n \n \nIII. RESULTS AND DISCUSSION \nA. RIXS and RIXS -MCD experimental spectra \n Figures 1(a) and 1(b) show the x -ray absorption spectr a (XAS) of V and Mn -L3 edges, \nrespectively, by means of the total electron yield recorded at 20 K [18]. The numbers above \nthe vertical bars on the XAS indicate the incidence photon energy hv in for RIXS spectra. \nFigure s 1(c) and 1(d) show the RIXS spectra for the V and the Mn L3-edges at room \ntemperature, respectively, measured by parallel ( μ+: blue) and antiparallel ( μ-: red ) \nconfigurations between the light helicity and the direction of the magnetic field as a function \nof the energy loss given by the horizontal axis. In the figures, strong intensity peak s without \nany energy loss are always observed and they are called the elastic component. In the larger 6 \n energy loss region, the so-called fluorescence peaks are observed with their energy loss \nincreasing linearly with hv in. The relative height of the fluorescence peak is noticeably smaller \nthan that of the elastic peak in the case of V L 3-edge excitation. The difference of μ+ and μ- is \nclearly observed in Mn L 3-edge excitation. \nFigure s 2(a) and 2(b) show the intensity maps of the RIXS spectra for the V L 3-edge \nobtained by Fig. 1(c). The horizontal axis shows the energy loss from the incident photon \nenergy hvin. The spectral differences of the RIXS between the parallel and antiparallel \nconfigurations , RIXS -MCD, are shown in F ig. 2(c). The corresponding results for the Mn \nL3-edge are shown in Fig s. 2(d)- 2(f), respectively. From the comparison of the se spectra \nbetween the V and the Mn L 3-edges, several significant features are recognized . The most \ncharacteristic feature is that the fluorescence peak associated with the V L3-edge does not \nbranch off from the elastic peak . On the other hand, t here is almost no gap in the energy loss \nbetween the elastic peak and the appearance of the fluorescence peak for the Mn L3-edge . In \naddition, weak structure is observed around the constant energy l oss ∼2 eV for a wide \nexcitation region above hvin ∼515 eV for V L3-edge . This inelastic energy loss feature is \nconsidered to be associated with the d- d excitation because the existence at a constant energy \nloss cannot be due to any fluorescence feature. If we compare th e fluorescence MCD for the \nV and Mn L3-edge s in Figs. 2(c) and 2(f), it is also recognized as the negative Mn \nfluorescence MCD is spread in the photon energy range in hvin = 638 - 639 eV , while very \nweak positive V fluorescence MCD is observed around hv in = 513 eV . \nTypical RIXS spectra and its MCD at hv in = 512.5 eV for the V L3-edge and hvin = 638.6 \neV for the Mn L3-edge are reproduced in Fig s. 3(a) and 3(b), respectively , by solid lines . The \nRIXS -MCD features are observed in both cases of V and Mn L3-edge excitation s, and the sign \nof the MCD of the major fluorescence feature is found to be opposite between the cases of V \nand Mn in agreement with the ferrimagnetic character of this material, Mn 2VA l . Here, one \nnotices that a broad peak is observed in the V L 3-edge excitation, while double peak features \nsplit by 1.0 ~ 1.2 eV are observed in Fig . 3(b) in the case of the Mn L 3-edge excitation beside \nthe elastic peak. Detailed RIXS -MCD for the Mn L3 threshold excitation with changing hv in is \nshown later in Figs. 4(a1) -4(a6). These characteristic features must be closely correlated with \nthe electronic structures of V and Mn. \n \nB. Theoretical basis \nFor interpret ing the observed RIXS and MCD spectra, theoretical calculations were \nperformed by means of the density functional theory ( DFT ). Based on the DFT, spin resolved \npartial density of states (PDOS) of the V , Mn and Al are calculated as shown in Fig. 3(c) 7 \n together with the total density of states ( DOSs). The e g and t2g derived components are \nseparately shown in Figs. 3(d) and 3(e) for V and Mn, respectively. First of all, the PDOS of \nAl is almost negligible near EF. In the case of the V 3 d states, the PDOS are found to be very \nsmall in the region of E F ± 0.6 eV . High PDOS of V of the t 2g occupied states are around - 1.5 \neV and those of the unoccupied t2g and eg states are located at around +1.6 eV . In the case of \nthe Mn 3d states, however, rather complex electronic structures strongly dependent on the \nspin are predicted. Although the occupied valence bands below - 0.4 eV are composed of both \nt2g and eg states with both spin up and down states, clear differences are recognized above this \nenergy. Namely, spin down t 2g states are crossing E F with high PDOS and the spin up t 2g states \nhave negligible PDOS between - 0.5 eV and +0.7 eV . In the case of the Mn e g states, on the \nother hand, up spin states have negligible PDOS between -0.4 and +0.4 eV , though down spin \nstates have a certain PDOS between -0.4 and +1 eV before showing high PDOS around +1.4 \neV . Thus half metallic PDOS behavior so far predicted is re confirmed in Fig. 3(e). \nAs already pointed out in Figs. 2 (a) and 2(b), the fluorescence component of RIXS in V \nL3-edge does not branch off from the elastic peak. The absence of any additional fluorescence \npeak between the elastic peak and the peak around ∼2 eV for the L 3 threshold excitation is \nconsistent with the weakness or negligible PDOSs of the V 3d states around EF for both spin \nup and spin down V 3d states. The energy splitting between the V unoccupied eg+t2g states \nPDOS and the occupied t 2g PDOS ran ges from 2 to 4 eV . Although this predicted d- d splitting \nenergy is slightly larger than the experimental d- d splitting energy roughly estimated as 2 -3 \neV in Fig. 3 (d), the constant loss energy feature in Figs. 2(a) and 2(b), is unambiguously \nascribed to the genuine RIXS feature due to the d- d excitation. On the other hand, RIXS \nfeature for the Mn L3-edge excitation branches off really from the elastic peak and moves \nlinearly with hv in revealing its fluorescence origin, sugges ting the finite PDOS at E F of the Mn \n3d states. In the PDOS for the Mn shown in the Fig. 3(e), finite PDOS exists around the E F in \nthe down spin state, supporting the experimental results. \nThe RIXS -fluorescence spectra were simulated based on the Kramers -Heisenberg \nformula as shown in equation (8) in the Supplementa l Material [31], where details of the \ncalculations are explained. Since the correlation between the fluorescence RIXS -MCD and \nthe band structures must be discussed, more detailed calculation with taking the \nmagnetization into account must be performed. Under the external magnetic field of 0.25 T, \nmagnetization is almost saturated with the magnetic moment along the magneti c field [11,18] . \nSimulated RIXS -fluores cence spectra and the MCD based on the DFT for V and Mn L 3-edges \nare shown by dotted lines in the Fig s. 3(a) and 3(b), respectively, in addition to the \nexperimental spectra. It is seen that the simulated RIXS -MCD qualitatively reproduces the 8 \n experimental feature, for example, such as the double peak feature of the fluorescence in the \nMn L3-edge. If we compare the spectra of theoretically predicted fluorescence and its MCD \nwith corresponding experimental results , one notices that the predicted spectra are more \nwidely energy spread than the experimental results. This may be due to the larger electron \ncorrelation energy between the Mn 3d electrons beyond that employed in the ordinary DFT \ncalculation. \n \nC. Detection of the spin polarized Mn 3d electronic states \nIn order to further discuss the Mn 3d states around EF, RIXS and the MCD were \nsystematically investigated in the Mn L3-edge threshold excitation region. Figures 4(a1)-4(a6) \nshow the experimental RIXS spectra for μ+ and μ- configurations and their MCD with \nchanging the hvin (638.2 ≤ hvin ≤ 639.2 eV). Figure 5 shows schematic views of the Mn L3 \nRIXS-MCD processes by considering the P DOS for the half -metallic magnetic system. The \ndifference of the photon energy hvin from the energy E2p0 between the E F and the m j = -3/2 Mn \n2p core state is defined hereafter by hv1. PDOS of the up (down) spin state is schematically \nshown on the left (right) hand side of the energy axis (vertical axis) in each figure. The \nZeeman splitting of the Mn 2p core level states of around 0.5 eV due to the effective magnetic \nfield induced by the spin- polarized 3d states is taken into account in the present simulation. \nThe excitation to the empty conduction band states is shown in Fig. 5 by the upward arrows \nwith the filled circles with assuming the gap of 0.3 eV from the E F to the bottom of the DOS \nin the majority up spin state of the Mn 3d states. \nFive energy ranges of hν 1 = hνin- E2p0 can be considered for the excitation as 1) below \n0.16 eV, 2) 0.16 - 0.46 eV, 3) 0.46 - 0.63 eV, 4) 0.63 - 0.8 eV and 5) above 0.8 eV . Since the \nexperimental results were obtained at 300 K, one should note that the energy broadening of \naround 0.1 eV cannot be neglected in the comparison between the experimental results and \ntheoretical prediction. In both the 2p core excitation and the 3 d-2p fluorescence decay \nprocesses , the spin is conserved in the dipole transition. However, the hole spin c an be relaxed \nbefore the fluo rescence decay in the core hole with mj = -1/2 and 1/2 states which are \ncomposed of both spin up and down states due to the spin- orbit coupling. So even when the m j \n= -1/2 state with spin down state is excited to the empty conduction band, the core hole spin \ncan be partially relaxed to the spin up state before the fluorescence takes place. This means \nthat one should take into account the fluorescence in both spin down and up channels w ith the \nfixed relative weight given in the initial core hole states when the fluorescence decay to the \ncore holes with m j = -1/2 or 1/2 states are calculated . In Fig. 5, t he width of the arrows \ncorresponds to the transition probability. 9 \n The fluorescence spectra for the Mn L 3-edge excitation are calculated as shown in Fig s. \n4(b1)-4(b6). In the experimental MCD spectra in Fig s. 4(a1)-4(a6), the intensity increases \nwith increasing hv in showing the double peak future, which becomes less clear above hvin = \n638.8 eV. Such a tendency is qualitatively reproduced by the calculated spectra. Figure s \n4(c1)-4(c6) show the predicted MCD spectra of fluorescence to the Mn 2p m j = -3/2, -1/2, 1/2 \nand 3/2 states. It is clear that the fluorescence decay to the m j = -3/2 state is dominating a t the \nlow hv threshold. Since the down spin t 2g states have substantial PDOS of the unoccupied \nstates from EF to +0.8 eV, and the down spin eg states have high PDOS around +1.4 eV, t he \npure down spin m j = -3/2 states can be continuously excited at least up to hv 1 ∼ 0.8 eV in \naddition to the region in +1.2 - 1.6 eV inducing the remarkable negative fluorescence MCD in \nFigs. 4(c1)-4(c6). Since the PDOS of the occupied Mn 3d states in the region from EF down to \n-0.4 eV for the down spin state is rather high, noticeable fluorescence and its clear MCD is \npredicted just from the energy loss of 0 eV as clearly seen in Figs. 4(b1) and 4(c1) , 4(b2) and \n4(c2). The fluorescence feature with an energy loss peak near 0.3 eV in Figs. 4(b1) and 4 (b2) \nis thought to reflect the high PDOS of the down spin occupied t2g states in the above \nmentioned region. \nWith increasing hv1 up to ∼0.46 eV, excitation from the mj = -1/2 state becomes gradually \nfeasible in addition to the excitation from the mj = -3/2 state , providing negative but small \nMCD of the fluorescence to the m j = -1/2 state as shown in Figs. 4(c2) and 4(c3) . The origin \nof the doublet feature of the fluorescence and its MCD separated by ∼1 eV (Figs. 4(a2), 4(a3) \nand 4(a4)) is relatively well predicted by the calculation as shown in Figs. 4(b1)-4(b4) and \n4(c1) -4(c4). Both of the doublet features must be due to the f luorescence transition into the \ndown spin mj = -3/2 core hole state. Judging from the sum of the Mn t 2g and eg down spin \nstates shown in Fig. 3( e), which is equal to the Mn down spin PDOS in Fig. 3(c), two peaks of \nthe occupied down spin Mn 3d PDOS are recognizable at ∼0.8 and ∼1.8 eV. These two peaks \nare surely contributing to the doublet fluorescence features of Fig. 3(b) as well as Figs. \n4(a2)-4(a4). \nFor 0.46 eV < hv1 < 0.63 eV, both down and up spin states are excited from the mj = -1/2 \ncore states . For 0.63 eV < hv1 < 0.8 eV, the similar situation takes place for the mj = +1/2 core \nstates. In each case, both spin down and up states are excited and fluorescence takes place \nafter the relaxation of the core hole spin. With the increase in the hv 1, the relative weight of \nthe transition to the m j = -1/2, +1/2 and +3/2 states increase gradually with the increase of the \nmagnitude of the individual fluorescence MCD. As a result, the total magnitude of the \npredicted fluorescence MCD decreases relatively as seen in the series of Fig s. 4(b1) to 4(b6) \nin consistence with the experimental results in Fig. 4(a3) to 4(a6). 10 \n Various investigation s to directly check the half-metallic electronic structure have been \nperformed up to now . Ultraviolet- photoemission spectroscopy (UPS) and hard x -ray angle \nresolved photoelectron spectroscopy (HAXARPES ) [32-36] are such examples . In the case of \nUPS, it is known that the surface electronic structure accessible by UPS is noticeably different \nfrom that in the bulk. The HAXARPES is much more bulk sensitive . However, its orders of \nmagnitude reduced photoionization cross sections for the valence band electronic states \ncompared with the UPS strongly hinder the usefulness of HAXARPES studies. Moreover the \npossible recoil shift effects for the valence electron states spoil reliable discussions on the \nmost important electronic structures near the E F in Mn 2VA l [32,35,36] . Although \nspin- polarized and angle -resolved photoelectron spectroscopy (SP -ARPES) will be desired to \ndetect the half -metallic ity, its low detection efficiency (orders of 10-4 or less) makes it almost \nimpossible to make reliable experiment. Therefore, the electronic structure, especially \nhalf-metallicity of the bulk Heusler alloys has not yet been fully clarified. In the present study, \ndetailed hvin dependence of RIXS -MCD was clearly observed, providing the spin- dependent \ninformation. The qualitative agreement obtained in the present experiments and theoretical \nanalyses confirmed the half -metallicity of Mn 2V Al, demonstrating that the RIXS and \nRIXS -MCD are extremely powerful for the study of the electronic structures of the \nhalf-metallic ferromagnetic or ferrimagnetic materials. \n \n \nIV. CONCLUSION \nResonant inelastic x -ray scattering ( RIXS) measurement in magnetic field was performed \non the single crystal half -metallic Heusler alloy, Mn 2V Al, for th e first time in order to obtain \nreliable information on the bulk electronic state of the 3 d electrons. The d -d excitation due to \nthe t2g-eg splitting is clearly observed for V in Mn 2V Al under the L 3-edge excitation. The loss \nenergy of the V d-d RIXS maximum is found to be about 2 eV , being comparable to the \nsplitting energy between the theoretically predicted e g and t2g states. The delayed branching \noff in the V 3 d-2p fluorescenc e peak from the elastic peak demonstrat es the nearly absen t V \n3d PDOS around EF. The clear appearance of the t 2g-eg RIXS of V reflect the rather localized \ncharacter of the V 3 d states . The RIXS -MCD of the fluorescence peaks of Mn 3d- 2p transition \nunder the L 3-edge excitation shows negative sign with clear hvin dependence . The sign and the \nshape of the RIXS -MCD are qualitatively reproduced in consistence with the DFT \ncalculations and confirmed the absence of the up spin Mn 3d PDOS at the E F, demonstrating \nthe half -metallicity of Mn 2V Al Heusler alloy. Thus the bulk sensitive RIXS studies under \nexternal magnetic field are confirmed to be essential to study the detailed electronic structures 11 \n of various Heusler alloys and family materials. \n \n \nACKNOWLEDGMENTS \nWe sincerely thank K. Yubuta, T. Sugawara and Y . Murakami for their helps to make single \ncrystals and EPMA experiments, and Prof. T. Kanomata for valuable discussions . This \nresearch was supported by 1) Precursory Research for Embryonic Science and Technology \n(PRESTO), Japan , 2)Science and Technology Agency (JST), Japan and 3)Grant -in-Aid for \nChallenging Exploratory Research from the Japan Society for the Promotion of Science \n(JSPS) . This w ork was carried out under the approval of BL07LSU at SPring -8 Synchrotron \nRadiation Research Organization and the Institute for Solid State Physics, T he University of \nTokyo (Proposal No. 2016BG05, 2016B7512). Partial fundamental measurements were carried o ut at the Center for Low Temperature Science, Institute for Materials Research, \nTohoku University. \n \n 12 \n References \n[1] R. A. de Groot , F. M. Mueller , P. G. van Engen , and K. H. J. Buschow , Phys. Rev. Lett. \n50, 2024 (1983). \n[2] J. J. Kübler, A. R. Williams , and C. B. Sommers, Phys. Rev. B 28, 1745 (1983). \n[3] S. Ishida, S. Akazawa, Y. Kubo, and J. Ishida , J. Phys. 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In (c) and \n(d), the RIXS spectra were measured by parallel (μ+) and antiparallel ( μ-) configurations \nbetween the light helicity and the direction of the magnetic field. The numbers above the \nvertical bars on the XAS indicate the excitation photon energy hv in for the RIX S spectra. \n \n \n \n16 \n \n \n \n \n \nFIG. 2 Intensity maps of hv in-dependent RIXS and MCD at V and Mn L 3-edges in Mn 2VAl. \n(a) and (b) are the intensity maps of the RIXS of V as a function of hv in obtained at room \ntemperature in a magnetic field of 0.25 T for μ+ and μ- configurations . RIXS MCD is given by \nμ+ - μ- in (c). Figures (d), (e) and (f) are the corresponding results for the Mn L3-edge. \n \n \n \n17 \n \n \n \n \n \nFIG. 3 Representative RIXS spectra for V and Mn L 3-edges, and theoretically predicted \nPDOSs of Mn, V and Al in Mn 2VA l. RIXS spectra recorded for μ+ and μ- configurations at \nincoming photon energy, hvin of 512.5 eV (a) and 638.6 eV (b) at the V and Mn L 3-edges, \nrespectively, together with the MCD. Figures ( a) and ( b) include simulated spectra based on \nthe DFT for V and Mn L3-edge, respectively. Figure ( c) shows the theoretical prediction of \nthe spin dependent total DOS as well as PDOSs of Mn, V and Al. Figures ( d) and ( e) \ncorrespond to the eg and t 2g components of the partial DOSs of V and Mn, respectively . \n \n18 \n \n \nFIG. 4 RIXS and RIXS -MCD spectra for Mn L 3-edges. Figures (a1)-(a6) show the experimental RIXS results for the \nμ+ and μ- configurations and their MCD spectra for the Mn L3-edge as a function of the energy loss. Theoretically \ncalculated RIXS and RIXS -MCD spectra based on DFT calculation are in (b1) -(b6). The definition of h ν1 is given in \nthe text. Figures (c1) -(c6) show m j resolved RIXS -MCD spectra for Mn L3-edge . \n \n19 \n \n \n \n \nFIG. 5 Schematic views of the RIXS -MCD processes in the case of half -metallic density of \nstates. Excitation from the Zeeman split 2 p mj core states is considered in the present \nsimulation as indicated in the table and t he Zeeman splitting is assumed to be around 0.5 eV \ndue to the effective magnetic field. The difference of the photon energy ( hvin) from the energy \nE2p0 between the Fermi energy (EF) and the Mn 2 p mj = -3/2 state is given here by hv1. PDOS \nof the up (down) spins state is schematically shown on the left (right) hand side of the energy \naxis (vertical axis) in each figure. A gap of 0.3 eV from the E F to the bottom of the DOS in \nthe majority up spin state is also assumed to exist. T he spin is thought to be fully down and up \nin the m j = -3/2 and 3/2 states , respectively . On the other hand, the spin at m j = -1/2 and 1/2 \nstates is composed of both spin up and down states due to the spin- orbit coupling . The width \nof the arrows corresponds to the transition probability . \n \n \n \n20 \n SUPPLEMENTAL MATERIAL \nSA. C RYSTAL STRUCTURE OF Mn 2VAl HEUS LER ALLOY \n \nFIG. S1 . Crystal structure of Mn 2VAl Heusler alloy with L21-type [1]. If the Mn 2V Al orders \ncompletely, Mn atoms (purple) occupy the Wyckoff position 8c , V atoms (red) 4b, and Al \natoms (grey) 4a with the space group Fm3m. \n \nSB. EXPERIMENTAL GEOMETRY \n \n \n \nFIG. S2. Schematic view of the scattering geometry in SX -RIXS measurements. The single \ncrystal of Mn 2V Al specimen was tilted 45º from the direction of the incident x -ray. Magnitude \nof the magnetic field in the central region of the yoke -type magnet is about 0.25 T, which is \nenough to saturate the magnetization for Mn 2VAl. The right and left helicity circularly \npolarized incident photon is used for RIXS e xcitation and the magnetic circular dichroism \n(MCD) is defined by the difference of the RIXS spectra for the μ+ and μ- configurations. \n \n21 \n SC. S IMULATION BASED ON THE DENSITY FUNCTIONAL THEORY \n \nThe electronic structure calculation based on DFT has been performed using the HiLAPW \ncode, which is based on the all -electron full -potential augmented plane -wave (FLAPW) \nmethod [2]. The generalized gradient approximation (GGA) using the \nPerdew -Burke -Ernzerhof scheme has been used for the exchange correlation potential [3,4]. \nThe relativistic effects are considered for the 2 p core states including the spin -orbit coupling. \nOn the other hand, the spin- orbit coupling for the 3d states are negligible compared with the \n2p states, since the orbital magnetic moment of Mn and V were estimated as 0.026 μ B/Mn and \n0.037 μB/V by XMCD measurements, respectively [5]. Plane -wave expansion cutoffs were set \nto 20 Ry for the wave functions and 80 Ry for the charge density and potential functions. The muffin -tin sphere radius was chosen as 1.1 Å for all elements. For the Brillouin- zone \nintegration, a 16×16×16 uniform mesh was used with the tetrahedron integration technique. \nThe atoms were placed on the general form X\n2YZ of the L 21 structure with X at 8c site in the \nWyckoff position, Y at 4b site, and Z at 4a site. The lattice constant was set to 5.875 Å [6]. \nThe RIXS spectra were simulated by using the Kramers -Heisenberg formula [7,8] \ndescribed as \n \n \nwhere a, b, and i denote the initial, final, and intermediate states having the energy of E a, Eb \nand Ei, respectively. The incoming and outgoing photon energies are described as hv in and \nhvout, respectively. Lifetime broadening of a core hole is given by Γi of 0.36 (0.28) eV for Mn \n(V) [9], and the dipole transition operator is described as )(μFˆ for the photon helicity ( μ) of \nthe circularly polarized light. The electron configuration of the initial, intermediate and final \nstates with | a ⟩ = |2p6v⟩, |i ⟩ = |2p5ve⟩, and | b ⟩ = |2p6v -1e′⟩ are taken into account with the \nrelative energy of E a = 0, Ei = εe - E2p, and E b = eε′-1-vε[10]. Here v and e represent the \nvalence electrons and the electron in the empty conduction band, and v-1 and 1-vε stand for \nthe valence electron state with one hole induced by the decay into the 2 p core hole state and \nthe energy of valence electrons with one hole . Then the denominator of equation (1) can be \nexpressed by considering these energies and t he content of the δ function as \n \n \nIf we assume ε e =eε′, where no change is considered for the electron excited into the empty (1) \n \n \n22 \n conduction band on the fluorescence decay , the denominator of equation (1) is further \napproximated as \n \n \nOn the same assumption the transition matrix elements are transformed as \n \nThe summation for the intermediate states | i ⟩ can be expressed by the two integrals as follows \nand the RIXS intensity can be described as, \n \nIf we assume that the square of the matrix element in the numerator is proportional to the \npartial density of states of the occupied (unoccupied) states denoted as Docc (Dunocc) at the \nenergy of 1-vε (εe) of occupied (unoccupied) valence states multiplied by the transition \nprobability )(μ\nsmjjmw between the 2 p and 3d states depending on the helicity ( μ) [10,11,12] , we \nobtain \n \nHere j, mj, ms and those with’ stand for the total angular momentum of the 2p core states , its z \ncomponent and its spin, while m, and m s as well as those with’ denote the magnetic quantum \nnumber and the spin of the 3d states under consideration. Here m = mj - ms is assumed for the \n2p states. In the present calculation, the relativistic effects are considered for the 2 pj=3/2 core \nstates including the spin -orbit interaction for the states with m j = ±3/2 and ±1/2, while the 3 d \nspin is well defined by ms. Furthermore, we take into account the Zeeman splitting of the 2 p \nstates due to the effective magnetic field of the 3 d states. The weight coefficient )(μ\nsmjjmw is \ngiven by multiplication of the Clebsch -Goldan coeffcient and the Gaunt coefficient . \nFinally we obtained the RIXS intensity as \n (3) \n(4) \n(5) \n(6) \n(7) (2) \n23 \n \n \nIn the simulation, we practically give the energy of unoccupied valence states ε e using the \nenergy offset ( hv1) from Fermi energy (EF) as εe = EF + hv1. Then, we further transform the \nequation ( 8) as \n \nIn this model, the magnitude of the RIXS -MCD is proportional to the Dunocc or the quantity of \nthe core holes and the line shape reflects the energy distribution of the Docc. Moreover, the \nspin polarization of the 2 p core hole plays an essential role to the spin selective transition with \ndipole selection rule. Thus, the RIXS -MCD is clearly observed at the pre -edge of the L3 XAS \nsince the excitation from the spin -polarized mj = -3/2 states is dominant. In the case of the m j \n= ±1/2, however, the spins of the core hole are mixed, and the core hole spins can be relaxed \nbetween spin up and down before the fluorescence takes place. This allows the optical path in \nwhich the spins in the absorption process and emission process are diff erent. The fluorescence \nspectra thus obtained are shown in Figs. 5(b) and 5(c). Note that the transition probability for \nthe circular polarized photon was taken into account in the absorption process only, but it was \naveraged in the emission process since the polarization of the outgoing photon was not measured in the experiment. \n \n \n (8). \n(9)\n \n24 \n References \n[1] K. Momma and F. Izumi, J. Appl. Crystallogr. 95, 1272 (2011). \n[2] E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Phys. Rev. B 24, 864 (1981). \n[3] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). \n[4] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78, 1396 (1997). \n[5] K. Nagai, H. Fujiwara, H. Aratani, S. Fujioka, H. Yomosa, Y. Nakatani, T. Kiss, A. \nSekiyama, F. Kuroda, H. Fujii, T. Oguchi, A. Tanaka, J. Miyawaki, Y. Harada, Y. \nTakeda, Y. Saitoh, S. Suga , and R.Y. Umetsu, Phys. Rev. B 97, 035143 (2018). \n[6] R. Weht and W. E. Pickett, Phys. Rev. B 60, 13006 (1999). \n[7] F. de Groot and A. Kotani, CRC Press, Boca Raton, FL, ( 2008). \n[8] L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, \nRev. Mod. Phys . 83, 705 (2011). \n[9] J. L. Campbell and T. Papp, Atomic Data and Nuclear Data Tables 77, 1 (2001). \n[10] J. Jiménez -Mier , J. van Ek, D. L. Ederer, T. A. Callcott, J. J. Jia, J. Carlisle, L. \nTerminello, A. Asfaw, and R. C. Perera, Phys. Rev. B 59, 2649 (1999). \n[11] K. O. Kvashnina, Y. O. Kvashnin, J. R. Vegelius, A. Bosak, P. M. Martin , and S. M. \nButorin, Anal. Chem. 87, 8772 (2015). \n[12] G. van der Laan and A. I. Figueroa, Coord. Chem. Rev. 95, 277 (2014). \n \n " }, { "title": "2308.16806v1.Tunable_magnetic_domains_in_ferrimagnetic_MnSb__2_Te__4_.pdf", "content": "Tunable magnetic domains in ferrimagnetic MnSb 2Te4\nTatiana A. Webb,1Afrin N. Tamanna,2Xiaxin Ding,2Jikai Xu,1Lia\nKrusin-Elbaum,2,∗Cory R. Dean,1,†Dmitri N. Basov,1,‡and Abhay N. Pasupathy1, 3,§\n1Department of Physics, Columbia University, New York, NY 10027, USA\n2Department of Physics, The City College of New York, New York, NY 10027, USA\n3Condensed Matter Physics and Materials Science Division,\nBrookhaven National Laboratory, Upton, New York 11973, USA\nHighly tunable properties make Mn(Bi,Sb) 2Te4a rich playground for exploring the interplay\nbetween band topology and magnetism: On one end, MnBi 2Te4is an antiferromagnetic topological\ninsulator, while the magnetic structure of MnSb 2Te4(MST) can be tuned between antiferromagnetic\nand ferrimagnetic. Motivated to control electronic properties through real-space magnetic textures,\nwe use magnetic force microscopy (MFM) to image the domains of ferrimagnetic MST. We find\nthat magnetic field tunes between stripe and bubble domain morphologies, raising the possibility of\ntopological spin textures. Moreover, we combine in situ transport with domain manipulation and\nimaging to both write MST device properties and directly measure the scaling of the Hall response\nwith domain area. This work demonstrates measurement of the local anomalous Hall response using\nMFM, and opens the door to reconfigurable domain-based devices in the M(B,S)T family.\nThe recent discovery of MnBi 2Te4(MBT) [1–5] was\na breakthrough to realize the quantum anomalous Hall\neffect in a stoichiometric crystal, avoiding the need for\ndisorder-inducing magnetic dopants [6–9]. In addition,\ncrystals are exfoliatable down to few layer thicknesses\nenabling integration into Van der Waals heterostruc-\ntures with well developed fabrication techniques [1, 10–\n12]. This discovery was rapidly followed by work ex-\ntending MBT into a family of materials with highly\ntunable properties via crystallographic and chemical\nparadigms [4, 5, 13–21]. Substituting Sb for Bi changes\nthe doping from n-type to p-type [16, 22]. But surpris-\ningly, within MST, the magnetic order can also be tuned\n(via the concentration of magnetic defects) from A-type\nantiferromagnetic seen in MBT, where planes of Mn mo-\nments are aligned ferromagnetically (antiferromagneti-\ncally) within the plane (between planes), to ferrimag-\nnetic with net out of plane magnetization [19, 21, 23–25].\nThe ability to tune the effective inter-plane coupling from\nantiferromagnetic to ferromagnetic strongly suggests the\npresence of magnetic frustration in M(B,S)T, raising the\npossibility of stabilizing other interesting magnetic or-\nders [26, 27].\nThe ability to tune magnetic order in the M(B,S)T\nfamily opens more conventional applications of magnetic\nmaterials, where intense efforts have gone into develop-\ning materials structures with interdependent magnetic\nand electronic properties for control of charge and spin\ntransport (e.g. magnetic data storage and spintron-\nics). Growing evidence suggests that the low energy\nbands of MST are sensitive to the details of magnetic\norder [3, 19, 24, 28–30], but we do not yet have a de-\ntailed understanding of the correlation between electronic\n∗krusin@sci.ccny.cuny.edu\n†cd2478@columbia.edu\n‡db3056@columbia.edu\n§apn2108@columbia.eduproperties and real-space magnetic textures in MST. So\nfar, the use of magnetism to control electronic prop-\nerties in magnetic topological materials has been ex-\nplored primarily in terms of topological phase transi-\ntions (e.g. [10, 12, 28, 31]) and manipulating chiral edge\nmodes of the quantum anomalous Hall effect [11, 32, 33].\nEdge mode manipulation has been demonstrated via\nmagnetic domains in Cr-doped (Bi,Sb) 2Te3[32, 33] and\nvia layer-dependent magnetization in antiferromagnetic\nMBT [11], but to the best of our knowledge, the abil-\nity to write arbitrary-shaped domains in ferrimagnetic\nM(B,S)T compounds has yet to be investigated. In this\nwork, we therefore set out to investigate what magnetic\ntextures can be realized in MST, and the prospects for\nmanipulating the local magnetization to create config-\nurable devices. Specifically, we use the domain imag-\ning and writing capabilities of magnetic force microscopy\n(MFM) combined with in situ transport to directly mea-\nsure the device response to local changes in magnetiza-\ntion.\nOur interdependent transport and magnetic measure-\nments were performed on an exfoliated flake of ferrimag-\nnetic MST (Figure 1a) with average thickness 84 nm and\n±10 nm variations (Supporting Information SII). Four\ngold contacts were used to measure longitudinal Rxxand\nHall Rxyresistance. Immediately after device fabrica-\ntion, Rxxshowed a peak at 27 K on cooling, consistent\nwith typical Curie temperatures seen in MST. At 2K,\nthe hysteretic loop in Rxyand peaks in Rxxas a func-\ntion of magnetic field showed a coercive field near 10 mT\n(Supporting Information SIV). Refer to Supporting In-\nformation SI for additional details of sample fabrication\nand characterization.\nTo characterize the magnetic domains in MST, we per-\nformed MFM in a cryogenic atomic force microscope\n(AFM) with variable magnetic field Bextnormal to the\nsample surface. Figure 1c-i shows a MFM image of the\nzero-field cooled (ZFC) sample at 5 K. Because the co-\nercive field of MST is so low, we quenched the supercon-arXiv:2308.16806v1 [cond-mat.str-el] 31 Aug 20232\n12 3\n4\n10 μm(a)\n(e)\n(b)\n50\n25\n025 50\nB (mT)1\n01Rxy( )\nbefore MFM\nafter MFMiiiiiiiv\nvivΔf\n1.5 Hz\n-20 mT (v)\n(iv) 50 mT\n(vi) -30 mT\n(c) ZFC\n(ii) 12 mT\n (i)\n2 μmBext = 0 mT\n(iii) 24 mT\nxyxy(d)\n0 mT -20 mT FT\nkxky\n0 20\n|k| ( m1)\n0.00.51.01.52.0Normalized FT amplitude\n0mT ZFC\n-20.0mT~0mT FC\nFIG. 1. Evolution of stripe domains under Bext:(a)Optical micrograph of the MST device showing the MST flake with\n4 contacts for transport measurements in a Van der Pauw geometry. The arrows show the scan axes for the MFM images. (b)\nMagnetic field Bextdependence of the Hall resistance Rxymeasured at 5 K starting from a zero-field cool. Measurements were\nrecorded before (open symbols) and after (filled symbols) MFM imaging. The light orange lines are guides to the eye showing\nthe order of data acquisition. (c)Constant height MFM measurements of the magnetic domains at the center of the MST\ndevice recorded at 5 K after zero-field cooling. Measurements were interspersed with RxyandRxxmeasurements shown in (b)\nand in the Supporting Information SIV. The arrows indicate the order of data acquisition. The color scale on all images is\n1.5 Hz, but the zero values have been offset. The tip was lifted 300 nm above the SiO 2surface. (d)Amplitude of the Fourier\ntransforms of (c-i) and (c-v) after mean value subtraction. (e)Angular averaged amplitude of the Fourier transformed MFM\ndata.\nducting magnet prior to cooling the sample to ensure a\ntrue ZFC with no trapped flux. The MFM image is a\nmeasurement of ∆ f, the resonance frequency shift of the\nAFM cantilever due to the interaction of the sample’s\nstray fields with the cantilever’s magnetic tip, so we ex-\npect images to primarily detect the domain structure of\nthe ferromagnetically aligned components of MST’s fer-\nrimagnetic ordering [34, 35]. Correspondingly, the ZFC\nimage shows disordered maze-like stripe domains (Fig-\nure 1) consistent with ferromagnetic ordering in the out\nof plane direction, similar to domain images from Ge et\nal [25]. Applying magnetic field Bextnormal to the sam-\nple surface polarizes the sample (c-i to iv), increasing the\narea of the domains aligned with the field until at 50 mT,\nonly a single domain remains, giving a uniform MFM sig-\nnal. In situ transport measurements show an associated\nincrease in Rxyfrom 0.04 to 1.42 Ω. Over this range of\nBext, the contribution to Rxyfrom the linear Hall effect\nis negligible, so the change in Rxyis primarily due to the\nanomalous Hall Effect (AHE) [36] (Supporting Informa-\ntion SVII).\nReversing the magnetic field (Figure 1c-iv to vi), we\nobserve the reformation of stripe domains at -20 mT\nasRxydrops and changes sign, indicating the reversal\nof the magnetization. These field-reversed domains are\nsignificantly less disordered than the ZFC domains. To\nquantify the difference, we examine the Fourier Trans-\nforms (FT) of the ZFC and -20 mT images, shown in\nFigure 1d. Both exhibit a ring shape, or a peak in the\nangular-averaged FT (Figure 1e), indicating the domainshave a characteristic length scale, as expected from the\nenergetics of domain formation [37–39]. The peak occurs\nat wavevector |k|6.3µm−1with standard deviation σ\n4.8µm−1for the ZFC domains and |k|= 5.7 µm−1with\nσ= 2.1 µm−1for the -20 mT domains. The broader peak\nassociated with the ZFC domains indicates that during\ncooling the domains form features with a wider range of\nlength scales compared to during magnetization reversal\nat low temperature.\nTo further explore how an external field can tune the\ndomain morphology, we cooled the MST device from 35 K\nbelow Tcunder |Bext|up to 15 mT (Figure 2). With\n|Bext|larger than 10 mT, a single domain forms across\nthe entire MST flake. However, when we nominally zero\nthe magnet’s current such that a small Bext≈0 exists\nonly from trapped flux, we see circular features in the\nMFM, indicating the formation of bubble rather than\nstripe domains. MST is thus remarkably sensitive to\nsmall magnetic fields. At intermediate |Bext|(10 mT),\nthe domains formed are not uniform in size and shape,\nand it is not clear if they are intrinsically bubbles or\nstripes. We now focus on Bext≈0, where bubbles are\nclearly observed.\nThe bubble domains are highly disordered (Figure 2e).\nThe nearly isotropic Fourier transform (f) shows no\nevidence for lattice organization, and the distribution\nof wavevectors centered at |k|= 11.3 µm−1with\nσ= 7µm−1is extremely broad. Correspondingly, the\ncircular MFM features range in size from below 100 nm\nto above 300 nm, as shown by the distribution of full3\n150 300450\nd (nm )1502503504502Bz/z2FWHM (a.u.)\n5 µm(a) Bext ~ 0 -10 mT -12 mT -15 mT (d) (c) (b)\nSiO2MSTΔf\nkxky (f)\n(e)\n2 µm\n(g) (h) (i)\n(j)(k)(l)\nx (nm )250\n0 2501\n01f(Hz)\n0 100 200 300 400\nMFM FWHM (nm )0102030Count\ndz\nt\n332 nmFWHM\n97 nm197 nmxy\nFIG. 2. Field cooled domain structures. (a-d) Constant height MFM images of the magnetic domains in the MST device\nunder field cooling with the indicated Bext. The tip was lifted 300 nm above the SiO 2surface. The range and offset of the\ncolor scale has been chosen for each image independently. Color scale range: 3.7 Hz (a), 1.5 Hz (b), 1.4 Hz (c), 1.5 Hz (d).\nTemperature: 5 K (a), 10 K (b-d). (a)Bext∼0 indicates that a small unknown residual flux from the superconducting magnet\nwas present. (e)Smaller scale constant height MFM image after cooling under Bext∼0 showing clear bubble shaped domains.\nColor scale range: 4.4 Hz. Temperature: 5 K. The images in (a) and (e) are from separate cooling cycles with the same Bext.\n(f)Amplitude of the Fourier transform of (e) after mean value subtraction. (g-i) Zooms of 3 regions in (e) showing a large,\nmedium, and small size bubble. (j)Profiles through the large, medium, and small size bubbles from (g-i). (k)Histogram of the\nfull-width-at-half-max (FWHM) of the bubbles imaged via MFM, determined from horizontal and vertical profiles through all\nresolvable bubbles in (e). The MFM FWHM is not directly interpretable as the bubble domain size. (l)FWHM of ∂2Bz/∂z2\nfor the stray magnetic field generated by a bubble domain with diameter d, using sample thickness t= 81.2 nm and z= 300 nm\nmeasured from the bottom of the material.\nwidth at half maxima (FWHM, Figure 2k). The size of\nthe features seen in MFM cannot directly be interpreted\nas the size of the domains in the MST. Approximating\nthe tip as a point dipole with small oscillation amplitude,\nthe MFM image ∆ fis proportional to ∂2Bz/∂z2arising\nfrom the sample’s stray field [34, 35]. To help interpret\nthe MFM features, we model the stray field for a single\ncylindrical bubble domain at a representative height z.\nAs the domain diameter decreases below z, the spatial\npeak in ∂2Bz/∂z2decreases in intensity ( Supporting In-\nformation SV) and the FWHM saturates at a lower limit\nnear 150 nm (Figure 2l). The FWHM does not decrease\nlinearly in the domain diameter for small bubbles. Re-\nturning to the MFM data, we therefore expect small bub-\nbles may not be detectable due to weak intensity, and for\nslightly larger bubbles, the apparent size in MFM may\nsaturate at a lower limit larger than the domain diame-\nter. The MFM data, however shows bubbles with FWHM\nbelow the expected 150 nm cuttoff, likely because the\nwidth of the low intensity bubbles can be dominated by\nthe positions of the neighboring bubbles. Under repeat\ncooling, we find that some but not all bubbles form in\nthe same location (Supporting Information SIX), whichalong with their disordered organization could indicate\nsignificant pinning, either due to crystal inhomogeneity,\nor extrinsic factors such as strain. The observation of\nbubble domains under field cooling, but not when sweep-\ningBextat low temperature suggests that the bubble and\nstripe morphologies are separated by a significant energy\nbarrier, likely associated with nucleating a domain wall.\nThe domain wall structure (i.e. Bloch or N´ eel) can pro-\nduce topologically non-trivial chiral spin textures on bub-\nble domains [40], and topological bubble and skyrmion\nphases have been reported in other (Bi,Sb) 2Te3-based\nmaterials [41, 42]. While our detection of bubble do-\nmains opens the possibility of stabilizing topological spin\ntextures in M(B,S)T, our MFM measurements do not al-\nlow us to draw a conclusion about the topology of the\nbubbles. We observed no evidence of a topological Hall\neffect (THE) – a deflection of carriers due to the real\nspace Berry curvature of topological spin textures – in\ntheRxyhysteresis loops when sweeping Bextto flip the\nsample magnetization at low temperature. However, un-\nlike many skyrmion materials that display a THE as the\nskyrmion phase is formed over a finite range of Bat con-\nstant temperature, the MFM data in Figure 1 does not4\n0 2 4 6 8\nTip distance ( m)\n0.02.5V13(nV)\n(b)\nIV13\n12 3\n4\n(d)\n(c)\n(a)\nΔf V13\n12 nV1.13 Hz\nxy\n2 μm\nFIG. 3. Impact of a single domain on transport (a)\nMFM image showing a domain flipping during the scan, in-\ndicated by the black arrow. Same as Figure 1(c-vi). (b)V13\nmeasured simultaneously with (a), using a 500 nA amplitude\nAC source current. The black arrow indicates the location of\nthe domain flip, identical to the arrow in (a). Inset: Schematic\nof the V13measurement. (c)Zoom of (a) on the area show-\ning the domain flip. The fast scan direction is vertical. The\ndomain abruptly disappears from one vertical scan line to the\nnext. (d)V13averaged vertically along the fast scan direc-\ntion to show the jump in value that occurred as the domain\nflipped. The value from the first scan line has been subtracted\nto show the change ∆ V13.\nshow the bubble morphology when sweeping Bextat low\ntemperature. Further work is therefore required to de-\ntermine if the bubble domains formed on field cooling are\ntopological or trivial.\nWe have seen how the domain morphology can be con-\ntrolled with Bext; now we investigate the possibility of\nusing the stray field from the magnetic MFM tip, Btip,\nto locally manipulate the domains in MST. To reduce\nthe influence of Btipon the sample, the domain imag-\ning discussed so far was done with the tip lifted high\n(roughly 200-230 nm) above the MST surface. However,\nwhen Bextis near the coercive field, small changes in\nthe magnetic field can have a large influence on the sam-\nple magnetization, and even 200 nm from the tip, Btip\ncould be on the order of 10 mT [43], comparable to the\ncoercive field. Correspondingly, small changes in Rxy\nandRxxduring MFM imaging (Figure 1b, Supporting\nInformation SIV) demonstrate that the tip mildly influ-\nenced the sample magnetization. Moreover, tip-induced\ndomain flips are seen in some images as a domain that\nabruptly disappears partway through imaging. To quan-\ntify the tip’s influence, we applied an AC current between\ncontacts 2 and 4, and measured the induced transverse\nvoltage V13across contacts 1 and 3 during MFM imag-\ning. The MFM image in Figure 3a shows a domain flip,\nand the simultaneously acquired V13image (b,d) shows\nan abrupt change by more than 2 nV at the same loca-\ntion, demonstrating a measurable impact of the domain\nflip on the device transport. Interestingly the tip-induceddomain flips are not always in the sense of aligning the\ndomain with the tip, suggesting that the spatial gradient\nor time-dependence of Btipmay be equally important or\nmore important for overcoming energy barriers compared\nto the local Zeeman energy term.\nWe can harness the tip’s influence to controllably write\ndomains by bringing the tip close to the MST surface, in-\ncreasing Btip. For this purpose, we first used Bextto pre-\npare the sample with magnetization anti-aligned to the\ntip (Figure 4c). After zeroing Bext, we then brought the\ntip into amplitude-controlled feedback on the MST sur-\nface ( Btipon the order of 50 mT [43]), and moved the tip\nacross the surface to write a domain aligned with the tip.\nIn Figure 4, we show both linear (e) and square (h) areas\nwritten with the MFM tip, demonstrating that both nar-\nrow 1D-like and 2D domains can be written. During the\nwrite process, the square area formed a mixed domain\nstate, suggesting that because the mixed domain state is\nenergetically favored at Bext= 0, there is a maximum\nsingle-domain area of roughly several µm2(Supporting\nInformation SXI) that can be written. Decreasing tem-\nperature to increase the importance of the domain wall\nnucleation energy may increase that area.\nBy inverting the magnetization of a small area locally\nwith our MFM tip, we can directly probe that area’s\nimpact on the AHE. During domain writing we there-\nfore recorded V13as a proxy for Rxy(Supporting Infor-\nmation SVIII). While writing the line domain, V13de-\ncreased linearly (Figure 4d), matching the area-scaling\nthat one would expect for AHE contributions [36] that\nscale linearly with the sample average magnetization. V13\nrecorded while writing the square domain is also consis-\ntent with area scaling. Here, V13forms two 2D images\nfor forward (Figure 4f) and backward scans–to write the\ndomain, the tip moved up and down along each scan line\nbefore advancing one pixel at a time left to right. Typi-\ncally, V13has a finite slope on the forward pass (the blue\nhistogram in i is peaked at 0.6 nV /µm), confirming that\nthe tip is writing a magnetization, but not on the back-\nward pass (orange histogram, peaked at zero). Consid-\nering that typically each scan line advances the domain\nwall by one pixel width (53 nm), we can quantify the local\nAHE: 11 nV/( µm)2. This value is quantitatively consis-\ntent with both: (1) the linear V13seen when writing the\nline domain, and (2) the ratio of the change in V13from\nbefore to after the write step to the domain area imaged\nvia MFM (Supporting Information SXI). Moreover, the\nentire evolution of V13during the square write can be un-\nderstood in detail as a linear decrease (gray dashed line\nin g) from writing the red domain plus deviations from\nforming the inner blue domain, in abrupt steps initially\nbut then more smoothly near the end of the write.\nWe have therefore demonstrated a direct measurement\nof the scaling of the anomalous Hall effect with domain\narea. The technique can also in principle measure devi-\nations from this area scaling to probe local properties of\ninhomogenous devices (spatially varying magnetization\nor Berry curvature). Within homogenous materials, the5\n5 μm\n2 μm\nΔf(c) (a)\n(b)\n(d)\n(e)0 5 10\nTip distance ( m )\n50\n0V13(nV)\n(i)\n(j)\n10\n5\n0 5 10\ndV13/ds(nV / m)\n0102030Count\nbwd\nfwd\n5 μm\n2 μm(f)\n(g)\n(h)\nV13\n626 nV\n0 2 4 6 8\nTip distance ( m )\n0.5\n0.0V13( V)\nfwd\nbwd\nimagedomain\t\nwriteI1\n2 34V13x\ny\n0.5\n0.0 0.5\nDistance ( m)\n0.60.81.01.21.41.61.8f (Hz)\nFIG. 4. Writing magnetic domains. (a) Schematic of the device and the V13measurement used as an approximation of\nthe Hall response during domain writing. (b)Schematic of the magnetic tip over the MST sample during domain writing, with\nthe tip close to the surface, and during domain imaging, with the tip lifted high above the surface. (c)MFM showing the MST\ndevice has no domains, and has magnetization anti-aligned with the tip after ramping Bextto -50 mT and then 0 mT. Constant\nheight imaging was done with the tip lifted 300 nm above the SiO 2surface. The overlays show the approximate locations of:\nthe electrical contacts (orange lines), the tip trajectory for writing the line domain (orange arrow), and the MFM image of the\nline domain (gray box). Color scale range: 1.1 Hz. (d)V13measured while writing the line domain. 500 nA amplitude AC\nsource current. (e)Constant height MFM image after writing the line domain, with the tip lifted approximately 200 nm above\nthe MST surface. The red line indicates the location of the cut shown in (j). Color scale range: 1.9 Hz. (f)V13recorded as a\nfunction of the tip position while attempting to write a square area. See main text. 500 nA amplitude AC source current. (g)\nV13from (f) averaged vertically along the fast scan direction. The value of the first point was subtracted to show the change\n∆V13. The gray dashed line is the expected linear trend if the tip were fully polarizing the sample. (h)Constant height MFM\nimage after writing the square area. Tip lifted 300 nm above the SiO 2surface. The white lines indicate the locations of the cuts\nshown in (j). Color scale range: 1.6 Hz. (i)Histograms of the slopes of V13during each forward and backward vertical scan\nline while writing the square area (f). The vertical lines mark the slope while writing the line domain (red), and the expected\nper-pixel slope (blue) calculated from the overall change in V13and the domain area in (h). (j)Line cuts through the MFM\nimages of the line domain (red) and of the square area (gray). All cuts have been offset for comparison, and the cuts from the\nsquare domain have been inverted. Temperature: 10 K.\narea-scaling contributions and deviations represent bulk\nand boundary contributions, meaning that this technique\ncan be used to probe topological effects such as dissipa-\ntionless chiral edge conduction in a Chern insulator or\nthe topological Hall effect from chiral spin textures at\ndomain walls.\nThis work opens the door to making programmable\nmagnetic devices within ferrimagnetic compounds in the\nM(B,S)T family. M(B,S)T could be a platform for\nwritable chiral currents (e.g. [11, 32, 33]) either in a mag-\nnetic Weyl semimetal or Chern insulating state (mul-\ntiple potential band topologies have been predicted in\nMST [3, 16, 19, 22, 29, 30]). The ability to tune mag-\nnetic domains in a compound that retains magnetic or-\nder when exfoliated to the few layer limit [1, 16, 31, 44]\nraises the possibility of using M(B,S)T to introduce pro-\ngrammable magnetic landscapes (e.g. supperlattices orboundaries made of magnetic gradients) on length scales\nof 100s of nanometers to micrometers into van der Waals\nheterostructures. Generically, the tip writing process\nallows us to locally move between different metastable\nmagnetic configurations that are separated by energetic\nbarriers. So beyond writing individual domains of uni-\nform magnetization explicitly, the tip influence could be\ncombined with external fields and temperature to stabi-\nlize and write areas of non-uniform spin textures (just as\nthe mixed domain state formed in our square area was\nnot uniform) in order to create functional devices based\non boundaries between magnetic phases.6\nSUPPORTING INFORMATION\nAdditional experimental details, characterization of\nthe sample topography, images of stripe domains, elec-\ntrical transport measurements, analysis and modelling of\ndomain length scales, analysis of the repeatability of bub-\nble domain locations, and analysis of the scaling of the\nAHE with domain area (PDF)\nACKNOWLEDGMENTS\nWe thank Zachariah Addison and Nishchhal Verma\nfor helpful discussions. This work was supported by theAir Force Office of Scientific Research via grant FA9550-\n21-1-0378 (T.A.W., A.N.P.) and by NSF grants DMR-\n2210186 (D.N.B) and HRD-2112550 (L.K.-E.). 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Pasupathy1, 3\n1Department of Physics, Columbia University, New York, NY 10027, USA\n2Department of Physics, The City College of New York, New York, NY 10027, USA\n3Condensed Matter Physics and Materials Science Division,\nBrookhaven National Laboratory, Upton, New York 11973, USA\nCONTENTS\nSI. Methods 2\nA. Crystal growth and structural characterization. 2\nB. Magnetic force microscopy 2\nC. Electrical transport measurements 2\nD. Modeling bubble domains 2\nSII. Sample topography 4\nSIII. Evolution of stripe domains under Bext 5\nSIV. Transport measurements 6\nSV. Modeling bubble domains 7\nSVI. Domain length scales 8\nSVII. Area of stripe domains 9\nSVIII. V13as a proxy for Rxy 10\nSIX. Repeatability of bubble locations 11\nSX. Bubble and stripe domains in bulk MST 12\nSXI. Area scaling of the Anomalous Hall Effect 14\nReferences 14arXiv:2308.16806v1 [cond-mat.str-el] 31 Aug 20232\nSI. METHODS\nA. Crystal growth and structural characterization.\nCrystals of nominally MnSb 2Te4were grown out of a Sb–Te flux [1, 2]. Mixtures of Mn (Alfa Aesar, 99.99%) and\nSb pieces (Alfa Aesar, 99.999%), and Te shot (Alfa Aesar, 99.9999%) in the molar ratio of 1:10:16 (MnTe:Sb2Te3 =\n1:5) were placed in a 2 ml alumina growth crucible and heated to 900 C and held for 12 h. After slowly cooling across\na∼10 degree window below 600 C in two weeks, the excess flux was removed by centrifugation above the melting\ntemperature of Sb 2Te3(≥620 C). Crystals produced by this flux method were typically a few mm on a side and often\ngrew in thick, block-like forms with thicknesses up to 2 mm but were easily delaminated.\nEDX: Energy Dispersive X-ray (EDX) microanalysis was performed in the Zeiss Supra 55, a field emission SEM with\na maximum resolution of 1 nm. The ferrimagnetic (FM) MST stoichiometry was determined as Mn:Sb:Te=1.3 : 2.9\n: 5.8. XRD: X-ray diffraction of crystals was performed in a Panalytical diffractometer using Cu Ka ( λ= 1.5405 ˚A)\nline from Philips high intensity ceramic sealed tube (3 kW) X-ray source with a Soller slit (0.04 rad) incident and\ndiffracted beam optics. The determined c-axis parameter 40.898 ˚Awas consistent with space group, R-3m as reported\nin literature. Magnetization: D. c. magnetization measurements were performed using Quantum Design SQUID\nMagnetometer in up to 5.5 T fields. In all magnetic measurements the samples were supported in gelcaps without\nany substrates. From the fits to Curie-Weiss law, the magnetic moment in as-grown crystals was determined as\nµeff= 5.36µB/mole.\nB. Magnetic force microscopy\nMagnetic force microscopy (MFM) measurements were performed in an Attocube cantilever-based cryogenic atomic\nforce microscope (attoAFM I with attoLIQUID 2000 cryostat), where the microscope sits in a helium exchange gas\nat low temperature. Nanosensors PPP-MFMR probes with hard magnetic coating, resonant frequencies near 75 kHz,\nand force constants near 2.8 N /m were used for all measurements. All low temperature measurements on the MST\ndevice were were performed using a single AFM probe with cantilever resonance at 78 kHz. MFM images record the\nresonant frequency shift of the probe cantilever (∆ f). Constant height MFM images were taken with the tip at a fixed\nheight above the plane of the SiO 2surface. Constant lift MFM images were taken by passing twice over each line.\nThe first pass, in amplitude-controlled feedback, recorded the surface topography. On the second pass, the tip was\nlifted by a constant offset from the topographic pass to record ∆ f. In this way, topographic and MFM images were\nrecorded over the same field of view in a line-by-line interleaved fashion. During MFM imaging, the tip oscillation\namplitude was typically 35 nm to 55 nm. MFM data shown is raw data unless otherwise noted.\nC. Electrical transport measurements\nEx situ transport measurements were performed in a 14 Tesla Quantum Design Physical property measurement\nsystem (PPMS) in 1 Torr (at low temperature) of He gas. Crystals were mechanically exfoliated onto 285 nm SiO 2/Si\nwafers. Electrical contacts in the van der Pauw (vdP) configuration were photo-lithographically patterned and a\nsputtered Au metallurgy was used. Conformal Au coating amply covered side surfaces in order to make good contacts\nto top and bottom surfaces. The vdP DC measurements were carried out using a custom-configured electronic system\nin which four measurement configurations are switched by a Keithley scanner, with the current direction reversal\nemployed for each measurement to minimize thermal emf.\nIn situ electrical tranport measurements were done using a Signal Recovery model 7265 lock-in amplifier to source\na 17.777 Hz AC voltage, and measure the response voltage at the same frequency. A 10 MΩ resistor was used to\nconvert the voltage source into a current source. According to the vdP technique [3], Rxxwas calculated from four\nmeasurements, V23,14,V12,43,V41,32, and V34,21, and Rxyfrom two measurements, V31,24andV24,13, where Vab,cd is\nthe voltage measured from c to d while applying the source current from a to b, and the contacts are numbered in\nFigure 1a of the main text. For brevity, we refer to V31,24andV24,13asV24andV13, respectively.\nD. Modeling bubble domains\nCalculations of the stray magnetic field arising from a bubble domain were done using Magpylib [4]. The sample\nmodel consisted of a rectangular slab of uniform magnetization with a cylinder of the same thickness located at the3\ncenter. The cylinder had twice the magnetization with the opposite sign, such that the net magnetization inside the\ncylinder was equal to that of the rectangular slab, but opposite in direction. The lateral size of the rectangular slab\nwas taken to be 100 mm in both directions, large enough to avoid edge effects in the vicinity of the bubble domain.\nAfter calculating the stray magnetic field above the sample surface, numerical differentiation was used to calculate\nthe stray field gradients.4\nSII. SAMPLE TOPOGRAPHY\n0 50 100\nZ (nm)0246Count\n70 75 80 85 90\nZ (nm)012Count\n17 nm\n2 μm\n129 nm(a) (b) (c)\n(d)\n2 μm\nFIG. S1. (a, b) AFM topographies (tapping mode) of the full MST device at ambient (a) conditions, and of the center\nof the device, with the 4 contacts at the corners of the image, at 35 K (b). Background subtraction: Line-by-line constant\nsubtraction based on the median of the difference between consecutive lines, followed by plane subtraction to level the image.\n(c)Histogram of (a), showing 4 peaks associated with the SiO 2surface, the gold contacts on SiO 2, the MST surface, and the\ngold contacts on MST. (d)Zoom of (c) showing just the peak from the MST surface. For the MST surface only, the mean\nheight is 83 nm with standard deviation 4 nm. Considering only the center of the MST device, where most MFM imaging was\nperformed, the mean height is 81 nm with 4 nm standard deviation.5\nSIII. EVOLUTION OF STRIPE DOMAINS UNDER Bext\n(n) -30 mT \n(k) 50 mT\n(l) -10 mT\n(m) -20 mT\n(c) 8 mT\n(e) 14 mT\n(d) 12 mT\n(b) 4 mT\n(a) 0 mT (ZFC)\n(f) 16 mT\n (g) 18 mT\nΔf1.5 Hz\n(h) 20 mT\n (i) 24 mT2 μm\n(j) 32 mT\nFIG. S2. (a-n) Constant height MFM images of the magnetic domains at the center of the MST device recorded at 5 K after\nzero-field cooling. Measurements were interspersed with RxyandRxxmeasurements shown in Figure 1b and in Figure S3d.\nThe arrows indicate the order of data acquisition. The color scale on all images is 1.5 Hz, but the zero values have been offset.\nThe tip was lifted 300 nm above the SiO 2surface.6\nSIV. TRANSPORT MEASUREMENTS\n50\n25\n0 25 50\nB (mT)1.5\n1.0\n0.5\n0.00.51.01.5Rxy( )\n50\n25\n0 25 50\nB (mT)125.2125.4125.6125.8126.0126.2Rxx( )\n50\n25\n0 25 50\nB (mT)125.0125.2125.4125.6125.8Rxx( )\n8\n6\n4\n2\n0\nB (T)3\n2\n1\n01Rxy( )\n8\n6\n4\n2\n0\nB (T)123.0123.5124.0124.5125.0125.5126.0Rxx( )\n0 50 100 150 200 250 300\nT (K)140160180200Rxx( )\nex-situ PPMS\nMFM in-situ(a)\n(f) (e) (d)(c) (b)\nbefore MFM\nafter MFM\nFIG. S3. (a)Temperature dependence of Rxx, measured ex situ just after sample fabrication (gray line). The orange symbols\narein situ measurements in the the MFM using AC source current with amplitude between 100 nA and 500 nA. (b-f) Magnetic\nfield dependence of RxxandRxyat low temperature measured ex situ just after sample fabrication at 2 K (gray line), and in\nsituin the MFM at 5 K (orange symbols). (e) and (f) are the same data as (b) and (c), but displayed over a smaller field range\nin order to make the hysteresis visible. (d)AC source current amplitude: 500 nA.7\nSV. MODELING BUBBLE DOMAINS\n100 150 200 250 300 350 400 450\n2Bz/z2FWHM (nm)\n012342Bz/z2(a.u.)\n100 200 300 400\nMFM bubble FWHM (nm)1.0\n0.5\n0.00.51.0MFM bubble intensity (Hz)\n051015202530\n(a)(b)\n(c)\n400\n200\n0 200 400\nx(nm)1\n01f(Hz)\nFWHM=97nm\n197nm\n332nm(d) FWHM=197nm\n200\n0 200\nx (nm)1.5\n1.0\n0.5\n0.02Bz/z2(a.u.)\n(e) FWHM=332nm\n200\n0 200\nx (nm)1.0\n0.5\n0.02Bz/z2(a.u.)\n(f)\nFIG. S4. (a) Constant height MFM image repeated from Figure 2e, with the bubble locations and perimeters used in\nFigure 2k overlaid. The points show the bubble locations determined as the local minima of the image. The ellipses have\nmajor and minor axes terminated by the 4 half-minima positions in the x- and y-directions. The size of the major and minor\naxes are the FWHM in the x- and y-directions. Points without ellipses denote bubbles at the edge of the image for which the\nFWHM could not be reliably determined, and these bubbles were therefore ignored in the data analysis. While most bubbles\nare well-described by the ellipses, there are several where the representation is inaccurate due to the influence of neighboring\nbubbles on the x- and y-direction linecuts. (b)Histogram of the intensities and FWHM of the bubbles in (a). The vertical\nand horizontal FWHM were included independently, such that each bubble is counted twice. (c)Calculated ∂2Bz/∂z2peak\nintensity as a function of FWHM for a ferromagnetic cylindrical bubble domain. Note that the FWHM in ∂2Bz/∂z2on the x\naxis is related to the domain diameter by Figure 2l. The height zabove the ferromagnet, and the thickness hof the ferromagnet\nare related by z+h= 300nm for comparison to the MFM data. The thicknesses are h= 81.2nm (black), h= 76.7nm,85.6nm\n(orange), h= 71.2nm,91.4nm (blue), which correspond to the mean, 15.9, 84.1, 0.5, and 99.5 percentiles of the MST’s height\ndistribution in the approximate location of (a). (d)Linecuts through 3 bubbles in the MFM image repeated from Figure 2j.\n(e, f) Linecuts through the peak in ∂2Bz/∂z2for the same handzvalues as in (c), and for two of the FWHM values shown\nin (d). The model is a good description of the mid-size bubbles (e), but the model shows a distinct flat top for large bubbles\n(d), which differs from the MFM data. Discrepancies between the model and data could arise from: (1) The size and shape\nof the tip’s magnetic coating, (2) The presence of neighboring bubbles, (3) The finite amplitude of the tip oscillation, (4) The\nfinite size of domain walls.8\nSVI. DOMAIN LENGTH SCALES\n0 10 20 30 40 50\nk (μm-1)0250500750MFM FT (a.u.)\n0200400MFM FT (a.u.)\n050100MFM FT (a.u.)\n(a)\n(b)\n(c)\nFIG. S5. Angular averaged amplitude of the Fourier transforms of MFM-imaged stripe domains during magnetization reversal\natBext=−20mT (a) and after zero-field cooling (b), as well as MFM-imaged bubble domains after field-cooling (c). The\noriginal MFM images are Figures 1c-v, 1c-i, and 2e of the main text. The blue lines are fits to a Gaussian peak with linear\nbackground giving the peak locations and standard deviations listed in the main text.9\nSVII. AREA OF STRIPE DOMAINS\n(a) (b)\n40\n20\n0 20 40\nB (mT)1\n01Rxy( )\n0.00.20.40.60.81.0\nMFM fraction\n50\n0 50\nThreshold offset (mHz)0.00.20.40.60.81.0MFM fraction\n(c) 0 mT (ZFC) (d) 4 mT (e) 8 mT (f) 12 mT (g) 14 mT\n(h) 16 mT (i) 18 mT (j) 20 mT (k) 24 mT (l) 32 mT\n(m) -20 mT (n) -30 mT2 μm\nFIG. S6. (a)Comparison of Rxy(orange) to the fraction of the MFM images covered by red domains (colored points) for\nthe data shown in Figure 1 and Figure S2. The area fractions were estimated by applying a threshold to the MFM data\nafter subtracting a 2nd degree polynomial background and applying Gaussian smoothing ( σ= 1 pixel). The threshold is the\nmidpoint between the 0.1 and 99.9 percentiles of the image. The error bars show the range of values obtained by offsetting the\nthreshold from -80 mHz to 80 mHz. Other sources of error are not represented. For the 50 mT and -10 mT images where no\ndomain contrast is visible, the area fractions were manually chosen to be 1.0. (b)MFM area fractions calculated for a range of\nthreshold values. Each color represents a single image, matching the points in (a). (c-n) Binarized MFM images obtained by\napplying the threshold to calculate the points in a. The 50 mT and -10 mT images are not included because the images show\nno domain contrast.10\nSVIII. V13AS A PROXY FOR Rxy\n32.5\n32.0\n31.5\n31.0\n30.5\n30.0\nV42( V)\n30.030.531.031.532.032.5V13( V)\nFIG. S7. V13andV42measurements for two 5 K data sets (blue: stripe domains under changing Bext; orange: bubble\ndomains under nominal Bext= 0 before and after domain manipulation with the MFM tip on 2 cools). Rxyis calculated\nasRxy= (V13+V42)/2I, where Iis the source current amplitude, so changes that appear symmetrically in V13andV42are\nattributable to changes in Rxy. The approximately linear with slope near 1 relationship between V13andV42confirms that V13\nis a reasonable proxy for monitoring changes in Rxyduring domain manipulation. AC source current with 500 nA amplitude.11\nSIX. REPEATABILITY OF BUBBLE LOCATIONS\n(a)\n(c)\n2 μm\n(e)\n(d)(b)\nFIG. S8. To check the repeatability of the bubble domain locations, we cooled twice from 35 K to 5 K under nominal\nBext= 0, without changing Bextin between cools, such that the residual field from the superconducting magnet should have\nbeen identical for both cools. (a)Large scale constant height MFM image of the resulting bubble domains (cool #1). The\nsmearing on the right side is from piezo drift at the beginning of the image. Color scale range: 4.4 Hz. (b)The bubble locations\ndetermined as the locations of all local minima of the MFM image within the area of the flake, overlaid on the MFM image.\n(c-d) Same as a and b, but for cool #2. The MFM image is repeated from Figure 2a of the main text. Color scale range:\n3.7 Hz. (e)Bubble locations from the two cools overlaid on one another. The images have been shifted and scaled slightly\nin order to line up the edges of the flake from the two images (lines). Some bubble locations are identical between the cools,\nothers do not match. For MFM imaging, the tip was lifted 300 nm above the SiO 2surface. Temperature: 5 K.12\nSX. BUBBLE AND STRIPE DOMAINS IN BULK MST\n(a) T = 35 KB ~ 0 T\n(b) 25 K (c) 15 K (d) 5 K\n1 μm 2 μm 5 μm 1 μm\n12 nm\n(e) B = 0.2 T(f) 0.05 T (g) 0.02 T (h) 0.0 T\n2 μmT = 5 K\n43 nm 37 nm 160 nm 577 nm54 nm 31 nm 36 nmΔf28.4 Hz\nIn this section, we show temperature-dependent and Bext-dependent MFM imaging of a bulk MST crystal. We\ncleaved the MST crystal in air before loading into the MFM. We note several limitations of the MFM images shown\nhere: (1) The constant lift MFM imaging technique means that MFM imaging is done after topographic scanning, so\nit is possible that the magnetic domains have been influenced by the tip; (2) The appearance of magnetic features in\nthe topographic scans means that the lift height during the MFM imaging is not accurate with respect to the sample\nsurface. Nonetheless the data shown confirms the presence of bubble domains (under small Bextcooling), and stripe\ndomains (under Bextsweep at low temperature) in bulk MST.13\nFIG. S9. (a-d) Constant lift MFM images and their simultaneously recorded topographic images at temperatures cooling\nfrom 35 K to 5 K. No magnetic contrast is visible at 35 K, above Tc. Below Tc, bubble domains are observed. The topographic\nimages show a combination of topographic features and magnetic features due to the strong tip-sample interaction. (e-h)\nConstant lift MFM images and their simultaneously recorded topographic images at 5 K, starting at 0.2 T, where no domains\nare seen, and lowering to 0 T. At 0.02 T and 0 T, stripe domains are observed. Again, when domains are present, magnetic\nfeatures appear in the nominally topographic images due to the strong tip-sample interaction. All MFM images are shown\nwith the same color scale range 28.4 Hz, but the zero value of the images have been offset. Topographic images were levelled\nby subtracting a planar background.14\nSXI. AREA SCALING OF THE ANOMALOUS HALL EFFECT\nTABLE I. Multiple measurements of the V13response to domain writing. stipis the (linear 1D) distance written by the tip.\nAis the area of domains aligned with the tip.\nExperiment dV13/dstipdV13/dA Measured as\n(nV/µm) (nV/ µm2)\nLine domain 8.4 Slope of V13during write\nSquare area 8.4 Slope of V13during the first forward scan line.\nSquare area 0.6 11 dV13/dstip: Peak of V13slope histogram for forward scan\nlines. dV13/dA=dV13/dstip/∆x, where ∆ xis the pixel\nwidth, 53 nm\nSquare area 0.0 Peak of V13slope histogram for backward scan lines\nSquare area 12 Overall change in V13during write, divided by the area\nof red domains in Figure 4h.\nThis section provides additional details about the domain writing experiments shown in Figure 4 of the main\ntext. We discuss: (1) additional details about the Hall response observed when writing the square area, as shown in\nFigure 4f-h of the main text, and (2) the consistency between multiple measurements of the Hall response to domain\narea.\nWe attempted to write a 8 µm square. The post-write imaging (Figure 4h) shows that the square was not uniformly\nmagnetized, but instead formed a mixed domain state, with an inner blue domain inside the red domain. We examine\nV13recorded during the write step to understand how this pattern formed dynamically (f, g). During the write, the\nfast and slow scan directions are vertical and horizontal, respectively, meaning that forward (f) and backward (not\nshown) V13images were recorded as the tip moved up and down along each pixel before advancing one pixel at a\ntime from left to right. The V13evolution has both smooth changes as well as abrupt jumps, which can be seen more\nclearly in (g) after averaging along the vertical fast scan direction.\nTo quantify the write process, we extract dV13/dstipas the slope of each vertical scan line for both forward and\nbackward passes (Figure 4i), where stipis the linear (1D) distance written by the tip. The forward slopes are peaked\naround 0.6 nV/ µm, while the backward slopes are peaked around 0 nV/ µm, matching the expectation that the tip\nshould typically flip the local magnetization on the forward pass and have a random influence on the backward pass.\nHowever, the forward pass of the first scan line has a much larger dV13/dstip= 8.4nV/µm, which instead matches the\nslope seen during the line writing experiment (Figure 4d). The order of magnitude difference between the slope of the\nfirst and subsequent scan lines can be explained by the width being written by the tip: the first line creates a domain\nof a finite width on the order of hundreds of nm (Figure 4j), while subsequent scan lines advance the domain wall by\none pixel width (53 nm in this case). The consistency across measurements of the V13slope when taking into account\nthe width being written strongly supports that V13changes linearly with the area of the domains aligned with the tip.\nWe quantify this response, dV13/dA, in two ways: (1) 11 nV/( µm)2, by dividing the peak slope of the forward scan\nlines by the pixel width, and (2) 12 ±1 nV/( µm)2, by taking the ratio of the overall change in V13during the write to\nthe area of red domains imaged in Figure 4h. The values of the Hall response to domain area are compiled in Table I.\nWith the relationship between V13and the sample magnetization firmly established, we return to the question of\nhow the mixed domain state of Figure 4h formed. If each scan line is fully polarized by the tip, then V13should\nchange according to the linear trend shown in gray in Figure 4g. As previously discussed, the first scan line shows\na sharp drop in V13associated with writing a finite width domain. Then during the next 4 µm,V13changes largely\nas expected, implying that the written area is nearly fully polarized, with the exception of a few upward jumps,\nsuggesting abrupt formation of areas anti-aligned with the tip (inner blue domains of h). During the later part of\nthe scan frame, the V13slope shows a larger deviation from the expectation. We therefore suggest that the initial\nformation of the inner blue domains within the write area occurs primarily in abrupt steps. But later, the domains\ngrow more smoothly. There may therefore be a maximum size, on the order of a few µm2(based on the location of\nthe first abrupt deviation) that can be uniformly polarized with our procedure at 10 K.\n[1] H. Deng, L. Zhao, K. Park, J. Yan, K. Sobczak, A. Lakra, E. Buzi, and L. Krusin-Elbaum, Topological surface currents\naccessed through reversible hydrogenation of the three-dimensional bulk, Nat. Commun. 13, 2308 (2022).\n[2] J.-Q. Yan, S. Okamoto, M. A. McGuire, A. F. May, R. J. McQueeney, and B. C. Sales, Evolution of structural, magnetic,\nand transport properties in MnBi 2−xSbxTe4, Phys. Rev. B 100, 104409 (2019).15\n[3] L. J. van der Pauw, A method of measuring specific resistivity and Hall effect of discs of arbitrary shape, Philips Res. Repts.\n13, 1 (1958).\n[4] M. Ortner and L. G. Coliado Bandeira, Magpylib: A free python package for magnetic field computation, SoftwareX\n10.1016/j.softx.2020.100466 (2020)." }, { "title": "1308.3701v2.Theory_of_metallic_double_perovskites_with_spin_orbit_coupling_and_strong_correlations__application_to_ferrimagnetic_Ba2FeReO6.pdf", "content": "Theory of metallic double perovskites with spin orbit coupling and strong\ncorrelations; application to ferrimagnetic Ba 2FeReO 6\nAshley Cook1and Arun Paramekanti1;2\n1Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 and\n2Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada\nWe consider a model of the double perovskite Ba 2FeReO 6, a room temperature ferrimagnet with\ncorrelated and spin-orbit coupled Re t 2gelectrons moving in the background of Fe moments stabi-\nlized by Hund's coupling. We show that for such 3d/5d double perovskites, strong correlations on\nthe 5d-element (Re) are essential in driving a half-metallic ground state. Incorporating both strong\nspin-orbit coupling and the Hubbard repulsion on Re leads to a band structure consistent with ab\ninitio calculations. Using our model, we \fnd a large spin polarization at the Fermi level, and obtain\na semi-quantitative understanding of the saturation magnetization of Ba 2FeReO 6, as well as X-ray\nmagnetic circular dichroism data indicating a signi\fcant orbital magnetization. Based on the or-\nbital populations obtained in our theory, we predict a speci\fc doping dependence to the tetragonal\ndistortion accompanying ferrimagnetic order. Finally, the combination of a net magnetization and\nspin-orbit interactions is shown to induce Weyl nodes in the band structure, and we predict a signif-\nicant intrinsic anomalous Hall e\u000bect in hole-doped Ba 2FeReO 6. The uncovered interplay of strong\ncorrelations and spin-orbit coupling lends partial support to our previous work, which used a local\nmoment description to capture the spin wave dispersion found in neutron scattering measurements.\nOur work is of interest in the broader context of understanding metallic double perovskites which\nare of fundamental importance and of possible relevance to spintronic applications.\nI. INTRODUCTION\nDouble perovskite (DP) materials A 2BB'O 6, where\nthe transition metal ions B and B' reside on the two\nsublattices of a cubic lattice, can realize many complex\nphases.1Metallic variants, such as Sr 2FeMoO 6, provide\nus with the simplest multi-orbital examples of ferrimag-\nnetic order2kinetically stabilized by the Pauli exclusion\nprinciple.3{11Insulating variants where only the B'-site\nion is magnetic, such as Ba 2YMoO 6and La 2LiMoO 6,\nprovide material examples of quantum mechanical mo-\nments living on the geometrically frustrated face-centered\ncubic lattice.12{16Metallic DPs, such as Sr 2FeMoO 6,\nare also of signi\fcant technological importance, being\nroom temperature ferrimagnets with half-metallic band\nstructures and a large spin polarization which is useful\nfor spintronic applications.17,18Metallic 3d/5d DPs are\nof particular interest in this regard since they appear\nto have strongly reduced B/B' site mixing; samples of\nBa2FeReO 6studied in previous work19have low<1%\nanti-site disorder. Such anti-site disorder, which is com-\nmon in other DPs and which is detrimental to spintronic\napplications, appears to be alleviated in 3d/5d DPs by\nthe B/B' ionic size mismatch suggesting that they might\nbe better suited for applications. However, such 3d/5d\nDPs require us to confront the twin aspects of strong\ncorrelations and strong spin-orbit coupling, topics at the\nforefront of fundamental research20motivated by the pos-\nsibility of stabilizing states such as fractionalized topo-\nlogical insulators (TIs),21{24or Weyl semimetals.25{28\nIn this paper, we focus on metallic ordered DPs with\nmixed 3d/5d transition metal ions on the B/B' sites,\nspeci\fcally the Ba 2FeReO 6material,29,30with the struc-\nture as shown in Fig.1. we obtain the following main\nresults. (i) We consider a model of the ordered dou-\nxyzBaFe\nRe\nOd\ndc\naFIG. 1: Crystal structure of Ba 2FeReO 6showing the choice\nof axes, the unit vectors for the elementary triclinic unit\ncell (green arrows), and magnetic moments in the ferrimag-\nnetic ground state. Also shown is the enlarged body-centered\ntetragonal unit cell with lattice parameters daanddc=dap\n2.\nble perovskite Ba 2FeReO 6(see Fig.1) retaining the rel-\nevant electronic states in the vicinity of the Fermi level.\nThis model, after taking spin-orbit coupling as well as\ncorrelations e\u000bects into account within a self-consistent\nmean \feld theory, is shown to reproduce previous ab\ninitio electronic structure results31in the ferrimagnetic\nground state. Our model accounts for the dominant en-\nergy scales in this material: (a) the strong Hund's cou-\npling on Fe, the Hubbard repulsion on Re, and the Fe-Re\ncharge transfer energy (all on the scale of \u00181eV), (b)\nthe strong spin-orbit coupling on Re ( \u00180:5eV), and (c)\nthe nearest neighbor Re-Fe hopping terms which leads toarXiv:1308.3701v2 [cond-mat.str-el] 28 Nov 20132\nelectron itinerancy ( \u00180:3eV). In addition, we include\nweaker terms such as inter-orbital mixing and second\nneighbor hopping which are required to reproduce the\nband degeneracies at high symmetry points in the Bril-\nlouin zone found in earlier ab initio studies. (ii) Our the-\nory accounts semi-quantitatively for the measured sat-\nuration magnetization32, as well as X-ray magnetic cir-\ncular dichroism (XMCD) experiments which \fnd a sig-\nni\fcant orbital contribution to the Re magnetization in\nthe ordered state.33,34(iii) Based on the orbital occu-\npations in the magnetically ordered state, we predict a\ntetragonal distortion, with c-axis compression accompa-\nnying magnetic order, in agreement with experimental\ndata.33,34We also predict a speci\fc doping dependence\nto this orbital order and distortion which could be tested\nin future experiments. (iv) The strong correlations on\nRe, inferred from our study, lends partial support to ear-\nlier work which showed that a local moment description\nof the ferrimagnetic state provides a reasonably good de-\nscription of the magnetic dynamic structure factor ob-\ntained using inelastic neutron scattering experiments.19\nThis importance of strong correlation e\u000bects and lo-\ncal moment physics on the 5d element is in agreement\nwith previous ab initio studies11that discussed the emer-\ngence of local moments of closely related Cr-based 3d/5d\nDPs Sr 2CrB'O 6upon progressing through the series with\nB'=W,Re,Os. (v) From our computed band dispersion,\nwe show the appearance of Weyl nodes in such metallic\nferrimagnetic DPs. This is in line with the general under-\nstanding that in the presence of spin-orbit coupling, such\nWeyl nodes are expected to be induced by breaking of\ntime-reversal symmetry or inversion symmetry.35,36(vi)\nUsing the Kubo formula for the spin-orbit coupled bands,\nwe \fnd that Ba 2FeReO 6itself appears to have only a\nsmall intrinsic anomalous Hall e\u000bect (AHE) in the or-\ndered ferrimagnetic state at low temperature, but the\nAHE is signi\fcant in hole doped systems, and we spec-\nulate that it might also be signi\fcant at intermediate\ntemperatures below the ferrimagnetic Tcin Ba 2FeReO 6.\nTaking a broader viewpoint, Re-based layered quasi-\ntwo-dimensional oxides or heterostructures may be more\nstrongly correlated than the three-dimensional DPs, and\nmay lead to interesting Mott physics14,16beyond the iri-\ndates due to the local competition between interactions\nand spin-orbit coupling due to the d2con\fguration of\nRe5+. Furthermore, one can carry out detailed inelastic\nneutron scattering studies in Re-based oxides, thus allow-\ning for the possibility to explore the magnetism in more\ndetail than in the iridates. This may prove to be useful\nin future studies of exotic variants of Re-based oxides.\nII. MODEL\nThe simple charge counting for Ba 2FeReO 6suggests\nRe5+and Fe3+valence states on the transition metal\nions. In this state, the \fve 3d-electrons on Fe are ex-\npected to be locked into a spin-5 =2 moment due to strongHund's coupling in the half-\flled d-shell. Here, we will\ntreat this magnetic moment as a classical vector. The\ntwo 5d-electrons in the Re t2gorbital are mobile, able to\nhop on and o\u000b the Fe sites subject to a charge transfer\nenergy \u0001 =EFe\u0000ERe>0, and Pauli exclusion which\nconstrains electrons arriving on Fe to be antiparallel to\nthe direction of the local Fe moment. For a general direc-\ntion of the Fe moment, ~F= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012)\nat a given site, we must project the added electrons onto\nthe allowed direction to satisfy Pauli exclusion, locally\nsettingf\"= sin\u0012\n2e\u0000i\u001e=2fandf#=\u0000cos\u0012\n2ei\u001e=2f, ef-\nfectively \\stripping\" the electron of its spin degree of\nfreedom. Such models have been proposed for other DP\nmaterials,3{8,10,11and shown to capture the phenomenol-\nogy of Sr 2FeMoO 6including thermal phase transitions\nand disorder e\u000bects.37{39However, most of these previ-\nous studies, with the notable exception of Ref. 11 have\nignored spin-orbit coupling e\u000bects, which are expected to\nbe extremely important for 5d transition metal oxides.\nOur model does not explicitly account for additional\nsuperexchange interactions between the Fe local moment\nand the emerging local moments on the Re sites which\nis explicitly taken into account as a separate term in\nsome previous studies (for example, Ref. 11); however,\nwe think such terms should emerge more naturally from\nan e\u000bective tight-binding model when strong correlations\nare incorporated, as might be relevant to Mott insulat-\ning oxides like Sr 2CrOsO 6. Fe-Fe superexchange terms\nwhich we omit, since they are not necessary to drive the\nferrimagnetic state observed in Ba2FeReO6, may prove\nto be important in understanding the complete magnetic\nphase diagram as a function of doping which is not ad-\ndressed in this paper. However, they are likely to be small\ngiven the Fe-Fe separation in the DP structure. Further\ndi\u000berences between the results of Ref. 11 and our work\nstem from the fact that their model is for d3con\fgura-\ntion of Cr, as opposed to our d5state on Fe; while both\nspin components of the itinerant electrons are permitted\non Cr (since the e gorbital is available), only one spin\nprojection is allowed for itinerant electrons on Fe due to\nthe Pauli exclusion.\nA. Non-interacting tight binding model\nThe model describing Re electrons moving in the pres-\nence of Fe moments then takes the form H0=Hhop+\nHso+Hct. Here, the Hamiltonian Hhopdescribes intra-\norbital hopping of electrons on the lattice, from Re to Fe\n(nearest-neighbor) and from Re to Re (next-neighbor), as\nwell as inter-orbital hopping of electrons between next-\nneighbor Re sites; Hsois the atomic spin-orbit coupling\non Re, projected to the t2gmanifold, of strength \u0015; \f-\nnally,Hctdescribes the charge transfer energy o\u000bset \u0001\nbetween Re and Fe sites. For simplicity, we only fo-\ncus on the case of a uniform magnetization on the Fe\nsite, assuming ( \u0012;\u001e) which describe the Fe moment to\nbe site-independent; it is straightforward to generalize3\nour work to a nonuniform spatially varying magnetiza-\ntion. We use the simple triclinic unit cell, with one Re\nand one Fe atom, as shown in Fig.1 to study the model\nHamiltonian; however in order to facilitate a comparison\nwith published ab initio electronic structure calculations,\nwe will later assume a body-centered tetragonal unit cell\ncontaining two Re and two Fe atoms, with lattice con-\nstantsda=db=dc=p\n2 as shown in Fig. 1, and use or-\nthorhombic notation to plot the band dispersion of the\neighteen bands in the Brillouin zone.\nWe label the electrons on the Fe and Re sites by f`and\nd`\u001brespectively, with `=(1\u0011yz;2\u0011xz;3\u0011xy) denoting\nthe orbital, and \u001b=\";#being the spin. The Hamilto-\nnian takes the following form in momentum space, where\nwe assume implicit summation over repeated spin and\norbital indices,\nHhop=X\nk(\u0011`(k)g\u001b(\u0012;\u001e)dy\n`\u001b(k)f`(k) + h:c:)\n+X\nk\u000f`(k)(dy\n`\u001b(k)d`\u001b(k) +\u000bffy\n`(k)f`(k))\n+X\nk(`6=`0)\r``0(k)(dy\n`\u001b(k)d`0\u001b(k)+\u000bffy\n`(k)f`0(k)) (1)\nHso=i\u0015\n2X\nk\"`mn\u001cn\n\u001b\u001b0dy\n`\u001b(k)dm\u001b0(k) (2)\nHct=\u0001X\nkfy\n`(k)f`(k) (3)\nHere, in light of our previous discussion, we have only\nretained a single spin projection on the Fe site, with\ng\"(\u0012;\u001e) = sin\u0012\n2e\u0000i\u001e=2andg#(\u0012;\u001e) =\u0000cos\u0012\n2ei\u001e=2. The\nvarious hopping processes are schematically illustrated in\nFig. 2. The \frst term in Hhopdescribes nearest-neighbor\nintra-orbital hopping from Re to Fe, parameterized by\nt\u0019;t\u000e. The next two terms in Hhopcharacterize next-\nneighbor hopping processes, with the ratio of Fe-Fe hop-\npings to Re-Re hoppings being \u000bf; we will \fx \u000bf= 0:5.\nWhile the second term captures intra-orbital hopping be-\ntween closest pairs of Re atoms or Fe atoms (parame-\nterized by t0;t00), the third term captures inter-orbital\nhopping between closest pairs of Re atoms or Fe atoms\n(parameterized by tm). Many of these hopping processes\n(t\u000e;tm;t00) have a small energy scale; however they are\nimportant to reproduce the band degeneracies found in\nab initio calculations at high symmetry points in the Bril-\nlouin zone. The explicit momentum dependence of the\ndispersion coe\u000ecients appearing in Hhopis given in Ap-\npendix A.\nB. Interaction e\u000bects\nElectron-electron interactions are partially accounted\nfor byH0in the previous section | in part, by the charge\ntransfer gap \u0001, and, in part, by the implicit Hund's cou-\npling which locks the Fe electrons into a high-spin state.\nHowever, electronic interactions on Re have been omitted\nx’π\ntδz\nxy’t\n+\n++\n+\nRe FeFe\nt\"t’ Redyz tm\n+\n+\n+++++\n+\nFeRe\nyRedxyFIG. 2: Symmetry-allowed hopping matrix elements for dou-\nble perovskites A 2BB'O 6(e.g., Ba 2FeReO 6), indicated for a\nfew orbitals. t\u0019;t\u000eare B-B' (Fe-Re) intraorbital hoppings,\nt0;t00are B'-B' (Re-Re) intraorbital hoppings, and tmdenotes\nthe interorbital B'-B' (Re-Re) hopping. All processes related\nto these by cubic symmetry are allowed. The Fe-Fe hop-\npings are identical to Re-Re hoppings, but scaled by a factor\n\u000bf= 0:5. Also shown are the rotated axes (compass) for the\ntetragonal unit cell of Ba 2FeReO 6, withx0-y0(dashed lines)\nbeing the original cubic axes for de\fning the orbitals.\ninH0. We next include these local Hubbard interactions\non Re. The interaction Hamiltonian in the t 2gorbitals\nof Re takes the form40\nHint=UX\ni`\u000bni`\"ni`#+ (U\u00005JH\n2)X\n`<`0ni`ni`0\n\u00002JHX\n`<`0~Si`\u0001~Si`0+JHX\n`6=`0dy\ni`\"dy\ni`#di`0#di`0\"(4)\nwhereilabels the Re sites, and ~Si`=1\n2dy\ni`\u000b~ \u001b\u000b\fdi`\fis\nthe spin at site iin orbital`. We wish to then study\nthe full Hamiltonian H=H0+Hint. For simplicity,\nwe only retain only the dominant intra-orbital Coulomb\nrepulsion, treating it at mean \feld (Hartree) level, as\nHint\u0019UX\ni`\u0014\u001a`\n2(ni`\"+ni`#)\u00002~ m`\u0001~Si`\u0000\u001a2\n`\n4+~ m`\u0001~ m`\u0015\n(5)\nwhere\u001a`=hni`\"+ni`#i,~ m`=h~Si`i, and we set\n~ m`=\u0000m`(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012), withm`>0, so\nthat~ m`is anti-parallel to the Fe moment ~F. For simplic-\nity, we only focus on the case \u0012=\u001e= 0, so the Fe sites\ncan only accommodate itinerant spin- #electrons. We\nthen numerically determine m`and\u001a`in a self-consistent\nfashion, using the non-interacting ground state as the\nstarting point for the iterative solution, while ensuring4\nthat the choice of the chemical potential lead to a total\nof two electrons per unit cell (i.e., per Re atom). Such\na mean \feld treatment of electron-electron interactions\ndoes not capture all aspects of the strong correlation\nphysics, e.g. bandwidth renormalization and mass en-\nhancement. Nevertheless, recognizing this caveat, we use\nthe self-consistent solution of the mean \feld equations to\nstudy the e\u000bects of interactions and spin-orbit coupling\non the reorganization of the nine electronic bands, com-\npare the physical properties with experimental results,\nand make qualitative predictions for future experiments.\nIII. PHYSICAL PROPERTIES\nWe begin by discussing the e\u000bect of electronic correla-\ntions in the DPs in the absence of spin-orbit coupling. We\nshow that such correlation e\u000bects appear to be crucial to\nstabilize a half-metallic state with complete polarization\nin the 5d perovskites, due to the large second-neighbor\nRe-Re hopping which otherwise prevents a half-metallic\nstate. We then turn to the e\u000bect of spin-orbit coupling,\nand show that it reorganizes the band structure, yielding\nresults which are in reasonable agreement with previous\nab initio electronic structure studies.31(As pointed out\nearlier, the band dispersions discussed below are plot-\nted using the orthorhombic notation with an enlarged\nunit cell containing two Fe and two Re atoms, leading\nto eighteen electronic bands instead of nine.) Finally, we\ncompare the mean \feld result for the saturation magne-\ntization with experiments, and the spin and orbital mag-\nnetization on the Re site with previous XMCD data, and\ndiscuss other physical properties such as tetragonal lat-\ntice distortion and predictions for the AHE. Throughout\nthis discussion, we will assume a ferromagnetic order of\nthe Fe moments - a more complete study of the magnetic\nphase diagram as a function of doping and temperature\nwill be the subject of future numerical investigations.\nA. Correlations stabilize a half-metal\nIf we ignore Re correlations entirely, setting U= 0,\nand also ignore spin-orbit coupling by setting \u0015= 0, the\nband structure shown in Fig.3(a) has decoupled spin- \"\nand spin-#bands. The twelve spin- #bands correspond-\ning to electrons which can delocalize on Re and Fe. By\ncontrast, the six spin- \"bands corresponds to purely Re\nstates. Working in units where t\u0019= 1, we \fnd that to\nmake a reasonable comparison with the ab initio calcu-\nlations, we have to choose a signi\fcant t0= 0:3 (Re-Re\nhopping), but all other hoppings can be assumed to be\nsmall; for simplicity, we \fx t\u000e=t00=tm= 0:1. Finally,\nwe have to assume a moderate charge transfer energy\n\u0001 = 3 which splits the spin- #states into two groups: 6\nlower energy Re-Fe hybridized spin- #states (dominant\nRe character) which form a broad band, and 6 higher\nenergy dominantly Re-Fe hybridized spin- #states (dom-\n-6-4-2 0 2 4 6-6-4-2 0 2 4 6\n-6-4-2 0 2 4 6\n-6-4-2 0 2 4 6Z Γ X S Y Γ Z Γ\nX S Y Γ Z Γ X S Y Γ Z Γ(a) (b)\n(c)X\n(d)S Y ΓFIG. 3: Band dispersion in the orthorhombic notation for the\nRe and Fe electronic states for di\u000berent choices of Hubbard\ninteraction Uand spin-orbit coupling \u0015, with energy on the y-\naxis in units of t\u0019. The solid black line indicates the chemical\npotential. For no spin-orbit coupling, (a) U= 0 and\u0015= 0\nand (b)U= 8t\u0019and\u0015= 0, we \fnd decoupled spin- #(red,\nsolid) and spin-\"(blue, dashed) states. Comparing (a) and\n(b), we see that correlations on Re push the spin- \"states\nto higher energy, leading to the stabilization of a half-metal\nground state. A nonzero spin-orbit coupling, (c) U= 0 and\n\u0015= 2t\u0019, and (d)U= 8t\u0019and\u0015= 2t\u0019, leads to mixed-spin\nstates and splits degeneracies, but for a physically reasonable\nvalueU= 8t\u0019preserves signi\fcant spin polarization \u001890%\nfor states at the Fermi level.\ninant Fe character) which form a narrow band. Finally,\nthe remaining 6 Re- \"states form a narrow dispersing\nband, crossing the chemical potential and overlapping in\nenergy with the broad spin- #band. ForU= 0, the sys-\ntem thus contains both spin states at the Fermi level.\nWhen we incorporate a Hubbard repulsion U= 8t\u0019at\nmean \feld level, we see from Fig. 3(b) that its main ef-\nfect is to self-consistently shift the spin- \"bands higher in\nenergy, leaving only spin- #states at the Fermi level. The\nresulting band dispersion is in reasonably good agree-\nment with LDA+U calculations. Although we have not\nattempted a detailed quantitative \ftting to the LDA+U\nband structure, the features noted below are robust. (i)\nA rough comparison with the overall bandwidth in the ab\ninitio calculations without spin-orbit coupling31suggests\nthatt\u0019\u0019330meV. This is somewhat larger than esti-\nmates for Sr 2FeMoO 6in the literature3,4,10(\u0018270 meV).\n(ii) We estimate the interaction energy scale on Re to\nbeU\u00192:5eV, smaller by a factor of two compared with\ntypical values for 3d transition metals. (iii) There is a sig-\nni\fcant Re-Re hopping, t0=t\u0019\u00180:3, we need to include in5\norder to be able to capture the bandwidths of the spin- \"\nand spin-#bands. All these observations are reasonable\ngiven the more extended nature of Re orbitals when com-\npared with 3d or 4d transition metal ions. The presence\nof appreciable Re-Re hoppings has been pointed out in\nprevious work,6,41although they did not take correlation\ne\u000bects on Re into account. More recent work has also ar-\nrived at similar conclusions regarding signi\fcant Re-Re\nhoppings.11\nTo summarize, we have obtained a tight-binding de-\nscription including interactions of DPs with spin-orbit\ncoupling. In contrast to 3d/4d DP materials like\nSr2FeMoO 6, we \fnd that 3d/5d DPs have a signi\fcant\nsecond neighbor hopping; strong correlations on the 5d\nelement (Re) therefore play a crucial role in stabilizing a\nhalf-metallic ground state in the 3d/5d DPs .\nB. Spin-orbit coupling: Band reconstruction,\nspin/orbital magnetization, and comparison with\nmagnetization and XMCD experiments\nWe next turn to the e\u000bect of incorporating both spin-\norbit coupling and Hubbard interactions on Re, solving\nthe mean \feld equations in case of a nonzero U. From\nFig.3(c) and (d), where we have set \u0015= 2t\u0019(\u0018660meV\nfor our estimated t\u0019), we see that spin-orbit coupling\nclearly eliminates the degeneracies occurring at the \u0000-\npoint for\u0015= 0. It also signi\fcantly reconstructs the dis-\npersion of the eighteen bands, leading to reasonably good\nagreement with published ab initio calculations which in-\nclude spin-orbit coupling.31In the next section, we will\ndiscuss the resulting appearance of Weyl nodes in the\nband dispersion and the intrinsic anomalous Hall e\u000bect\nin the ordered state. Here, we will use the mean \feld\nsolution to estimate the average Fe valence, the Fe or-\ndered moment, and the spin and orbital contributions to\nthe Re moment. In the ground state with correlations,\nwe \fnd that the average valence of Fe shifts from the\nnaive charge counting value Fe3+to Fe2:6+, and the Fe\nmoment is lowered to an e\u000bective value Fz\u00192:3 (cor-\nresponding to 4 :6\u0016B). Quantum spin \ructuations be-\nyond the mean \feld result might further slightly sup-\npress this value. On Re, we \fnd an ordered spin moment\nSz\u00190:78 and an orbital moment Lz\u00190:48; taking\nthe g-factor into account, and undoing the sign change\nof the orbital angular momentum which appears upon\nprojection to the t2gHamiltonian, this implies a ratio of\nmagnetic moments \u0016orb\nRe=\u0016spin\nRe\u0019\u00000:31, remarkably close\nto the experimentally measured XMCD result \u0019\u00000:29.\nWe \fnd that the actual value of the spin magnetic mo-\nment,\u0016spin\nRe\u00191:56\u0016B, is larger than the experimentally\nreported XMCD value \u00191:08\u0016B. This discrepancy might\nbe partly due to the fact that (i) the experimental re-\nsults are on powder samples, and hence might appear to\nbe smaller simply due to averaging over grain orienta-\ntions, and (ii) the method to extract the individual spin\nor orbital magnetic moments relies on additional assump-tions, while the ratio is apparently more reliable.33We\nmust contrast these results with the case where we ignore\nRe correlations entirely; in that case, the Fe moment is\nnot much a\u000bected, FU=0\nz\u00192:4, but the Re moments\nare strongly suppressed, yielding SU=0\nz\u00190:15 and an or-\nbital momentLU=0\nz\u00190:09 which would lead to a much\nsmaller\u0016spin\nRe(U= 0)\u00190:3\u0016Bthan is experimentally\nestimated, as well as a much larger saturation magne-\ntization, 4:6\u0016B, than the measured value30,32which is\n\u00193:2-3:3\u0016B. Our estimates in the presence of correla-\ntions, by contrast, yield msat\u00193:5\u0016B, in much better\nagreement with the data. Finally, we use our solution to\nestimate the polarization, de\fned as the degree of mag-\nnetization for states near the Fermi level. We \fnd that\nwhile the correlated half-metal state in the absence of\nspin-orbit coupling exhibits (obviously) 100% polariza-\ntion, using \u0015= 2t\u0019reduces the polarization to \u001890%.\nHowever, if we only take spin-orbit coupling into account\nand ignore strong correlations, the states near the Fermi\nlevel are nearly unpolarized.\nIn 3d/5d DP materials, spin orbit coupling and strong\ncorrelations are both crucial to obtain the experimentally\nobserved spin and orbital magnetization and their lock-\ning, and to explain the experimentally observed satura-\ntion magnetization and XMCD signal. Spin-orbit cou-\npling leads to a slight decrease of the correlation-induced\nspin polarization at the Fermi level.\nC. Orbital order, tetragonal distortion in\nferrimagnetic state, and doping dependence\nIn the converged mean \feld state, with the magneti-\nzation along the z-axis, we \fnd that the density on Re\nin the three orbitals are di\u000berent, with \u001axy\u00190:60 and\n\u001axz=\u001ayz\u00190:53. This orbital imbalance is induced in the\nz-ferrimagnetic state due the spin-orbit coupling. The\nlarger extent of the xy-orbital in the xy-plane, compared\nwith its smaller extent along the z-direction, implies that\nthis orbital charge imbalance would lead to a tetragonal\ndistortion of the lattice, to occur coincident with ferri-\nmagnetic ordering and with a shrinking of the c-axis, as\nhas indeed been observed to occur experimentally. The\nprecise extent of this distortion, which is observed33to be\n\u00180:1%, depends on details such as the lattice sti\u000bness,\nand is beyond the scope of our calculation.\nWhen we solve the self-consistent equations at vari-\nous dopings \u000e(excess electrons per Re) assuming persis-\ntent ferrimagnetic order, the extent of this orbital im-\nbalance, characterized by a tetragonal order parameter\n\u0011tet=1\n\u001a(\u001axy\u0000\u001axz=2\u0000\u001ayz=2), changes systematically\nas shown in Fig. 4(a). Light electron doping leads to a\nslightly larger orbital population imbalance and should\nenhance the c-axis compression, while a larger electron\ndoping leads to a gradual decrease of \u0011tet. Hole doping\nbeyond &0:25 holes/Re leads to \u0011tet<0, which should\ncause elongation along the c-axis. The spin contribution\nto the magnetization on Re, arising from the di\u000berent6\n-0.5-0.45-0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.05 0\n-1-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.03-0.02-0.010.000.010.020.030.040.05\n-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60(a) (b)2ρ ρ\n2xyρ( /ρxz yz )\nDoping (δ) Doping (δ)LzMagnetization on ReS(xy)\nz\nSz(xz) = Sz(yz)\nFIG. 4: (a) Relative orbital occupancy in the ferrimagnetic\nstate of Ba 2FeReO 6incorporating mean-\feld interactions and\nstrong spin-orbit coupling at a doping of \u000eexcess electrons per\nRe. The parameters used are the same as those for Fig. 3(b),\nnamelyU= 8t\u0019and\u0015= 2t\u0019. This orbital order implies a\ntetragonal distortion of the lattice, with c-axis compression\nfor\u000e&\u00000:25, and c-axis elongation for \u000e.\u00000:25. (b)\nDoping dependence of orbital magnetization on Re, and the\nvarious orbital components of the spin magnetization on Re.\norbitals, also shows a similar doping trend as seen from\nFig. 4(b), while the orbital contribution to the magneti-\nzation on Re has the largest magnitude at zero doping.\nThese results could be possibly be explored experimen-\ntally by partially substituting Ba by trivalent La (elec-\ntron doping), or by Cs or other monovalent ions (hole\ndoping).\nThus, in 3d/5d DP materials, spin orbit coupling and\nthe ferrimagnetic order of itinerant electrons leads to or-\nbital ordering. This, in turn, should lead to a compres-\nsion along the c-axis, consistent with the experimentally\nobserved tetragonal distortion, and we predict a speci\fc\ndoping dependence to this structural distortion.\nD. Doping-dependent anomalous Hall e\u000bect\nWe next turn to the intrinsic AHE in the ferrimag-\nnetic state of such 3d/5d DPs. As pointed out in recent\nwork, for pyrochlore iridates with all-in-all-out order un-\nder uniaxial pressure,27as well the ferromagnetic in\fnite-\nlayer ruthenate SrRuO 3,42this intrinsic AHE contains\ntwo contributions: (i) a surface contribution arising from\nFermi arc states25associated with Weyl nodes in the\ndispersion, and (ii) a bulk contribution from carriers\nnear the Fermi surface. A pair of such Weyl nodes for\nBa2FeReO 6is shown in Fig. 5 obtained from the inter-\nacting band dispersion.46\nBoth contributions to the intrinsic AHE are captured\nby the momentum-dependent Berry curvature43,44of the\nspin-orbit coupled bands, which is, in turn, obtained from\nthe Kubo formula\n\u001bxy=e2~Zd3k\n(2\u0019)3X\nm6=nf(\"km)\u0000f(\"kn)\n(\"km\u0000\"kn)2Im(vx\nmnvy\nnm):(6)\nHere,\"kmis the single-particle energy at momentum k\nand bandm,v\u000b=1\n~(@Hmf=@k\u000b) are components of the\nvelocity operator with Hmfbeing the self-consistently\n-3-2-1 0 1 2 3 4 5 6\n-3 -2 -1 0 1 2 3-250-200-150-100-50 050\n-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0k =1.9242y\nkz\nσxy cm(Ω )−1 −1\nDoping (δ)(a)\nx(b)\nk =0.0393FIG. 5: (a) Band dispersion of Ba 2FeReO 6in the presence of\ninteractions and spin-orbit coupling plotted along kzat \fxed\n(kx=0:0393;ky=1:9242), showing the nine bands and a pair\nof Weyl nodes (with energy on y-axis in units of t\u0019). The\nparameters used are the same as those for Fig. 3(b), namely\nU= 8t\u0019and\u0015= 2t\u0019. (b) Doping dependence of the intrinsic\nanomalous Hall conductivity \u001bxy.\ndetermined mean-\feld Hamiltonian matrix, and f(:) is\nthe Fermi function.47From the mean \feld solution cor-\nresponding to two electrons per Re, as appropriate for\nBa2FeReO 6, we \fnd that \u001bxyat zero temperature is\nsmall,\u001bxy\u001810\u00003e2\n~dcwheredc\u00198\u0017Ais the lattice con-\nstant in Fig. 1. This translates into \u001bxy\u00185\n\u00001cm\u00001.\nIn order to explore \u001bxyover a larger space of param-\neters, we consider its variation with doping. Rather\nthan simply shifting the chemical potential, we solve the\nHartree mean \feld equations over a range of electron\ndensities, and then compute \u001bxyin the resulting self-\nconsistent band structure. We \fnd that electron dop-\ning does not signi\fcantly enhance the AHE, but a hole\ndoping of about 0 :5-0:8 holes/Re leads to a larger AHE\n\u001bxy\u0018\u0000100\n\u00001cm\u00001to\u0000250\n\u00001cm\u00001. Even this signif-\nicant AHE is small in natural units ( \u00180:1e2\n~dc) atT= 0,\nwhich we attribute to the large spin polarization in the\ncompletely ordered ferromagnet. It is possible that the\nAHE is a non-monotonic function of temperature, peaked\nat some intermediate temperature below the magnetic Tc\neven in the undoped compound.\nThus, in 3d/5d DP materials, spin orbit coupling and\nthe ferrimagnetic order is expected to lead to an intrin-\nsic AHE. The AHE appears likely to be larger for hole\ndoped systems compared to an expected small value for\nBa2FeReO 6and is likely, in Ba 2FeReO 6, to be peaked at\nintermediate temperatures below Tc.\nIV. CONCLUSION\nWe have obtained a tight-binding description of the\nmetallic DPs, including spin-orbit coupling and strong\ncorrelation e\u000bects. Although we have here only applied\nit to Ba 2FeReO 6, \fnding good agreement with a broad\nvariety of experiments and with electronic structure cal-\nculations, our work should be broadly applicable to other\n3d/4d and 3d/5d DP materials as well. Our \fnding that\nstrong correlation e\u000bects are needed to explain many of7\nthe experimental observations also lends partial justi\f-\ncation to our previous theoretical work which modelled\nthe measured spin wave spectrum using a local moment\nmodel. Further theoretical work is needed to study the\nthermal \ructuation e\u000bects of the Fe moments, clarify\nwhat factors control the doping dependence of \u001bxy, and to\nseparate the bulk and surface contributions to the AHE.\nFurthermore, it would be useful to investigate if ferrimag-\nnetic order in fact survives over a wide range of doping\nusing an unbiased numerical approach. In future experi-\nments, it would be useful to test our predictions for the\ndoping dependence of the structural distortion and the\nAHE. Given that most DP materials are in the form of\npowder samples, measuring the AHE and separating the\nintrinsic contribution from extrinsic contributions would\nbe experimentally challenging; nevertheless systematic\ndoping studies of the various properties of such 3d/5d\nDPs would be valuable. 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Millis, K. Ramesha,\nB. Hannoyer, and G. Marest, Phys. Rev. B 62, 9538 (2000).\n42Y. Chen, D. L. Bergman, and A. A. Burkov, ArXiv e-prints\n(2013), 1305.0183.\n43G. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999).\n44N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).\n45T. Fukui, Y. Hatsugai, and H. Suzuki, Journal of the Phys-\nical Society of Japan 74, 1674 (2005).\n46The full set of Weyl nodes - their location, charges, and\ndependence on the direction of the magnetization vector -\nwill be discussed elsewhere. (A. M. Cook, A. A. Burkov,\nand A. Paramekanti, work in progress)\n47An equivalent route to computing \u001bxyis to view the 3D\nband dispersion \"k`as a sequence of 2D band structures pa-\nrameterized by the momentum k3along one direction,26,42\nwriting it as \"[k3]\n`(k1;k2). Each such 2D band can have\na momentum dependent Berry curvature and a nonzero\nChern number, thus yielding a quantum Hall insulator for\na \flled band. Weyl nodes, which act as `monopoles' in mo-\nmentum space - sources or sinks of integer quanta of Berry\n\rux25{27- correspond to quantum Hall transitions in mo-\nmentum space where the Chern number jumps as a func-\ntion ofk3. Integrating the Berry curvature, obtained by\nusing gauge invariant plaquette products of wavefunction\noverlaps,42,45over all the k3slices yields the total \u001bxy.\nAppendix A: Tight binding coe\u000ecients\nWhen we work with the triclinic unit cell, there is one\nFe atom and one Re atom in each unit cell. Going to mo-mentum space, the coe\u000ecients of the tight-binding hop-\nping Hamiltonian Hhopin Eq. 1 have intra-orbital terms\ngiven by\n\u000fxy=\u00002t0(coskxa+coskya) (A1)\n+ 8t00cos(kxa\n2) cos(kya\n2) cos(kzc\n2) (A2)\n\u000fxz= 2t00(coskxa+coskya)+4t00cos(kxa\u0000kya\n2) cos(kzc\n2)\n\u00004t0cos(kxa+kya\n2) coskzc\n2(A3)\n\u000fyz= 2t00(coskxa+coskya)+4t00cos(kxa+kya\n2) cos(kzc\n2)\n\u00004t0cos(kxa\u0000kya\n2) coskzc\n2; (A4)\nand\n\u0011xy= 4t\u0019coskxa\n2coskya\n2\u00002t\u000ecoskzc\n2(A5)\n\u0011xz= 2t\u0019cos(kxa+kya\n2) + 2t\u0019coskzc\n2(A6)\n\u00002t\u000ecos(kxa\u0000kya\n2) (A7)\n\u0011yz= 2t\u0019cos(kxa\u0000kya\n2) + 2t\u0019coskzc\n2(A8)\n\u00002t\u000ecos(kxa+kya\n2): (A9)\nThe intra-orbital terms take the form\n\rxz;yz =\u00002tm(coskxa\u0000coskya) (A10)\n\rxy;yz =\u00004tmsin(kxa+kya\n2) sinkzc\n2(A11)\n\rxy;xz = 4tmsin(kxa\u0000kya\n2) sinkzc\n2: (A12)" }, { "title": "1102.3244v1.Collapse_of_Ferrimagnetism_in_Two_Dimensional_Heisenberg_Antiferromagnet_due_to_Frustration.pdf", "content": "arXiv:1102.3244v1 [cond-mat.str-el] 16 Feb 2011Typeset with jpsj3.cls Letter\nCollapse of Ferrimagnetism in Two-Dimensional Heisenberg Antiferromagnet due to\nFrustration\nHirokiNakano∗, Tokuro Shimokawa , and Toru Sakai1\nGraduate School of Material Science, University of Hyogo, K outo 3-2-1, Kamigori, Ako-gun, Hyogo 678-1297, Japan\n1Japan Atomic Energy Agency, SPring-8, Kouto 1-1-1, Sayo, Hy ogo 679-5148, Japan\n(Received May 13, 2018)\nWe study ferrimagnetism in the ground state of the antiferro magnetic Heisenberg model\non the spatially anisotropic kagome lattice, in which ferri magnetism of the conventional Lieb-\nMattis type appears in the region of weak frustration wherea s the ground state is nonmagnetic\nin the isotropic case. Numerical diagonalizations of small finite-size clusters are carried out to\nexamine the spontaneous magnetization. We find that the spon taneous magnetization changes\ncontinuously in the intermediate region between conventio nal ferrimagnetism and the nonmag-\nnetic phase. Local magnetization of the intermediate state shows strong dependence on the site\nposition, which suggests non-Lieb-Mattis ferrimagnetism .\nKEYWORDS: antiferromagnetic Heisenberg spin model, ferri magnetism, frustration, numerical-\ndiagonalization method, Lanczos method\nFerrimagnetism has been studied extensively as an im-\nportant phenomenon that has both ferromagneticnature\nand antiferromagnetic nature at the same time. One of\nthe fundamental keys to understanding ferrimagnetism\nis the Marshall-Lieb-Mattis (MLM) theorem.1,2)This\ntheorem clarifies some of the magnetic properties in the\nground state of a system when the system has a bipar-\ntite lattice structure and when a spin on one sublattice\ninteracts antiferromagnetically with a spin on the other\nsublattice. Under the condition that the sum of the spin\namplitudesofspinsineachsublatticeisdifferentbetween\nthe two sublattices, one finds that the ground state of\nsuch a system exhibits ferrimagnetism. In this ferrimag-\nneticgroundstate,spontaneousmagnetizationisrealized\nand its magnitude is a simple fraction of the saturated\nmagnetization. We hereafter call ferrimagnetism of this\ntype the Lieb-Mattis (LM) type.\nSome studies in recent years, on the other hand, re-\nported cases when the magnitude of the spontaneous\nmagnetization of the ferrimagnetism is not a simple frac-\ntion of the saturated magnetization.3–9)The ferrimag-\nnetic ground state of this type is a nontrivial quantum\nstate whose behavior is difficult to explain well only\nwithin the classical picture. Ferrimagnetism of this type\nwas first predicted in ref. 10 using the quantum rotor\nmodel. The mechanism of this ferrimagnetism has not\nbeenunderstoodsufficientlyuptonow.Hereafter,wecall\nthis case the non-Lieb-Mattis (NLM) type. Note that in\nthecaseswhenNLMferrimagnetismispresent,thestruc-\nture of the lattices is limited to being one-dimensional.\nRecall that the above conditions of the MLM theorem\ndo not include the spatial dimension of the system; the\nMLM theorem holds irrespective of the spatial dimen-\nsionality. We are then faced with a question: can NLM\nferrimagnetism be realized when the spatial dimension is\nmore than one?\nThe purpose of this letter is to answer the above ques-\ntion concerning the existence of NLM ferrimagnetism\n∗E-mail address: hnakano@sci.u-hyogo.ac.jpin higher dimensions. In this letter, we consider a case\nwhen we introduce a frustrating interaction into a two-\ndimensional lattice whose interactions satisfy the con-\nditions of the MLM theorem. When the frustrating in-\nteraction is small, ferrimagnetism of the LM type sur-\nvives; however, the ferrimagnetism is destroyed with the\nincrease in the frustrating interaction and the system\nfinally becomes nonmagnetic due to the considerably\nlarge frustrating interaction. We examine the behavior\nof the collapse of the ferrimagnetism and the existence\nof an intermediate region between the LM ferrimagnetic\nand nonmagnetic phases by means of the numerical-\ndiagonalization method applied to finite-size clusters.\nOur study of the two-dimensional system successfully\nclarifies the existence of the intermediate phase and cap-\ntures a feature of NLM ferrimagnetism.\nFirst, we explain the model Hamiltonian examined in\nthis letter. The Hamiltonian is given by\nH=/summationdisplay\ni∈A,j∈BJ1Si·Sj+/summationdisplay\ni∈A,j∈B′J1Si·Sj\n+/summationdisplay\ni∈B,j∈B′J2Si·Sj, (1)\nwhereSidenotes an S= 1/2 spin operator at site i.\nSublattices A, B, and B′and the network of antiferro-\nmagnetic interactions J1andJ2are depicted in Fig. 1.\nHere, we consider the case of isotropic interactions. The\nsystem size is denoted by Ns; the saturation magneti-\nzation is Msat=Ns/2. Energies are measured in units\nofJ1; thus, we take J1= 1 hereafter. We examine the\nproperties of this model in the range of 0 < J2/J1≤1.\nNote that in the case of J2= 0, sublattices B and B′\nare combined into a single sublattice; the system satis-\nfies the above conditions of the MLM theorem. Thus,\nferrimagnetism of the LM type is exactly realized in this\ncase. In the case of J2=J1, on the other hand, the\nlattice of the system is reduced to the kagome lattice.\nThe ground state of the system on the kagome lattice\n12 J. Phys. Soc. Jpn. Letter Author Name\nFig. 1. Network of interactions in the system and sublattice s A,\nB, and B′. Black straight lines and green dotted lines denote\ninteractions of J1andJ2, respectively. Open circles at lattice\npoints represent S= 1/2 spins. The system of classical spins in\nthis lattice was studied by Monte Carlo simulations in ref. 1 1.\nwithout a magnetic field is known to be singlet from\nnumerical-diagonalization studies,12–15)which indicates\nthat the ground state is nonmagnetic. One thus finds\nthat LM ferrimagnetism collapses between J2= 0 and\nJ2=J1. Consequently, we survey the region between\nthe two cases.\nNext, we discuss the method we use here, which is nu-\nmerical diagonalization based on the Lanczos algorithm.\nItisknownthatthismethod isnonbiasedbeyondanyap-\nproximations and reliable for many-body problems such\nas the present model. A disadvantage of this method is\nthat the available system sizes are limited to being small\nbecause the dimension of the matrix grows exponentially\nwith respect to the system size. To treat systems that are\nas large as possible, we have developed parallelization in\nour numerical calculations using the OpenMP and MPI\ntechniques, either separately or in a hybrid way.16)\nIn this letter, we treat the finite-size clusters depicted\nin Fig. 2 when the system sizes are Ns= 12,Ns= 24,\nNs= 27, and Ns= 30 under the periodic boundary con-\ndition and Ns= 33 under the open boundary condition.\nNote that each of these clusters forms a regular square\nalthough clusters (b) and (d) are tilted. The next larger\nsize under the condition that a regular square is formed\nisNs= 48, which is too large to handle using the present\nmethod, even when one uses modern supercomputers.\nBefore our numerical-diagonalization results for the\nfinite-sizeclustersarepresented,letusconsiderthedirec-\ntions of the spins in the ground state within the classical\npicture. We here examine the spin directions of classi-\ncal vectors with length Sdepicted in Fig. 3. One ob-\ntains the energy of the spin state with angle θto be\nE/J1=−(2Ns/3)S2[2cos(π−θ)+(J2/J1)cos(2θ)]. This\nexpression of the energy indicates that for J2/J1≤1/2,\nthe state of θ= 0, namely, ferrimagnetism of the LM\ntype, is realized. Thus, the normalized magnetization of\nthis state is M/Msat= 1/3. One finds, on the other\nhand, that for J2/J1>1/2, the lowest-energy state is\nrealized for nonzero θwhenJ1/J2= 2cos( θ) is sat-\nFig. 2. Finite-size clusters: (a) Ns= 12, (b) Ns= 24, (c) Ns=\n27, and (d) Ns= 30 under the periodic boundary condition,\nwherereddashed linesdenote asinglefinite-sizecluster wi theach\nsystem size. Black straight lines and green dotted lines are the\nsame as in Fig. 1. Note that cluster (c) under the open boundar y\ncondition includes Ns= 33 spins.\nFig. 3. Ferrimagnetic spin direction in the classical pictu re.\nisfied. The normalized magnetization of this state is\nM/Msat= (J1\nJ2−1)/3. When J2/J1becomes unity, the\nmagnetization finally vanishes. This classical argument\nwill be compared with our finite-size results obtained\nfrom numerical diagonalizations.\nNow, let us present our numerical results for the quan-\ntum case.First,weshowourdataforthelowestenergyin\neach subspace of Stot\nz, which reveal the magnetization of\nthe systems. Figure 4 depicts our results for the system\nwithNs= 30 depicted in Fig. 2(d). Note that Msat= 15\nin this case. For J2/J1= 0.5, the energies from Stot\nz= 0\ntoStot\nz= 5 are numerically identical, which means that\nM/Msatbecomes 1/3and that ferrimagnetism of the LM\ntype is realized. For J2/J1= 1, the energy for Stot\nz= 0 is\nlower than the other energies for larger Stot\nz. The ground\nstate of this case is nonmagnetic. For J2/J1= 0.6, the\nenergies from Stot\nz= 0 toStot\nz= 2 are the same; thus,\nwe find that the spontaneous magnetization is M= 2,\nwhich is smaller than the value for ferrimagnetism of the\nLM type. One finds that a state with intermediate mag-\nnetization appears between LM-type ferrimagnetismand\nthe nonmagnetic state, at least according to the finite-\nsize calculations.\nNext, we examine the region of such an intermediateJ. Phys. Soc. Jpn. Letter Author Name 3\nFig. 4. Lowest energy in each subspace of Stot\nzfor the system of\nNs= 30 depicted in Fig. 2(d). Results for J2/J1= 1, 0.5, and\n0.6 are presented by black circles, blue triangles, and red s quares,\nrespectively. Inset: our data in the entire range of Stot\nzgiven for\nJ2/J1= 1 and 0.5.\nstate for various system sizes; our results are depicted in\nFig. 5. In the case of Ns= 12, the intermediate state\nFig. 5. Dependence of the spontaneous magnetization normal ized\nby the saturated magnetization on J1/J2. The solid line repre-\nsents the result for the magnetization within the classical picture\nshown in Fig. 3. Note that we take the J1/J2dependence as the\nabscissa because the classical magnetization shows linear depen-\ndence not on J2/J1but onJ1/J2. Results for Ns= 12, 24, 27,\nand 30 under the periodic boundary condition are presented b y\nblack pluses, violet crosses, green squares, and blue diamo nds,\nrespectively. Red circles denote results for Ns= 33 under the\nopen boundary condition.\nbetween LM-type ferrimagnetism and the nonmagnetic\nstate is absent; on the other hand, the intermediate re-\ngion exists for all the largersystems. Note that the width\nof the intermediate region increases for the cases under\nthe periodic boundary condition when Nsis increased.\nThis indicates that the intermediate phase is present in\nthe thermodynamic limit. One of the characteristics ob-\nserved is that the continuity of the magnetization im-\nproves with increasing Ns. In the cases under the open\nboundary condition, we successfully detect the interme-\ndiate phase although its width is relatively smaller. The\nwidth for the Ns= 33 case under the open boundarycondition is close to that for the Ns= 24 case under the\nperiodic boundary condition. This is consistent with the\nfact that there are 21 sites in the inner part of cluster\n(c) ofNs= 33 under the open boundary condition. Our\npresent results for both boundary conditions imply that\nthe presence of the intermediate phase is irrespective of\nthe boundary conditions.\nAn important characteristic of NLM ferrimagnetism is\nthat the local magnetization in sublattice exhibits long-\ndistance periodicity, which is absent in LM-type ferri-\nmagnetism. Note that one cannot detect this periodicity\nin the cases under the periodic boundary condition. We\nthus examine the local magnetization in the intermedi-\nate phase for the case under the open boundary con-\ndition; the results for Ns= 33 are depicted in Fig. 6.\nForJ2/J1= 0.5 with LM-type ferrimagnetism, the local\nFig. 6. Local magnetizations of the cluster with Ns= 33 under\nthe open boundary condition. Results for J2/J1= 0.5, 0.53, and\n0.57 are presented for the sites surrounded by the blue recta ngles\nin the inset. Site numbers 1 to 6 correspond to the sites from l eft\nto right in the blue rectangle.\nmagnetization shows weak dependence on the position\nof sites, although /angbracketleftSz\ni/angbracketrightat edge sites 1 and 6 is slightly\nlargerthan those at interiorsites, wherethe site numbers\nare illustrated in the inset of Fig. 6. This small difference\noriginates from the edge effect due to the open bound-\nary condition. For J2/J1= 0.5, the edge effect does not\nseem to affect /angbracketleftSz\ni/angbracketrightat internal sites. For J2/J1= 0.53\nand 0.57, on the other hand, /angbracketleftSz\ni/angbracketrightat edge sites 1 and 6\nbecomes very small. The behavior of this appearance of\nthe edge effect is different from the case of J2/J1= 0.5.\nForJ2/J1= 0.53and0.57,onefindsastrongdependence\nof/angbracketleftSz\ni/angbracketrightonthe position ofthe site fromsite 2to site 5.For\nJ2/J1= 0.53,/angbracketleftSz\ni/angbracketrightat sites next to the edges seems to be\naffected bythe edge sites.It is unclearwhetherornot the\ncase ofJ2/J1= 0.53 corresponds to NLM type ferrimag-\nnetism at present. For J2/J1= 0.57, on the other hand,\nthe strong dependence on the site position suggests the\nexistenceoforiginsthataredifferentfromthe edgeeffect.\nThe system size Ns= 33 is sufficiently small for long-\ndistance periodicity to be observed clearly. Although the\npresent results are not decisive evidence of the periodic-\nity, our finding of the large change in /angbracketleftSz\ni/angbracketrightis considered\naspossible evidence. In orderto obtaindecisiveevidence,4 J. Phys. Soc. Jpn. Letter Author Name\ncalculationsonsystemsoflargersizesarerequired,which\nare unfortunately difficult at the present time. Instead of\nthe present two-dimensional kagome case, we are now\nexamining a quasi-one-dimensional system on a kagome\nstripe lattice. Both systems partly share the same lat-\ntice structure. The system on the kagome stripe lattice\nreveals the clear appearance of NLM ferrimagnetism in\ntheintermediateregion.18)Resultswillbepublished else-\nwhere.\nThe phenomenon of ground-state magnetization\nchanging continuously with respect to a continuous pa-\nrameter in a model Hamiltonian has been reported in\nother cases. Tonegawa and co-workers reported such a\nphenomenon in spin systems with anisotropic interac-\ntions.19–23)It is unclear at present whether or not the\nstates of this continuouslychanging magnetization in the\nanisotropic case show long-distance periodicity because\nthe behavior of local magnetization has not been inves-\ntigated yet. Since this phenomenon disappears in the\nisotropic case when the quantum effect is stronger than\nthat in the anisotropic case, this phenomenon is con-\nsidered to arise from the anisotropy. From this point of\nview, the origin of this phenomenon seems to be different\nfrom that of intermediate ferrimagnetism in the isotropic\ncase studied here. Another reported phenomenon is par-\ntialferromagnetismintheHubbardmodel24,25)whenthe\nsystem is hole-doped near the half-filled Mott insulator.\nThe origin of this phenomenon has been clarified to be\nthe formation of spin polarons around doped holes. The\nmechanismofthesetwocasesisdifferentfromthepresent\ncase of NLM ferrimagnetism.\nFinally, we briefly discuss possible future experiments.\nFor volborthite, eq. (1) was proposed as a model Hamil-\ntonian from the argument of its crystal structure,26,27)\nalthough NLM ferrimagnetism has not yet been ob-\nservedin this material.A theoreticalstudy on the spatial\nanisotropy of this material indicated that the deviation\nof the anisotropy from the isotropic kagome point is not\nparticularly large.28)This is consistent with our present\nresult because the nonmagnetic ground state is realized\naround the region of weak anisotropy as shown in Fig. 5.\nIn order to observe NLM ferrimagnetism experimentally,\nit is necessary to realize a case with larger anisotropy.\nThe measurement of volborthite under high pressure in\nthe direction of the a-axis or discoveriesof new materials\nmight lead to such an observation.\nIn summary, we have clearly shown the existence of a\ngroundstateofnon-Lieb-Mattistypeferrimagnetismina\ntwo-dimensional lattice that lies between the well-known\nLieb-Mattis type ferrimagnetic phase and the nonmag-\nnetic phase including the kagome-lattice system. The\nnontrivial ferrimagnetism we have found in the interme-\ndiate phase occurs as a consequence of magnetic frustra-\ntion. Our present result indicates that non-Lieb-Mattis\nferrimagnetism is a general phenomenon irrespective of\nthe spatial dimensionality.\nAcknowledgments\nWe wish to thank Professor K. Hida, Profes-\nsor T. Tonegawa, Professor S. Miyashita, ProfessorM. Imada, and Dr. Y. Okamoto for fruitful discus-\nsions. This work was partly supported by a Grant-in-Aid\n(No. 20340096)from the Ministry of Education, Culture,\nSports, Science and Technology of Japan. This work was\npartly supported by a Grant-in-Aid (No. 22014012) for\nScientific Research and Priority Areas “Novel States of\nMatter Induced by Frustration” from the Ministry of\nEducation, Culture, Sports, Science and Technology of\nJapan. Nonhybrid thread-parallel calculations in the nu-\nmerical diagonalizations were based on TITPACK ver.2,\ncoded by H. Nishimori. Part of the computations were\nperformed using the facilities of Information Technology\nCenter, Nagoya University; Department of Simulation\nScience, National Institute for Fusion Science; and the\nSupercomputer Center, Institute for Solid State Physics,\nUniversity of Tokyo.\n1) W. Marshall: Proc. R. Soc. London, Ser. A 232(1955) 48.\n2) E. Lieb and D. Mattis: J. Math. Phys. 3(1962) 749.\n3) N. B. Ivanov and J. Richter: Phys. Rev. B 69(2004) 214420.\n4) S. Yoshikawa and S. Miyashita: Proc. Statistical Physics\nof Quantum Systems: novel orders and dynamics,\nJ. Phys. Soc. Jpn. 74(2005) Suppl., p.71.\n5) K. Hida: J. Phys.: Condens. Matter 19(2007) 145225.\n6) K. Hida and K. Takano: Phys. Rev. B 78(2008) 064407.\n7) R. R. Montenegro-Filho and M. D. Coutinho-Filho:\nPhys. Rev. B 78(2008) 014418.\n8) K. Hida, K. Takano, and H. Suzuki: J. Phys. Soc. Jpn. 79\n(2010) 114703.\n9) T. Shimokawa and H. Nakano: to be published in\nJ. Phys. Soc. Jpn. 80(2011) No.4.\n10) S. Sachdev and T. Senthil: Ann. Phys. (N.Y.) 251(1996) 76.\n11) R. Kaneko, T. Misawa, and M. Imada: J. Phys. Soc. Jpn. 79\n(2010) 073708.\n12) P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and\nP. Sindzingre: Phys. Rev. B 56(1997) 2521.\n13) Ch. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier,\nP. Sindzingre, P. Lecheminant, and L. Pierre: Eur. Phys. J. B\n2(1998) 501.\n14) K. Hida: J. Phys. Soc. Jpn. 70(2001) 3673.\n15) H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 79(2010) 053707.\n16) This hybrid code was originally developed in a study on th e\nestimate of the Haldane gap.17)\n17) H. Nakano and A. Terai: J. Phys. Soc. Jpn. 78(2009) 014003.\n18) T. Shimokawa and H. Nakano: submitted to\nJ. Phys.: Conf. Series.\n19) T. Tonegawa, I. Harada, and J. Igarashi:\nProg. Theor. Phys. Suppl. 101(1990) 513.\n20) I. Harada and T. Tonegawa: J. Magn. Magn. Mater. 90-91\n(1990) 234.\n21) T. Tonegawa, H. Matsumoto, T. Hikihara, and M. Kaburagi:\nCan. J. Phys. 79(2001) 1581.\n22) T. Tonegawa and M. Kaburagi: J. Magn. Magn. Mater. 272-\n276(2004) 898.\n23) M. Kaburagi, T. Tonegawa, and M. Kang: J. Appl. Phys. 97\n(2005) 10B306.\n24) H. Nakano and Y. Takahashi: J. Phys. Soc. Jpn. 72(2003)\n1191.\n25) H. Nakano and Y. Takahashi: J. Phys. Soc. Jpn. 73(2004)\n983.\n26) M. A. Lafontaine, A. L. Bail, and G. F´ erey:\nJ. Solid State Chem. 85(1990) 220.\n27) Z. Hiroi, M. Hanawa, N. Kobayashi, M. Nohara, H. Takagi,\nY. Kato, and M. Takigawa: J. Phys. Soc. Jpn. 70(2001) 3377.\n28) P. Sindzingre: arXiv:0707.4264." }, { "title": "1310.5170v2.Ultrafast_thermally_induced_magnetic_switching_in_synthetic_ferrimagnets.pdf", "content": "Ultrafast thermally induced magnetic switching in synthetic ferrimagnets\nRichard F. L. Evans,1,\u0003Thomas A. Ostler,1Roy W. Chantrell,1Ilie Radu,2and Theo Rasing3\n1Department of Physics, University of York, Heslington, York YO10 5DD United Kingdom.\n2Institut f ur Methoden und Instrumentierung der Forschung mit Synchrotronstrahlung,\nHelmholtz-Zentrum Berlin f ur Materialien und Energie,\nGmbH, Albert-Einstein-Stra\u0019e 15, 12489 Berlin, Germany\n3Radboud University Nijmegen, Institute for Molecules and Materials,\nHeyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.\nSynthetic ferrimagnets are composite magnetic structures formed from two or more anti-\nferromagnetically coupled magnetic sublattices with di\u000berent magnetic moments. Here we report on\natomistic spin simulations of the laser-induced magnetization dynamics on such synthetic ferrimag-\nnets, and demonstrate that the application of ultrashort laser pulses leads to sub-picoscond magne-\ntization dynamics and all-optical switching in a similar manner as in ferrimagnetic alloys. Moreover,\nwe present the essential material properties for successful laser-induced switching, demonstrating\nthe feasibility of using a synthetic ferrimagnet as a high density magnetic storage element without\nthe need of a write \feld.\nThe dynamic response of magnetic materials to ultra-\nshort laser pulses is currently an area of fundamental\nand practical importance that is attracting a lot of atten-\ntion. Since the pioneering work of Beaurepaire et al [1] it\nhas been known that the magnetization can respond to\na femtosecond laser pulse on a sub-picosecond timescale.\nHowever studies of magnetic switching are more recent.\nIn this context an especially intriguing phenomenon is\nthat of all-optical switching, which uses the interaction of\nshort, intense pulses of light with a magnetic material to\nalter its magnetic state without the application of an ex-\nternal magnetic \feld[2, 3]. Recent experiments [4{6] and\ntheoretical calculations[5, 7{9] have demonstrated that\nthe origin of all-optical switching in ferrimagnetic alloys\nis due to ultrafast heating of the spin system. The mag-\nnetic switching arises due to a transfer of angular momen-\ntum between the two sublattices within the material[7, 8]\nand the resulting exchange-\feld induced precession[7].\nRemarkably, this e\u000bect occurs in the absence of any sym-\nmetry breaking magnetic \feld [5], and can be considered\nas Thermally Induced Magnetic Switching (TIMS). So\nfar TIMS has only been demonstrated experimentally in\nthe rare-earth transition metal (RE-TM) alloys GdFeCo\nand TbCo which, in addition to their strong magneto-\noptical response, have two essential properties for heat-\ninduced switching: antiferromagnetic coupling between\nthe RE and TM sublattices[10] and distinct demagneti-\nzation times of the two sublattices[4]. The antiferromag-\nnetic coupling allows for inertial magnetization dynam-\nics, while the distinct demagnetization times under the\naction of a heat pulse allow a transient imbalance in the\nangular momentum of the two sublattices, which initi-\nates a mutual high speed precession enabling ultrafast\nswitching to occur.\nAlthough GdFeCo has excellent switching properties,\nits potential use in magnetic data storage is limited by\nits low anisotropy and amorphous structure, precluding\nthe use of single magnetic domains typically less than10 nm in size, required for future high density magnetic\nrecording media. One intriguing possibility, and the fo-\ncus of this paper, would be the use of a synthetic fer-\nrimagnet (SFiM), consisting of two transition metal fer-\nromagnets anti-ferromagnetically exchange coupled by a\nnon-magnetic spacer[11], shown schematically in Fig 1.\nThe important but as yet unanswered question is whether\nall-optical switching would also work in such an arti\fcial\nstructure and what essential physical properties of the\ndesign are required. Such a composite magnet also has a\nnumber of distinct advantages over intrinsic rare-earth-\ntransition metal ferrimagnets: the dynamic properties of\neach sublattice may be separately selected by choice of\nmaterial, nano-patterning is possible in the sub-10 nm\nsize range due to their crystalline nature and the omis-\nsion of costly rare-earth metals. Importantly the compos-\nite design has the advantage of allowing the use of high\nanisotropy materials such as FePt or CoPt to enhance\nthe thermal stability of the medium. These advantages\ncould make such synthetic structures very promising can-\ndidates for magnetic data storage applications.\nIn this letter we present dynamic studies of such a syn-\nthetic ferrimagnet using an atomistic spin model. We in-\nvestigate the dynamic properties of the separate layers\nand show that the demagnetization time is determined\nprimarily by the local atomic spin moment and the in-\ntrinsic Gilbert damping of the material. We \fnally con-\nsider an exchange-coupled Fe/FePt synthetic ferrimagnet\nand show that a short heat-pulse is su\u000ecient to induce\nultrafast heat-induced switching of the material.\nThe dynamic properties of the SFiM are studied us-\ning an atomistic spin model using the vampire software\npackage[12, 13]. The energetics of the system are de-\nscribed using a Heisenberg spin Hamiltonian, which in\ncondensed form reads:\nH=\u0000X\niβ in the following discussion. \nThe rest of the paper is arranged as follows. In Se ct. 2, we determine the quantum \nphase diagram of the model by the ED method. In Sec t. 3, CCM is used to discuss the \nproperty of the model. In Sect. 4, the CCM results are shown. A summary is given in \nSect. 5. \n2. Quantum Phase Diagram of the Alternating Bond Di amond Chain \nTo check the results of the CCM in the next section , we first give the quantum \nphase diagram of the ABDC by ED. Firstly, we consid er some special cases of the \nHamiltonian (1). The first one is the case of 0 =α . In this case, the Lieb-Mattis \ntheorem implies that the magnitude of the total spin of the GS of the Hamilt onian (1), \ndefined by ∑\n==N\nll tol s s\n1rr, is 6 / N 50) . The ABDC possesses ferrimagnetic long-range \norder, and it is equivalent to an alternating bond antiferromagnetic mixed spin (1, 1/2) \nHeisenberg chain 51, 52) . It is reasonable to expect that the ferrimagnetic long-range \norder will extend to a small- α-parameter region. In contrast, in the limiting cas e of \n∞→α , the spins 1 3−isr and is3r form a singlet dimer and the GS is the disordered \ndimer state. As all spins 2 3−isr are decoupled from each other in the dimer state, there \nis a 3 /2N-fold degeneracy in that state. The GS energy per u nit cell for the dimer state \nis given by \nα75 . 0D− = e . (1) \n(2) \n 4The case of 1=α and 1=β is the third one that we will discuss. The system reduces \nto a uniform diamond chain (UDC) in that case 14, 53) . The spin cluster state called the \nTD state is the exact GS of UDC when 1 =α 14, 18) . As displayed in Fig. 2, the \nquadruplets 2 3−isr, 1 3−isr, is3r, and 1 3+isr (1 3+isr, 2 3+isr, 3 3+isr, and 4 3+isr) of spins form \nsinglet tetramers, and the pairs 2 3+isr and 3 3+isr (1 3−isr and is3r) of spins construct \nsinglet dimers in the TD state. Let us define the f ollowing composite spin operators: 16, \n21, 53) \n3 2 3 1 3 3 1 i i i i i q s s s s − − + = + + + r r r r r , \n3 1 3 i i i t s s −= + rr r . \nThen, the TD state can be represented as \n) (\n21 2\nTD1\nTD TD ψψ ψ ± =±, \nwhere 1\nTDψ and 2\nTDψ are degenerate and they have the forms \n∏∏\n=−=− −\n=== ==== =\n6 /\n12 2 1 22\nTD6 /\n12 1 2 1 21\nTD\n1 , 0 , 00 , 1 , 0\nN\nii i iN\nii i i\nt q tt t q\nψψ\n \nwhere \n) 2 2(\n121) (\n211, 0 , 0) (\n21) 2 2(\n1210 , 1 , 0\n4 3 3 3 2 3 1 3 4 3 3 3 2 3 1 34 3 3 3 2 3 1 3 4 3 3 3 2 3 1 3 4 3 3 3 2 3 1 34 3 3 3 2 3 1 3 3 1 3 3 1 3 2 2 1 23 3 2 3 3 3 2 31 3 3 1 3 2 3 1 3 3 1 3 2 3 1 3 3 1 3 2 3 1 3 3 1 3 2 31 3 3 1 3 2 3 1 3 3 1 3 2 3 2 1 2 1 2\n++++ ++++++++ ++++ ++++++++ − − −++ +++ −− + −− + −− + −−+ −− + −− − −\n↑↓↓↑+↓↑↑↓+↑↓↑↓−↓↑↓↑−↑↑↓↓−↓↓↑↑ −⊗↑↓−↓↑====↑↓−↓↑⊗↑↓↓↑+↓↑↑↓+↑↓↑↓−↓↑↓↑−↑↑↓↓−↓↓↑↑ −====\ni i i i i i i ii i i i i i i i i i i ii i i i i i i i i i ii i i ii i i i i i i i i i i i i i i ii i i i i i i i i i i\nt q tt t q\n, \nwhere ↑ and ↓ are the zs eigenstates. If the TD state is the exact GS of th e \nsystem, it is easy to obtain that the GS energy per unit cell is (3) \n, (5) \n(6) (4) \n 5βα 5 . 0 25 . 0 5 . 0TD −−− = e . \nCompare Eq. (2) with Eq. (7), and you will find tha t the dimer state may be the GS of \nthe Hamiltonian (1) only when βα+>1 . The case of ∞→β is the last special one \nthat needs to be discussed. In that case, the three spins 1 3−isr, is3r, and 1 3+isr form a \ntrimer. The GS wave functions of the ith trimer are \n( ) ) 2 / 1 ( 2\n61\n1 3 3 1 3 1 3 3 1 3 1 3 3 1 3 = ↑↑↓+↓↑↑−↑↓↑=+ − + − + −+ z\ntol i i i i i i i i i i s φ , \n( ) 3 1 3 3 1 3 1 3 3 1 3 1 3 3 1 12 ( 1/ 2) \n6z\ni i i i i i i i i i tol s φ−\n− + − + − + = ↓ ↑ ↓ − ↓ ↓ ↑ + ↑ ↓ ↓ =− , \nwhere z\ntol s is the z-component of the total spin of the ith trimer. Using the \npseudo-operator iTr\n with the magnitude 1/2, one can express Eqs. (8) a nd (9) as 54, 55) \n−+\n=⇓=⇑\ni ii i\nφφ\n, \nwhere i⇑ and i⇓ denote the eigenstate of z\niT with the eigenvalue 1/2 and the \neigenstate of z\niT with the eigenvalue -1/2, respectively. In the fou rth special case, the \n1J terms of the Hamiltonian (1) can be treated as per turbations. By using the \nfirst-order perturbation theory with respect to 1J, one can obtain the effective \nHamiltonian \n13 /\n1194\n+\n=∑⋅ − =iN\nii eff TT J Hrr\n. \nThis result means that the GS of the Hamiltonian (1 ) is also in the ferrimagnetic state \nin the case of ∞→β , just as in the first case discussed above. \n Next, we determine the phase diagram by ED. As 2\nitr, defined by ) 1 (2+=i i i t t tr\n, \ncommutes with the Hamiltonian H, we have a sequence of good quantum numbers \n} ,, , {3 / 2 1 N itt t tLL . Thus, the GS of the ABDC belongs to one of the su bspaces that are (7) \n(11) (8) \n(10) (9) \n 6specified by } ,, , {3 / 2 1 N itt t tLL 22). As the magnitude of the composite spin itr is 0 or \n1, the correlation function between the spin pairs 1 3−isr and is3r takes a value of -0.75 \nor 0.25. One can then calculate the short-range cor relation function >⋅<− i is s3 1 3rr to \ndetermine the phase diagram of the ABDC. Our ED res ults show that the value of \n>⋅<− i is s3 1 3rr is equal to 0.25, -0.25, or -0.75 in the entire pa rameter region. Thus, as \nshown in Fig. 3, the GS phase diagram of the ABDC i s composed of the ferrimagnetic \nstate, TD state, and dimer state. Finite-size effec ts on the position of the phase \nboundary are very minimal, as can be seen from the comparison of the results for \nN=12 and 30. At 1=β , the ED results show that two critical points sepa rate the TD \nstate from the (i) ferrimagnetic state and (ii) dim er state. For a system with N=24, our \nresults show that the two critical points are 909. 0=α and 2 respectively, which are \nconsistent with those given in Ref. (14). Fig. 3 sh ows that, as expected above, the \nABDC possesses the ferrimagnetic long-range order i n the small- α or large- β \n-parameter region. Moreover, the ferrimagnetic stat e is always the GS of the chain if \nα is less than a certain critical value TDα. TDα for a system with 30=N is \nshown in Fig. 3. When the parameter α exceeds that critical point, the TD phase \nappears in the phase diagram and it exists in a fin ite-parameter region. Besides TDα, \nthe other critical value is Dα ( 1D=α ), beyond which the dimer phase is also \nincluded in the phase diagram. The straight line 1−=αβ in Fig. 3 represents the \nexact boundary between the TD state and the dimer s tate. \n3. Coupled Cluster Method Applied to the Alternatin g Bond Diamond Chain \nIn this section, we discuss the properties of the q uantum TD state, dimer state, and \nferrimagnetic state of the system determined by CCM . Since details of the CCM \napplied to quantum spin systems have been given els ewhere 23, 26, 27) , we present \nonly a brief description of the method that we used to treat the ABDC. \nWe first describe how we analyze the properties of the TD state by CCM. The \n 7starting and key point for a CCM calculation is to choose a suitable normalized \nreference state φ. In the past, people often chose the classical sta te or the quantum \nstate (such as the dimer state) of the spin systems as the reference state of CCM to \ninvestigate the properties of the spin cluster stat e 8, 24) . Since the singlet tetramer and \nsinglet dimer appear along the chain alternately in the TD state, we use the collinear \nstate as shown in Fig. 1(a), but not the two types of state mentioned above, as the \nCCM reference state. As neighboring spins in the A and B sublattices are aligned \nparallel, whereas those in the C sublattice are aligned antiparallel, that referenc e state \nis also called the ferromagnetic-ferromagnetic-anti ferromagnetic (FFA) state in the \nfollowing discussion for convenience. After carryin g out a mathematical rotation of \nthe local axes of all the “up” spins: x xs s−→ , y ys s→ , and, z zs s−→ , all the \nspins in the reference state align along the negati ve z-axis. The reference state is then \ngiven by \nL L ⊗↓⊗↓⊗↓⊗↓⊗↓⊗↓=+ + + − − 3 3 2 3 1 3 3 1 3 2 3 i i i i i i φ , \nand the CCM parameterizations of the ket and bra GS s of model (1) are expressed as \n26, 27) \n1 2 1 2 \n1 2 , , \n1 , , ,\nl l \nlN\nS\ni i i i i i \nl i i i e S S s s s ψ φ + + + \n== = ∑ ∑ L\nLL , \n1 2 1 2 \n1 2 , , \n1 , , , 1 \nl l \nlN\nS\ni i i i i i \nl i i i Se S S s s s ψ φ − − − − \n== = + ∑ ∑ L\nL% % % % L . \nBecause it is impossible to consider all the spin c onfigurations in the S and S~ \ncorrelation operators in practice, we use the well- established LSUB n approximation \nscheme to truncate the expansions of S and S~ 26, 27) . Within the LSUB n \napproximation, only the configurations, including n or fewer correlated spins that \nspan a range of no more than n contiguous lattice sites, are taken into account. In this \npaper, we assume that the two sites are contiguous if they are connected by 1J, 2J, \nor 3J bonds. Although the number of fundamental configur ations contained in the (12) \n(13) \n 8LSUB n approximation grows rapidly with respect to the tr uncation index n, it can be \nreduced if we use the lattice symmetries and conser vation laws that pertain to the \nHamiltonian. Obviously, the LSUB n approximation becomes exact in the limit \n∞→n . \nNow, one can prove that the exact TD state of the H amiltonian (1) can be produced \nby CCM with the FFA reference state. If all correla tion coefficients contained in the \nket-state correlation operator S except those displayed in Fig. 4(a) \nare set equal to zero, S is reduced to \n∑ ∑ ∑ ∑\n=+\n++\n+\n=+\n+++\n++\n−\n=++\n−+\n−+\n−\n=+\n+++\n−+\n− ++ ++ + =6 /\n13 3 2 3 26 /\n11 3 3 1 3 1 3 26 /\n13 2 3 1 3 2 3 26 /\n11 3 3 1 3 2 3 4 ) ( ) (N\nii icN\nii i i ibN\nii i i iaN\nii i i i ss S s s ss S ss ss S s s ss SS . \nThe ket GS of model (1) is then given by \nLL\n⊗↑↑+↓↓⊗↑↓↑↓+↑↑↓↓+↓↑↓↑+↓↓↑↑+↑↑↑↑+↓↓↓↓⊗==\n++ +++ −− + −− + −−+ −− + −− + −−\n] [][\n3 3 2 3 2 3 3 2 31 3 3 1 3 2 3 2 1 3 3 1 3 2 3 2 1 3 3 1 3 2 3 21 3 3 1 3 2 3 2 1 3 3 1 3 2 3 4 1 3 3 1 3 2 3\ni ic\ni ii i i ib\ni i i ib\ni i i iai i i ia\ni i i i i i i iS\nSS S SS S eφψ\n. \nBy “re-rotating” the local axes of spins that point upward in the FFA reference state, \none can obtain the following state: 8) \nLL\n⊗↓↑−↑↓⊗↓↓↑↑−↓↑↓↑−↑↑↓↓−↑↓↑↓−↓↑↑↓+↑↓↓↑⊗=\n++ +++ −− + −− + −−+ −− + −− + −−\n] [][\n3 3 2 3 2 3 3 2 31 3 3 1 3 2 3 2 1 3 3 1 3 2 3 2 1 3 3 1 3 2 3 21 3 3 1 3 2 3 2 1 3 3 1 3 2 3 4 1 3 3 1 3 2 3\ni ic\ni ii i i ib\ni i i ib\ni i i iai i i ia\ni i i i i i i i\nSS S SS S ψ\n. \nIt is obvious that an exact TD state is given by 14=S , 5 . 02 2==b aS S , and 12=cS . \nMoreover, the GS energy per unit cell of the Hamilt onian (1) is written as \nβα α β φφ 5 . 0 25 . 05 . 0 25 . 0)5 . 0 25 . 0 ( )5 . 0 25 . 0 (3\n2 2 2 TD −−− = −−−+−−= =− c b a S SS S S HeeNe . \nIt is in agreement with Eq. (7). \n Next, we prove that the exact dimer state can als o be constructed within the CCM. \nTo achieve this goal, the state shown in Fig. 1(b) is chosen to be the CCM reference \nstate and we call it the ferromagnetic-ferromagneti c-ferromagnetic-I (FFFI) state. If \nonly the ket-state correlation coefficient shown in Fig. 4(b) is not equal to zero, one \ncan obtain the following ket-state within the CCM: (16) \n(17) (15) (14) \n 9L L ⊗↓↑−↑↓⊗↓⊗=− − − ] [3 1 3 2 3 1 3 2 3 i i i i i S ψ . \nApparently, the above state is only the dimer state if 12=S . The GS energy per unit \ncell of the Hamiltonian (1) obtained by CCM based o n the FFFI reference state is \nα α 75 . 0 )5 . 0 25 . 0(2 D − = −−= S e . \nIt is obvious that Eq. (19) is the same as Eq. (2). \nAlthough the above two short-range correlated spin cluster states are the exact GS \nof the Hamiltonian (1), the exact solution to the f errimagnetic state cannot be obtained \nowing to quantum fluctuation. In the ferrimagnetic state, a pair of 1 3−isr and is3r \nforms a triplet dimer, and the magnitude of the total spin of that state is 6 /N as \nmentioned above. Thus, we choose the state displaye d in Fig. 1(c) as the CCM \nreference state to analyze the properties of the fe rrimagnetic state. For convenience, \nthe above reference state is also called the \nferromagnetic-ferromagnetic-ferromagnetic-II (FFFII ) state in the following \ndiscussion. We also calculate, aside from the GS en ergy, the typical physical quantity \nof the ferrimagnetic state, that is, the sublattice magnetizations AM and C BM+ \nusing \n/3 \n3 2 \n11\n/ 3 N\nz\nA i \niM s Nψ ψ −\n==− ∑% , \n/3 /3 \n3 1 3 \n1 1 1 1 \n/ 3 / 3 N N \nz z \nB C i i \ni i M s s N N ψ ψ ψ ψ + − \n= = =− − ∑ ∑ % % . \nCCM can be well applied to investigating the proper ties of the lowest-lying excited \nstate as well as the GS. The excited state wave fun ction eψ is determined after \napplying an excitation operator eX linearly to the ket-state wave function. It is giv en \nby 26) \n∑ ∑\n=++ += =N\nl i i ii i i i i ie s e\ne\nll ls s s X eX\n1 , ,, ,\n2 12 1 2 1,\nLLL χ φ ψ . \nAnalogous to the GS, the LSUB n approximation scheme is also used to truncate the (21) (20) (18) \n(19) \n 10 expansion of the operator eX. One can then use CCM to calculate the spin gap Δ of \nthe spin systems. It is given by the lowest eigenva lue of the following LSUB n \neigenvalue equation: 26) \nφ φ χS e S\ni i ie\ni i i eXHe s s s\nl l],[\n2 1 2 1, ,− − − −= Δ LL . \n We calculate two types of spin gap by CCM in the present paper. One is the \nsingle-triplet energy gap STΔ, which is a representative physical quantity of th e \nHamiltonian (1) if its GS is the TD state. The spin gap STΔ is defined as \ng tol ST E sE −==Δ ) 1 (1 , \nwhere 1E and gE respectively denote the energy of the lowest-lying state with \n1=tols and the GS energy. In the ferrimagnetic phase, the ABDC possesses an \nantiferromagnetic character as well as a ferromagne tic character 51, 52) . Thus, the other \nspin gap determined by CCM is the antiferromagnetic gap, which is given by \ng tol N AF E N s E −+= =Δ+ ) 16 / (1 6 / , \nwhere 1 6 /+ NE and gE are the energy of the lowest-lying state with 1 6 /+=N stol \nand the energy of the GS, respectively. \n4. Results of the Coupled Cluster Method \nFirstly, we present our CCM results for the GS. As TD and the dimer state are the \nexact GSs of the Hamiltonian (1), we focus on the p roperties of the ferrimagnetic state. \nWhen 0=α and 1=β , the property of the Hamiltonian (1) is exactly eq uivalent to \nthat of the one-dimensional Heisenberg ferrimagneti c spin chain which has been \ninvestigated by various analytical and numerical me thods 51, 56-58) . Table I shows the \nresults of CCM in that case. One can find that CCM results for the GS physical \nquantities, such as the GS energy per unit cell and the sublattice magnetization, \nconverge very rapidly with an increase in the level of approximation. This \nphenomenon would be related to the short correlatio n length of the one-dimensional \nHeisenberg ferrimagnetic spin chain 58) . As a result, the energy per unit cell e and (22) \n(24) (23) \n 11 the sublattice magnetization AM given by CCM at the LSUB12 level of \napproximation are in agreement with four decimal pl aces with the best results of the \nnumerical method, namely, those of the DMRG method 56) . For clarity, we only show \nthe results of the above physical quantities at the LSUB12 level of approximation in \nthe following discussion 59) . To check the results of CCM, we also calculated t hose \nphysical quantities by ED and found that the result s obtained by ED also converge \nextremely fast. The physical quantities for a syste m with 30=N are shown in Table I. \nIt can be seen that they are very close to those of DMRG. Therefore, in the following \npart, the results of ED are also given for 30=N sites for comparison with those of \nCCM. \nFig. 5 shows the GS energy e as a function of β given by CCM on the basis of \nthe FFFII reference state for three distinct values of the parameter α: 0=α , 0.85 , \nand 1. It can be found that the CCM results are in good agreement with those of ED \nin all cases. e decreases monotonically with increasing in β in the first case, \nwhereas in the second (third) case, the energy dete rmined by CCM on the basis of the \nFFFII reference state and that given by Eq. (7) int ersect at two critical points (one \npoint). This finding proves that the transition bet ween the TD state and the \nferrimagnetic state belongs to the first-order tran sition. By CCM, we have obtained \nthe critical points at which the GS of the Hamilton ian (1) evolves from the \nferrimagnetic state to the TD state for any other p arameter α greater than TDα. \nThey are presented in Fig. 6. One can see that the boundary line between the TD state \nand the ferrimagnetic state determined by CCM and t hat obtained by ED for a system \nwith 30=N almost overlap. \nThe results for the sublattice magnetizations AM and C BM+ when 0=α are \npresented graphically in Fig. 7. As seen in that fi gure, the sublattice magnetization \ngiven by CCM coincides fairly well with that obtain ed by ED across the entire \nparameter range. AM and C BM+ both experience growth with the increase in β in \n 12 the region 1 0<<β . At 1=β , they reach their maximum at the same time. \nAfterwards, they decrease with further increase in β. The reason for the evolution of \nsublattice magnetizations with the parameter β is that the increase in 1 −β (β−1 ) \nhelps every three spins 1 3−isr, is3r, and 1 3+isr (2 3−isr, 1 3−isr, and is3r) form a trimer. As a \nresult, the magnetic long-range order of the Hamilt onian (1) is strongest when 1 =β . \nNext, we present CCM results for the single-triplet energy gap STΔ and the \nantiferrimagnetic gap AFΔ, using 1=α as an example. In that case, whether the GS \nof the ABDC is in the TD state or ferrimagnetic sta te depends on whether the \nparameter β is located in the region 18 . 1 0<<β or 18 . 1>β . To check the \nresults of CCM, STΔ and AFΔ were also obtained by ED. \nIn Fig. 8, the ED results for STΔ are displayed when 18 . 1 0<<β . One can find \nthat the single-triplet gap of a finite system with 12≥N reaches its value in the \nthermodynamic limit in the entire parameter region. Fig. 9 shows the single-triplet gap \nSTΔ given by CCM. Apparently, CCM LSUB n results for STΔ converge rapidly \nwith an increase in n, and the spin gap in the limit ∞→n is determined by CCM if \n10≥n . Results of the spin gap STΔ in some cases are shown in Table II. As seen in \nFig. 9 and Table II, the spin gap STΔ given by CCM with 10 ≥n equals that \nobtained by ED in the entire parameter region. The results of CCM and ED both show \nthat the single-triplet gap obviously appears when 0>β , and increases with β in \nthe region 1 0<<β . When 1 =β , it reaches its maximum. Although it then \ndecreases with an increase in β, it does not vanish when 18 . 1 <β . Hence, our \ncurrent findings as well as the results of previous research indicate that the TD state is \ngapful 6, 60) . \nFinally, we turn to our CCM results for the antifer romagnetic gap AFΔ. The AFΔ \n 13 values for 0=α and 1=β obtained from CCM are listed in Table I. One can s ee \nthat our CCM results for AFΔ are highly converged. The antiferromagnetic gap AFΔ \ngiven by CCM at the LSUB12 level of approximation i s in agreement up to three \ndecimal places with that obtained by QMC 51) . The antiferromagnetic gap is plotted as \na function of β in Fig. 10 when 1 =α and 18 . 1>β . AFΔ values obtained by ED \nfor a system with 30 =N are also displayed in that figure for comparison w ith the \ncorresponding CCM data. One can find that AFΔ increases with increasing in β, \nalthough the rate of increase gradually decreases. The size of the antiferromagnetic \ngap obtained by CCM is in good agreement with that given by ED in the entire \nparameter region. Thus, CCM can also be used to acc urately analyze the lowest-lying \nexcited-state properties of the ABDC. \n5. Conclusions \nIn this paper, the CCM method, a powerful analytica l tool for treating the frustrated \nHeisenberg chain in any dimension, was applied to t he ABDC. To verify the accuracy \nof CCM results, we have also investigated the prope rties of the ABDC by the ED \nmethod. The ED results show that the GS phase diagr am is composed of the TD state, \ndimer state, and ferrimagnetic state. We have shown that the former two exact spin \ncluster solid GSs can both be formed by CCM. Some p hysical quantities of the \nferrimagnetic state, such as the GS energy and subl attice magnetizations, have been \ndetermined by CCM up to high orders of approximatio n. The results of the above \nquantities obtained by CCM are compared with those given by numerical methods. \nThe case of 0 =α and 1=β is a typical example, in which the results of CCM are \nsufficiently accurate to be comparable to those of the numerical Monte Carlo or \nDMRG method. For any other parameter, the CCM resul ts are also in perfect \nagreement with the results of the numerical method, namely, those of ED. Thus, it is \nnatural to observe that CCM as well as ED can be us ed to precisely determine the \nphase boundary between the TD state and the ferrima gnetic state. \nWe have also calculated, aside from the GS physical quantities, the single-triplet \n 14 energy gap and antiferromagnetic gap of the ABDC by CCM and compared them with \nthose given by ED. Our results show that the single -triplet energy gap in the \nthermodynamic limit can be obtained by CCM. It is a lso found that the \nantiferromagnetic gap obtained by CCM is comparable to that determined by ED. \nTherefore, the properties of the ABDC can be precis ely analyzed by analytical \nCCM. Moreover, our findings provide a typical examp le of a powerful CCM \napplication to frustrated quantum spin systems, eve n though its GS is in the quantum \nstate with no classical analogy. \nAcknowledgments \nWe thank Dr. Damain Farnell for his help with the a pplication of CCM to spin \nsystems. This work was supported by the Natural Sci ence Foundation of Jiangsu \nProvince (No. BK20131428) and the Natural Science F oundation of the Jiangsu \nHigher Education Institutions (No.13KJD140003). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 15 Reference \n1) S. R. White and I. Affleck, Phys. Rev. B 54 , 9862 (1996). \n2) K. Okamoto and K. Nomura, Phys. Lett. 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J. Mikeska, and S. Yamamoto, J. Phys.: Condens. Matter 9, 3921 \n(1997). \n59) We also tried to extrapolate the ‘raw’ LSUB n results to the limit ∞→n . As there \nare no exact extrapolation rules, we performed the extrapolation according to \nempirical experience. However, we found that the ex trapolated results of CCM are not \nreliable, so we did not include them in the present paper. \n60) K. Sano and K. Takano, J. Phys. Soc. Jpn. 69 , 2710 (2000). \n 18 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n e AM AFΔ \nLSUB8 -1.454172 0.292129 1.760226 \nLSUB10 -1.454109 0.292403 1.759433 \nLSUB12 -1.454096 0.292472 1.759224 \nED ( N=30) -1.454095 0.292478 1.759174 \nLinear spin wave theory (SWT) 56) -1.436 0.195 1 \nSecond-order SWT 57) -1.454322 0.293884 - \nQMC 51, 58) −1.455 ± 0.001 0.29 1.75914 \nDMRG 56) −1.45408 0.29248 - Table I. Results obtained for the ABDC using the CC M in the case of 0=α and 1=β . The \nGS energy per unit cell e, the sublattice magnetization AM, and the antiferrimagnetic gap \nAFΔ obtained by CCM are shown. These results are compa red with those obtained by other \nmethods. \n \n 19 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n β LSUB8 LSUB10 LSUB12 ED \n0.10 0.090396 0.080051 0.080051 0.080051 \n0.20 0.166259 0.146264 0.146264 0.146264 \n0.30 0.228444 0.198044 0.198044 0.198044 \n0.40 0.276621 0.235127 0.235127 0.235127 \n0.50 0.310449 0.257644 0.257644 0.257644 \n0.60 0.329849 0.266132 0.266132 0.266132 \n0.70 0.335139 0.261484 0.261484 0.261484 \n0.80 0.327065 0.244852 0.244852 0.244852 \n0.90 0.306736 0.217528 0.217528 0.217528 \n1.00 0.275494 0.180828 0.180828 0.180828 \n1.10 0.234759 0.136015 0.136015 0.136015 \n1.15 0.211265 0.110931 0.110931 0.110931 Table II. Results of the single-triplet energy gap using CCM-LSUB n approximation with \nn={ 8, 10, 12} when 1=α . These results are compared with those obtained by ED for N=30 \nsites. \n 20 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure captions \nFig. 1. Sketches of the FFA reference state (a), FFFI reference state (b), and FFFII ref erence state \n(c) of the ABDC. \nFig. 3. GS phase diagram obtained by ED. The solid line 1−=αβ represents the exact \nboundary between the dimer phase and the TD phase. \nFig. 4. Illustration of fundamental configurations retained in the ket-state correlation operator \nS for CCM based on FFA reference state (a) or FFFI r eference state (b). The centers of the \nshaded circles mark the flipped spins with respect t o the reference state. \nFig. 8. Spin gap STΔ of the ABDC versus β using ED when 1=α . \nFig. 9. Spin gap STΔ of the ABDC versus β using CCM based on FFA reference state and ED \nwhen 1=α . Fig. 5. GS energy per site e versus β using CCM based on FFFII reference state, ED, and \nEq. (7) for different α values. \n \nFig. 10. Spin gap AFΔ versus β using CCM based on FFFII reference state and ED wh en 1=α . \n Fig. 7. Sublattice magnetizations AM and C BM+ versus β using CCM based on FFFII \nreference state and ED when 0=α . Fig. 2. Schematic picture of the TD state. The rect angles and ellipses represent the tetramers and \nsinglet dimers, respectively. \nFig. 6. Boundary line between the TD state and the ferrimagnetic state determined by CCM and \nED. \n \n 21 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 1J\n2J\n2 3−iS\n1 3−iSiS3\nA\nBC\n3J\n1J\n2J\nA\nBC\n3J\n2 3−iS\n1 3−iSiS32 3−iS\n1 3−iSiS3\n1J\n2J\nA\nBC\n3J(a) \n(b) \n(c) Figure 1 \n 22 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 3−iS\n1 3−iSiS3\n23+iS33+iS\n1 3+iS\n3 4 iS+\n1 3−iSiS3\n1 3+iS\n23+iS33+iSFigure 2 \n 23 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 3 \n 24 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\naS22 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\nbS2\n2 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\ncS22 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\n4S\n2 3−ii3\n1 3−i\n2S(a) \n(b) Figure 4 \n 25 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 5 \n 26 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 6 \n 27 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 7 \n 28 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 8 \n 29 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 9 \n 30 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 10 " }, { "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. 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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": "1706.00549v1.Coherent_Terahertz_Spin_Wave_Emission_Associated_with_Ferrimagnetic_Domain_Wall_Dynamics.pdf", "content": "arXiv:1706.00549v1 [cond-mat.mtrl-sci] 2 Jun 2017Coherent Terahertz Spin-Wave Emission Associated with\nFerrimagnetic Domain Wall Dynamics\nSe-Hyeok Oh,1,∗Se Kwon Kim,2,∗Dong-Kyu Lee,3Gyungchoon Go,3Kab-Jin\nKim,4,5Teruo Ono,5Yaroslav Tserkovnyak,2and Kyung-Jin Lee1,3,6,†\n1Department of Nano-Semiconductor and Engineering, Korea U niversity, Seoul 02841, Korea\n2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n3Department of Materials Science and Engineering, Korea Uni versity, Seoul 02841, Korea\n4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea\n5Institute for Chemical Research, Kyoto University, Kyoto 6 11-0011, Japan\n6KU-KIST Graduate School of Converging Science and Technolo gy, Korea University, Seoul 02841, Korea\n(Dated: June 5, 2017)\nWe theoretically study the dynamics of ferrimagnetic domai n walls in the presence of\nDzyaloshinskii-Moriya interaction. We find that an applica tion of a DC magnetic field can induce\nterahertz spin-wave emission by driving ferrimagnetic dom ain walls, which is not possible for ferro-\nmagnetic or antiferromagnetic domain walls. Dzyaloshinsk ii-Moriya interaction is shown tofacilitate\nthe teraherz spin-wave emission in wide ranges of net angula r momentum by increasing the Walker-\nbreakdown field. Moreover, we show that spin-orbit torque co mbined with Dzyaloshinskii-Moriya\ninteraction also drives a fast ferrimagnetic domain wall mo tion with emitting terahertz spin-waves\nin wide ranges of net angular momentum.\nIn modern communications, information is carried by\nelectromagnetic waves of which frequency is limited to\n≈0.1 terahertz (THz), the frequency of oscillating cir-\ncuits based on high-speed transistors [1]. On the other\nhand, semiconductor lasers generate coherent light with\nthe frequency >30 THz [2]. Terahertz gap refers to the\nfact that no relevant technology exists in the frequency\nrange between these two limits (0 .1∼30 THz). There-\nfore, it is of critical importance to find relevant physical\nphenomena that fill in the terahertz gap.\nIn this respect, antiferromagnets of which resonance\nfrequencies are in the THz ranges [3, 4] are of inter-\nest [5, 6]. It has been reported that coherent THz\nmagnons or spin-waves are generated in antiferromag-\nnets, drivenbyalaser[7,8]oranelectricalcurrent[9,10].\nHowever, THz spin-wave excitations by a DC magnetic\nfield are in principle not possible for antiferromagnets as\ntheir magnetic moments are compensated on an atomic\nscale. In this work, we theoretically showthat generation\nof coherent THz spin waves can be achieved by a field-\ndriven domain wall (DW) motion in ferrimagnet/heavy\nmetal bilayers in which the interfacial Dzyaloshinskii-\nMoriya interaction (DMI) is present.\nAs far as the terahertz gap is concerned, this DC field-\ndrivenschemecould be beneficialasit allowsalow-power\noperation by avoiding laser-induced or current-induced\nheating. It is alsofundamentally interesting as THz spin-\nwave emission is caused by an approximately relativistic\ndynamics of a ferrimagnetic DW. Relativistic kinemat-\nics refers to kinematics compatible with the theory of\nrelativity [11], of which key ingredient is the Lorentz\ninvariance with limiting velocity c, the speed of light.\nWhen the dispersion of a wave satisfies the Lorentz in-\nvariance, a quasiparticle corresponding to the wave fol-\nlows an analogous relativistic kinematics with replacingthe speed of light by the maximum group velocity of the\nwave. When the velocity of quasiparticle approaches the\nmaximum group velocity, it undergoes the Lorentz con-\ntraction and its speed saturates to the limiting velocity.\nAn example of such quasiparticles is an antiferromag-\nnetic DW[10, 12, 13]. When the DW velocityapproaches\nthe maximum spin-wave group velocity, the DW width\nshirinks with emitting spin-waves [10]. Similarly, the dy-\nnamics of a ferrimagnetic DW is also expected to exhibit\nthe features of relativistic kinematics provided that the\nnet magnetization and DMI, which break the Lorentz in-\nvariance of the system, are sufficiently ineffective.\nIn this Letter, we show that such an approximately\nrelativistic DW dynamics is achievable for a class of fer-\nrimagnets, rare earth (RE) and transition metal (TM)\ncompounds, inwhichthe spinmomentsareantiferromag-\nnetically coupled. As RE and TM elements have differ-\nent Land´ e-g factors [14], RE-TM ferrimagnets have two\ndistinct temperatures; the magnetic moment compensa-\ntion point TMwhere net magnetic moment vanishes, and\nthe angular momentum compensation point TAwhere\nnet angular momentum vanishes. For RE-TM ferrimag-\nnets, resonance [15, 16], switching [17–21], domain wall\nmotion [22–24], and skyrmion (or bubble domain) mo-\ntion [25–27] near these compensation points have been\nexplored experimentally and theoretically. In particu-\nlar, an experimental observation of a fast field-driven\nDW motion at TAin GdFeCo single-layeredferrimagnets\nwas recently reported [23]. This observation reveals two\ndistinguishing features of RE-TM ferrimagnets at TA.\nOne is that the spin dynamics is antiferromagnetic and\nthus fast because of zero net angular momentum at TA.\nThe other is that this fast antiferromagnetic dynamics is\nachieved by a field because the net magnetic moment is\nnonzero at TAand thus couples with a magnetic field.We begin with deriving the equations of motion for\na ferrimagnetic DW based on the collective coordinate\napproach [28]. The dynamics of a general collinear ferri-\nmagnet at sufficiently low temperatures can be described\nby the following Lagrangian density [27, 29]\nL=ρ˙n2/2−δsa[n]·˙n−U, (1)\nwherenis the unit vector along the collinear order, ρ\nparametrizes the inertia of the dynamics, δsis the spin\ndensity in the direction n,a[n] is the vector potential\ngenerated by a magnetic monopole of unit charge satis-\nfying∇n×a=n, andUis the potential-energy density.\nHere, the first term is the spin Berry phase associated\nwith the staggered spin density, which thus appears in\nthe Lagrangian for collinear antiferromagnets; the sec-\nond term is the Berry phase associated with the net spin\ndensityδs, which is used to describe the dynamics of\nuncompensated spins in ferrimagnets. We consider the\nfollowing potential-energy density:\nU=A(∇n)2/2−K(n·ˆz)2/2+κ(n·ˆx)2/2\n+Dˆy·(n×∂xn)/2−h·n.(2)\nHere, the first term is the exchange energy with A >0;\nthe second term is the easy-axis anisotropy along the z\naxis with K >0; the third term is the weaker DW hard-\naxis anisotropy along the xaxis with κ >0; the fourth\nterm is the interfacial DMI; the last term is the Zeeman\ncoupling with h=MH, whereMis the net magnetiza-\ntion in the direction n. The dissipation can be accounted\nfor by introducing (the spatial density of) the Rayleigh\ndissipation function, R=sα˙n2/2. Here, sαis a phe-\nnomenologicalparameterquantifyingtheenergyandspin\nloss due to the magnetic dynamics. For example, in the\nferromagnetic regime, i.e., away from TA, it can be con-\nsidered as the product of the effective Gilbert damping\nconstant and the net spin density.\nThe low-energy dynamics of a DW can be de-\nscribed by the two collective coordinates, the posi-\ntionX(t) and the azimuthal angle φ(t). We consider\nthe Walker ansatz [30] for the DW profile: n(x,t) =\n(sinθcosφ,sinθsinφ,cosθ), where θ= 2tan−1{exp[(x−\nX)/λ]}andλ=/radicalbig\nA/Kis the DW width. The equa-\ntions of motion can be derived from Eqs. (1) and (2) in\nconjunction with the Rayleigh dissipation function:\nM¨X+G˙φ+M˙X/τ=F, (3)\nI¨φ−G˙X+I˙φ/τ=−˜κsinφcosφ+˜Dsinφ,(4)\nwhereM= 2ρA/λis the mass, I= 2ρAλis the moment\nof inertia, G= 2δsAis the gyrotropic coefficient, τ=\nρ/sαis the relaxation time, F= 2hAis the force exerted\nby an external field, ˜ κ= 2κλA,˜D=πDA/2, andA\nis the cross-sectional area of the DW. From Eq. (3), we\nobtain the steady-state solution of the DW velocity:\nVDW=Mλ\nsαH, (5)whereHis the external field applied along the z-axis.\nIn this steady state, the DW moves at a constant veloc-\nityVDWwith a constant angle φ. When the field be-\ncomes sufficiently strong such that VDWexceeds a cer-\ntain threshold Vmax, the DW begins to precess, engen-\ndering the phenomenon known as the Walker Break-\ndown [31, 32]. The Walker Breakdown field HWBcan\nbe obtained from Eq. (4) by\nHWB=Vmaxsα\nMλ. (6)\nIn the absence of DMI ( D= 0), the threshold velocity\nis given by Vmax= ˜κ/2Gand thus HWB= ˜κsα/2GMλ.\nWhen DMI is much stronger than the DW anisotropy\nin theydirection, i.e., |˜D| ≫˜κ, then|Vmax|=|˜D|/G.\nIn this strong DMI limit, the Walker Breakdown field is\ngiven by\nHWB=|˜D|\nGsα\nMλ=π|D|\n4δssα\nMλ. (7)\nWe note that HWBis inversely proportional to Gand\nthus to the net spin density δs. As a result, the Walker\nbreakdown is absent at TAwhere the net spin density\nvanishes, δs= 0. This suppression of the Walker break-\ndown at TAcan be understood as a result of decoupling\nof the DW position Xand the angle φatTA[23].\nIt is worthwhile comparing Eq. (7) to the Walker\nbreakdown field for ferromagnetic DWs in the strong\nDMI limit [33]: HWB,FM=απDFM/2MFMλFM, which\ncan be obtained from Eq. (7) by taking the ferromagnetic\nlimit. From this comparison, one finds that in the vicin-\nity ofTA,HWBfor ferrimagnetic DWs is much larger\nthan that for ferromagnetic DWs because δs≈0 and\n|M| ≪MFM. Moreover, this very large HWBfor fer-\nrimagnetic DWs suggests that VDWcan reach the max-\nimum group velocity of spin-wave more easily without\nexperiencing the Walker breakdown and thus ferrimag-\nnetic DWs can generate THz spin-waves in wide ranges\nof net angular momentum δs. Finally, the time averaged\nvelocity ¯Vfor a one period far above the Walker Break-\ndown is given as\n¯V=Mλ\nsα+δ2s/sαH. (8)\nTo verify these theoretical predictions on the DW ve-\nlocity and THz spin-wave emission, we perform atom-\nistic model calculations [10, 34] for two-sublattice ferri-\nmagnets, which correspond to RE-TM compounds. Two\nsublattices possess the magnetization M1andM2, which\nare coupled by the antiferromagnetic exchange. The spin\ndensities are given by s1=M1/γ1ands2=M2/γ2,\nwhereγi=giµB//planckover2pi1isthegyromagneticratioofthelattice\ni,µBis the Bohr magneton, and giis the Land´ e-g fac-\ntor. The parametersin the abovedescriptions for general\nferrimagnets are given by δs=s1−s2,M=M1−M2,\nC\u000b \n \nD\u000b \nݔݖ\n-500 0500 1000 1500 \nMRE \nTemperature Mi (kA/m) MTM \nTAMTM - M RE \n-5 0510 15 \u0003G s (10 -7 J s/m 3)\n ݕ\nFIG. 1. (color online) (a) A schematic illustration of a\nferrimagnet in which neighboring spins are coupled antifer -\nromagnetically. (b) The assumed magnetic moments of TM\n(red) and RE (blue) elements as a function of the temper-\natureT. Black symbols represent net magnetic moment (=\nMTM−MRE), and dark yellow symbols represent net angular\nmomentum δs. Zeroδscorresponds to the angular momentum\ncompensation temperature TA(purple). These parameters\nare used for simulations shown in Figs. 2 and 3.\nandsα=α1s1+α2s2, whereαiis the Gilbert damp-\ning constant for the lattice i. The one-dimensional dis-\ncrete Hamiltonian that we use for numerical calculations\nis given by\nH=Asim/summationdisplay\niSi·Si+1−Ksim/summationdisplay\ni(Si·ˆz)2\n+κsim/summationdisplay\ni(Si·ˆx)2+Dsim/summationdisplay\niˆy·(Si×Si+1)\n−giµBµ0/summationdisplay\niH·Si, (9)\nwhereSiis the normalized spin moment vector at lattice\nsitei[i.e., an even (odd) icorresponds to a RE (TM)\natomic site], Asim,Ksim,κsim, andDsimdenote the ex-\nchange, easy-axis anisotropy, DW hard-axis anisotropy,\nand DMI constants, respectively, and His the exter-\nnal field. We numerically solve the atomistic Landau-\nLifshitz-Gilbert equation:\n∂Si\n∂t=−γiSi×Heff,i+αiSi×∂Si\n∂t,(10)\nwhereHeff,i=−1\nMi∂H\n∂Siis the effective field. We use the\nfollowingsimulationparameters: Asim=30meV, Ksim=\n0.4 meV, κsim= 0.2µeV, damping constant αTM=αRE\n= 0.002, the lattice constant d= 0.4 nm, and Land´ e\ng-factors gTM= 2.2 for TM, and gRE= 2 for RE ele-\nment [14]. Figure 1(b) shows the assumed temperature-\ndependent changein the magnetic moment Miandcorre-\nsponding δs. For simplicity, we assume other parameters\nare invariant with temperature.\nFigure 2(a) shows VDWforD= 0 as a function of H.\nBelowHWB,VDWincreaseslinearlywith H,inagreement\nwith Eq. (5) (solid lines). For H > H WB, the Walker\nbreakdownoccursexcept for T=TAat which VDWkeeps\nincreasing because of the absence of the Walker break-\ndown, as explained above. Figure 2(b) shows HWBas a\nfunction of δsat various DMIs. Two features are worth\nC\u000b \nD\u000b \n\nE\u000b \nF\u000b -1.0 -0.5 0.0 0.5 1.0 0150 300 450 600 D sim (meV) \n 0 \n 0.02 \n 0.1 HWB (mT) \nGs (10 -7 J s/m 3)Solid lines \n: Eq. (6) \n0 100 200 300 400 01234567Temperature \n< < = T A < NBEC, under certain\nconditions, which will be discussed below. The critical\nmagnons concentration NBECfor different magnetic sys-\ntems was calculated in [24, 25]. Particularly for easy\nplane antiferromagnets with wave spectrum\nεk=/radicalBig\nε2\n0+ε2ex(ak)2 (3)\nit reads:\nNBEC≃(kBT)2\n2π2ε0\na3ε3ex, (4)\nwherekBis a Boltzmann constant.\nThe critical magnon density can be reached at dynam-\nical deflection of magnetization on the angle about few\ndegrees. For antiferromagnetic superfluid3He this angle\nis very small, below 1◦[23]. In the case of ferromag-\nnetic resonance in YIG film, we will discuss in this ar-\nticle, the estimation of critical magnon density is more\ncomplicated. Indeed, as it was shown in [25], the critical\ndensity of magnons can be obtained at a magnetization\ndeflection on the angle about only 3◦.\nThe formation of a magnon BEC state was first ob-\nserved in antiferromagnetic superfluid3He-B as a for-\nmation of extremely Long Lived Induction Decay Signal\n(LLIDS) [4, 26]. The LLIDS manifests the condensa-\ntion of magnons in a common wave function in the whole\nsample with a common phase and frequency of preces-\nsion. The LLIDS obeys the entire requirement for BEC\nof quasiparticles, which later was postulated as an re-\nquirementofmagnonBECinwell-knownarticlebySnoke\n[27]. Magnon BEC has one to one analogy with the ex-\nperiments of atomic BEC [28]. Owing the slow magnons\nrelaxation, the number of magnons decreases, but the\nmagnons remains in a coherent state. It is important tonote that the BEC state is the eigen state for given den-\nsity of excited magnons. It was shown experimentally,\nthat the small RF pumping on a frequency of magnon\nBECωBECcan compensate the magnons relaxation. In\nthis case the magnons BEC may maintains permanently\nfor an infinite time [29]. The physics of excited magnons\nBEC states and phenomena of spin superfluidity are well\nestablished during the 30 years of investigations. The re-\nview of this investigations one can found, for example in\n[30, 31] and in the book [32].\nFIG.1: Thefrequencyshiftofmagnetic resonanceatanormal\nandtangential orientation ofmagnetic field(left) andthe s hift\nof resonance magnetic field for 9.26 GHz (right) as function\nof magnetization deflection angle β.\nIn this article we describe the similar experiments, we\nhave performed in a normally magnetized thin film of\nYttrium Iron Garnet (YIG). We have observed a similar\nLLID signal, as well the permanent magnon BEC state,\nstabilized by a small RF pumping. YIG film is character-\nized by a very small Gilbert [33] damping factor αabout\n10−5, the one of the best value for solid magnetic ma-\nterials. That is why the formation of magnon BEC in\nYIG and observation of spin supercurrent, like in3He,\nshould leads to a development of new branch of physics\n- Supermagnonics. However, the observation of a con-\nventional BEC in YIG has some fundamental difficulties\nassociated with a sufficiently large value of ferrimagnetic\nmagnetization. The magnetization of the YIG at room\ntemperature 4 πMSis about 1750 Oe, which is several\norders of magnitude greater than in all magnetic me-\ndia where BEC magnons were previously observed. High\nmagnetization leads to significant inhomogeneities in the\neffective magnetic field within the sample, which requires\na substantially larger magnon super current and spatial\nvariation of magnons density to equalize the phase of\nthe wave function. For this reason, the first experiments\non the observation of BEC in YIG were carried out in\nYIG magnetic films with a plane magnetization and with\n/vectork/negationslash= 0 [14, 34, 35].\nIn this article we describe the first observation of\nmagnon BEC and spin supercurrent in a normally mag-\nnetized YIG film. At these conditions the minimum of3\nenergy corresponds to /vectork= 0 like in atomic BEC and in\nsuperfluid3He. It was shown [36] that at these condi-\ntions the magnetization precession is stable in the field\nabove 2 kOe. The spectrum of the ground mode of mag-\nnetic resonance for a normally magnetized film for a first\napproximation, reads [37]:\nω=γ(H0−4πMScosβ), (5)\nwhereβis an angle of magnetization deflection. The\nfrequency of precession increases with the density of\nmagnons, which corresponds to repulsive interaction be-\ntween magnons.\nIn Fig.(1) is shown the frequency shift of magnetic\nresonance for /vectork= 0 for normal and tangential orienta-\ntion of magnetic field is shown as function of magnetiza-\ntion deflection angle βfor resonance frequency 9.26 GHz\n(left) as well the shift of resonance magnetic field for\n9.26 GHz (right). For the case of normal magnetization\nwe have a classical potential energy trap for magnons\nwith/vectork= 0. Furthermore, the magnetization precession\nis stable against the splitting on the spin waveswith non-\nzero/vectork. There are very nice conditions for magnons BEC\nformation, very similar for one in antiferromagnetic su-\nperfluid3He-A [38].\nPulsed FMR\nWe have used single-crystal YIG films of thickness\nabout 6µm in the shape of a disk with diameter of 0.5\nand 0.3 mm, which have been grown in the (111) crystal-\nlographic plane on a Gadolinium Gallium Garnet (GGG,\nGd3Ga5O12) substrate by liquid-phase epitaxy [39]. In\norderto overcomethe influence ofthe inhomogeneousde-\nmagnetization field, we modified the shape of the edges\nof the sample by chemical etching (see Methods).\nThe external magnetic field was oriented perpendicu-\nlar to the film. The RF field was oriented in plane of\nthe film. Following the procedure, developed for BEC\nformation in3He-A in aerogel [40], we have excited the\novercritical density of magnons by a relatively long RF\npulse atthe frequencyhigherthe resonanceone. We have\nobservedthe typicalLLIDS, whichringsonthe frequency\nof RF pumping and not on the resonance frequency. In\nFig.(2) are shown the signals at room temperature after\nthe RF pulse of about 20 Oe and duration of 400 ns. At\nresonance excitation (Fig.(2) a) the magnon BEC does\nnot form. The length of signal is about 200 ns and has\nthe same length as in the case of a short pulse (with a\ndecay time constant of 50 ns. It corresponds to the in-\nhomogeneity of the resonance line. In the case of a non-\nresonance excitation (Fig.(2) b-d) the signals are drasti-\ncally changes. The signals are characterized by a time\ndecay on an order of magnitude longer (see Fig.(3)). We\nhaveobservedtheLLIDS with time decayconstantabout\nFIG. 2: The spectroscopic records of LLID signal in normally\nmagnetized YIG film at different frequency shift of RF pulse\nfrom the resonance frequency. The lines corresponds to a sig -\nnals, measured with 0◦,90◦,180◦and 240◦phase shift. Point\n(a) corresponds to the end of RF pulse of 400 ns duration and\n(b) - the end of spectrometer dead time.\n0.8 - 1.6µs for different samples. The amplitude of LLID\nsignalsis comparablewith the amplitude ofinitial partof\nsignals. Its field dependence does not correspond to the\nshapeofCWsignal. We haveobservedthe LLIDS signals\nin different samples and at different temperatures from\nroom to 100 K. The formation of LLIDS is very robust.\nAll its properties well correspond to a BEC signals from\nother systems and particularly antiferromagnetic super-\nfluid3He-A. It is well known from previous investigations\nthat the LLIDS signals radiates by a coherent system of\nexcited magnons, which forms during a non-resonance\nexcitation. Its observation confirms the magnon BEC\nformation in the YIG film.4\nFIG. 3: The LLID signals at 11.2 (above) and 22.4 (below)\nMHz frequency shift. The decay time constant for this sample\nis 0.84µs.\nFIG. 4: The amplitude of CW signal of absorption at different\npower of RF pumping at sweep down of magnetic field. The\nenlarged scale is shown in the inset. Here and in the next\nfigures signals marked a - h corresponds to an RF power 80,\n40, 20, 10, 1, 0.4, 0.1 and 0.05 mWt.\nContinuous wave FMR\nWe have performed the detailed investigations of CW\nsignals in normally magnetized YIG film. The RF field\nwas applied in plain of the film and the magnetic field\nsweep down. At a small power we can see the linear\nresponse of the magnetic system (Fig.(4)). It is impor-\ntant to point out, that between the power of excitation\n0.4 mW and 1 mW the behavior of the signal drastically\nchanges. The frequency of FMR signal begins to follow\nto the frequency, corresponds to descended field. This\nchange appears at the region of field at a shift about 2Oe. At further increase of RF power we have observed\nthe region of fields, where FMR frequency follows to the\nfield as large as 90 Oe, which correspond to a frequency\nshift about 240 MHz.\nThe usual explanation of this phenomenon is well\nknown as a ferromagnetic resonance or be-stable reso-\nnance[41]. ThetheoryofFMRbasedonthe propertiesof\nnon-linear oscillator, which frequency increases with ex-\ncitation. There aresupposing that the higher RF power-\nthe biggerangleofdeflection and consequentlythe bigger\nfrequency shift. This model of FMR was tested exper-\nimentally [42]. There was found that the FMR theory\nworks well only in the limit of a small RF power. At rel-\natively big power the amplitude and frequency shift does\nnot follow to this model. Based on our previous investi-\ngations of magnon BEC in other system we may suggest\nthat the observed phenomena are the result of magnons\nBEC formation.\nFIG. 5: The energy dissipated by a magnon spin system at\ndifferent level of exciting power. The energy was calculated as\na product of absorption signal on the amplitude of magnetic\nfield. The enlarged scale is shown in the inset.\nTo test this hypothesis we have calculated the energy,\ndissipated in the sample at different level of excitation.\nFor this purpose we have multiplied the amplitude of ab-\nsorption signal on the RF field of excitation. The results\nare shown in Fig. (5). It is clearly seen that the dis-\nsipated energy does not depend on the RF power and\ndetermines only by a frequency shift and corresponding\nmagnetic field shift! But this is the property of magnon\nBEC, the eigen state of non-equilibrium magnons, which\ndensity correspond to a resonance at a given frequency\nshift. This effect for antiferromagnetic superfluid3He-\nA was investigated in [43]. The analogy is even more\nclear if we will consider the energy loses as function of an\nangle of magnetization deflection, recalculated from the\nfrequency shift. The results are shown in Fig. (6). There\nis the squaredependence ofenergy losses, similar to BEC5\nin antiferromagnetic3He-A [44], which was explained by\na relaxation due to a spin diffusion of a transverse mag-\nnetization. The signals at different excitations fall on a\nuniversal curve independent of the amplitude of the RF-\nfield. Thisdemonstratesthatatlarge βthemagnonBEC\nis self-consistent and is not sensitive to the amplitude of\nRF field; the latter is only needed for compensation of\nlosses.\nFIG. 6: The dissipated energy as a function of angle of deflec-\ntion. The fitting lines correspond to square dependence form\ndeflected magnetization.\nThe properties of magnon BEC\nLet us consider the basic principles of magnon BEC.\nThe atomic BEC state described by a wave function\nψ=/angb∇acketleftˆa0/angb∇acket∇ight=N1/2\n0eiµt+iα. (6)\nwhereN0is the density of particles and µis a chemi-\ncal potential of Bose condensate. The coherent magnons\nsystem described by a very similar impression:\nψ=/angb∇acketleftˆa0/angb∇acket∇ight=N1/2eiωt+iα=/radicalbigg\n2S\n/planckover2pi1sinβ\n2eiωt+iα.(7)\nwhereβis the angle of deflection in the mean field ap-\nproximation. The role of the global chemical potential\nof magnons µis played by the global frequency of the\ncoherent precession ω, i.e.µ≡ω. The frequency of\nlocal precession in a linear approximation (at a small\nexcitation) is determined by the sum of local external\nH0and demagnetization 4 πMSfields. The interaction\nbetween the excited magnons leads to dynamical fre-\nquency shift due to decrease of local demagnetization\nfield, which for the normally magnetized YIG film reads:\nωN= 4πMS(1−cosβ). It corresponds to a repulsive in-\nteraction and, consequently the magnons BEC formationis possible. In the opposite case of attractive interaction\nthe spatial profile of magnons density has a tendency to\nsplits on a spatially inhomogeneous distribution [45].\nThere are two approaches to study the magnons non-\nequilibrium systems: at fixed particle number NMor at\nfixed chemical potential µ. These two approaches cor-\nrespond to two different experimental conditions: the\npulsed and continuous wave (CW) resonance, respec-\ntively. In the first case the RF pulse excite a number\nof non-equilibrium magnons. If the density of magnons\nis higher than NBEC, the magnon BEC state should be\ncreated with the frequency ω0+ωN. But the local fre-\nquency of magnon BEC may be different for different\nparts of the sample due to the inhomogeneity of local\nfrequencyω0r. Owing to its inhomogeneity ∆ ωthe in-\nduction decay signal decreases at the time scale about\n1/∆ω. It was shown in the experiments with magnon\nBEC in3He-B that the spatial inhomogeneity leads to\nformation of a gradient of magnon BEC wave function\nand to a spin supercurrent of magnons [4, 26]. The spin\nsupercurrents redistribute the density of magnons. As\na result the spatial magnetic inhomogeneity is compen-\nsated by a spatial distribution of magnons density. Fi-\nnally the global BEC state appears with the frequency of\nprecessionωBEC=ω0r+ωNr=constthrows the entire\nsample and radiates a LLID signal. This phenomenon is\na magnon analogy of a global coherent states in other\ncoherent liquids, like superfluids and superconductors,\nwhere supercurrents leads to an inhomogeneities of the\nground state.\nThe signals of CW FMR well correspond to a prop-\nerties of magnon BEC. The critical density for magnon\nBEC in YIG film at room temperature was calculated in\n[25].NBEC≃M−Mz=M(1−cosβ) whereβ= 3◦.\nThis angle of deflection leads to a frequency shift of FMR\nof 7.1 MHz and field shift of about 2.5 Oe. These param-\neters are very close to the point of signals transforma-\ntion we have observed in our experiments. The collective\nquantum state is formed at higher density of magnons.\nIt is an eigen state of excited magnons. It does not de-\npendonthe powerofexcited RFfield, startingfromsome\ncritical value. Particularly this effect was investigated in\n3He-A [43, 46].\nThe BEC state formation is usually analyzed by min-\nimum of Gross-Pitaevskii (GP) equations for Ginzburg-\nLandau (GL) free energy. For magnons in normally mag-\nnetized YIG film it reads:\nF=/integraldisplay\nd3r/parenleftbigg|∇Ψ|2\n2mM+(ω0−ω)|Ψ|2+1\n2b|Ψ|4/parenrightbigg\n,(8)\nwhere parameter bis a repulsive magnon interaction,\nb=4πMS\n2S(9)\nAtω>ωo, magnon BEC must be formed with density\n|Ψ|2=ω−ω0\nb. (10)6\nThis corresponds to the following dependence of the fre-\nquency shift on tipping angle βof coherence precession:\nω−ω0=γ4πMS(1−cosβ) (11)\nIf the precession is induced by continuous wave FMR,\none should also add the interaction with the RF field,\nHRF, which is transverse to the applied constant field\nH0. In continuous wave FMR experiments the RF field\nprescribes the frequency of precession, ω=ωRF, and\nthus fixes the chemical potential µ=ω. In the preces-\nsion frame, where both the RF field and the spin Sare\nconstant, the interaction term is\nFRF=−γHRF·S=−γHRFS⊥cos(α−αRF),(12)\nwhereHRFandαRFare the amplitude and the phase of\nthe RF field. In the language of magnon BEC, this term\nsoftly breaks the U(1)-symmetry and serves as a source\nof magnons [47]:\nFRF(ψ) =−1\n2η(ψ+ψ∗), (13)\nThe phasedifference α−αRFis determined bythe energy\nlosses due to magnetic relaxation, which is compensated\nby the pumping of power from the CW RF field:\nW+=ωSHRFsinβsin(α−αRF),(14)\nthe phase difference between the condensate and the\nRF field is automatically adjusted to compensate the\nlosses. If dissipation is small, the phase shift is small,\nα−αRF≪1, and can be neglected. The neglected\n(α−αRF)2term leads to the nonzero mass of the Gold-\nstone boson – quantum of second sound waves (phonon)\nin the magnon superfluid [47]. The signal of magnon\nBEC collapses at the moment, when the RF power is not\nenough for compensating the magnons dissipation. Since\nthe pumping (14) is proportional to sin βsin(α−αRF),\na critical tipping angle βc, at which the pumping can-\nnot compensatethe losses, increaseswith increasing HRF\n(see Fig.(5)).\nIn a perfect homogeneous sample, the collapse occurs\nwhen the phase shifts α−αRFreaches about 90◦. How-\never, in real systems the phase shift α−αRFat the\ncollapse is smaller owing the inhomogeneity of magnons\ndessipation. This indicates that YIG sample is not ho-\nmogeneous but contains some regions with higher dissi-\npation. In this case collapse may start when the local\nvalueα(r)−αRFreaches 90◦within one of such regions.\nCandidates for regions with high dissipation could be the\nregions with high impurities density, or topological de-\nfects.\nDiscussion\nIn the case of a non-coherent precession, like in the\nmodeldescribedin[41]theabsorption-dispersionrelationshould have a form of a circle. Moreover, if local oscilla-\ntors are independent, then after the precession in regions\nwith high dissipation collapses, the precession will con-\ntinue in regions with smaller dissipation. This, however,\ndoes not occur. The collapse of precession in our exper-\niments is very sharp. The sharp feature of collapse and\ntheshapeoftheabsorption-dispersionhistogramindicate\nthe coherence between different parts of the sample: the\nspin supercurrents transfer the deflected magnetization\nbetween the parts of the cell.\nThe rough estimates of the magnitude of the absorp-\ntion and dispersion signals indicate that almost all the\nmagnetization of the film is deflected and precess at the\nfrequency of the RF pumping. It is clearly visible that\nthe signal amplitude is very large and practically has no\nnoise. This behavior is usual for magnon BEC signals\nbecause of their coherence. We have investigated the\ndependence of the magnitude of the transverse magneti-\nzation as a function of the frequency shift (shift of mag-\nnetic field). To do this, we plot in Fig.(7) the magnitude\nof absorbtion signals at the moments of it collapse. At\nthis condition the phase difference α(r)−αRFshould be\nabout 90◦. Thus, the absorption signal supposed to be\nproportional to the magnitude of the transverse magne-\ntization. The theoretical line in Fig.(7) is calculated for\nassumption that BEC state fills up the entire volume of\nthe samplewhereinthemagneticfield islessthan ωRF/γ.\nIn this case the signal should be proportionalto sin βand\nthis volume. In Fig.(8) is shown the distribution of an\neffective field in the sample as it was calculated by a mi-\ncromagnetic simulation. In result the calculated depen-\ndence shows the good agreement with the experimental\npoints. The contributions of the volume and angle of de-\nflection to the amplitude of the signals in shown in insert\nin Fig.(7).\nThe amplitude of the LLID signals turned out to be\nan order of magnitude smaller than should be in the case\nwhen it is formed by the complete magnetization of the\nsample. Their properties correspond to the signals of\nthe so-called Q-ball that was discovered in antiferromag-\nnetic superfluid3He-B [48] and later explained as a sta-\nble droplet of coherently precessed magnetization [49]. It\nforms as a result of BEC instability near the surface of\nthe sample [50, 51] which pushed the BEC to the central\npart of the sample. The Q-balls excited on the frequency\nof RF pulse. The time dependence of Q-ball frequency is\ndetermining by a dynamical frequency shift and the pres-\nsureofthe orderparametertexture. It is likelythat these\ntwo forces well compensate each other in our case. We\nhave registered a very week time dependence of Q-ball\nfrequency. The stable objects, which radiates a signal\non the frequency of RF excitation and not on the reso-\nnance frequency was first observed in a coupling nuclear-\nelectron precession in MnCO 3and CsMnF 3and named\n“Capturedecho signals”[52]. The observationsofsimilar\neffect in YIG films requires a more detailed study of this7\nFIG. 7: The amplitude of absorption signals at the moment of\ncollapse ofprecession (points)andtheoreticalcurvecalc ulated\nfrom the angle of magnetization deflection and the volume of\nthe sample with local frequency below the frequency of RF\npumping. The inset shows the contributions of the angle of\nthe magnetization deflection and the volume of the sample\nwith BEC from the magnetic field shift.\nFIG. 8: The distribution of a local magnetic field from the\ncenter of the sample and the frequency of BEC pumping\n(dashed line). The BEC state filled up all the region, where\nlocal field is below ω/γ.\nphenomenon take place in spin systems with a dynamical\nfrequency shift.\nConclusion\nIn conclusion, we have observed the CW and pulse\nFMR signals, which properties corresponds to a magnon\nBEC states early observed in other non-linear spin sys-\ntemswithrepulsiveinteractionbetweenmagnons. InCWFMR the signals correspond to a formation BEC in the\nvolume of the entire sample. In pulsed FMR the insta-\nbility of homogeneous precession near the edges leads to\nformation of the BEC droplet in the central part of the\nsample, like in superfluid3He-B. The magnon BEC in\nYIGfilmaddstotheothercoherentstatesofmagnonsob-\nserved in antiferromagnetic superfluid states of3He and\nin antiferromagnets with coupled nuclear-electron pre-\ncession [53, 54]. We expect to conduct new experiments\nto observe the spin supercurrent, Josephson phenomena\nand spin vortex in YIG. It would be interesting to search\nsimilar dynamic coherent states of excitations in other\ncondensed matter systems. Owing the exceptionally long\nlifetime ofmagnetic excitationsYIG is used in microwave\nand spintronic devices that can operate at room temper-\nature. It makes YIG as an ideal platform for the devel-\nopment of microwave magnetic technologies, which have\nalready resulted in the creation of the magnon transis-\ntor and the first magnon logic gate [55, 56]. There is\na significant interest to investigate quantum aspects of\nmagnon dynamics. The YIG can be used as the basis for\nnew solid-state quantum measurement and information\nprocessing technologies including cavity-based QED, op-\ntomagnonics, and optomechanics [57]. That is why the\nformationofmagnonBECinYIGandobservationofspin\nsupercurrent, like in3He, should leads to a development\nof new branch of physics - Supermagnonics.\nMETHODS\nThe sample\nAll samples for experiments were prepared from gar-\nnet ferrite films grown by liquid phase epitaxy on GGG\nsubstrates with the (111) orientation. To reduce the ef-\nfect of cubic anisotropy, we used scandium substituted\nLu1.5Y1.5Fe4.4Sc0.6O12ferrite garnet films; the introduc-\ntion of lutetium ions was necessary to match the param-\neters of the substrate and film gratings. It is known that\nthe introduction of scandium ions in such an amount re-\nduces the field of cubic anisotropy by more than an or-\nder ofmagnitude [58]. In addition, the used lutetium and\nscandium ions practically do not contribute to additional\nrelaxation in the YIG, therefore the width of the FMR\nline of the obtained samples did not exceed 1 Oe at a\nfrequency of 5 GHz. Samples were prepared in the form\nof a disk with a diameter of 500 and 300 µm and a thick-\nness of 6 microns. The disk was located on the front side\nof the substrate with a thickness of 500 µm. The disk\nwas made by photolithography to avoid pinning on the\nsurface of the sample was etched in hot phosphoric acid\n[59]. As a result, the edges of the disk had a slope of 45\ndegrees and had a smooth surface.8\nSpectrometry\nThe CW FMR experiments was performed on Varian\nE-12 X-band EPR spectrometer at the room tempera-\nture and the frequency 9.26 GHz. The amplitude and\nthe frequency of magnetic field modulation were 0.05 Oe\nand 100 kHz, respectively. This frequency is much lower\nthe estimated frequency of second sound of magnon BEC\n(Goldstoun mode). That is why we may consider these\nconditions as stationary.\nThe pulsed FMR experiments were performed on\nBruker ELEXSYS E-580 X-band spectrometer at a fre-\nquency about 9.76 GHz. We were able to use the tem-\nperature from a room to 100 K. The results were prac-\ntically the same since the relaxation processes changes\nvery a little in this region of temperature. 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Technical\nPhysics Letters 41,458461 (2015)." }, { "title": "2211.12247v1.Spatially_Nonuniform_Oscillations_in_Ferrimagnets_Based_on_an_Atomistic_Model.pdf", "content": " Spatial ly Nonuniform Oscillation s in Ferrimagnets \nBased on an Atomistic Model \nXue Zhang1†, Baofang Cai2, Jie Ren1, Zhen gping Yuan1, Zhengde Xu1, Yumeng Yang1,3, \nGengchia u Liang2, Zhifeng Zhu1,3† \n1School of Information Science and Technology, ShanghaiTech University, Shanghai, China \n201210 \n2Department of Electrical and Computer Engineering, National University of Singapore, \nSingapore 117576 \n3Shanghai Engineering Research Center of E nergy Efficient and Custom AI IC, Shanghai, \nChina 201210 \n \nAbstract \nThe ferrimagnet s, such as Gd xFeCo (1-x), can produce ultrafast magnetic switching and \noscillation due to the strong exchange field . The two-sublattice s macrospin model has been \nwidely used to explain the experimental results. However, it fails in describ ing the spatial \nnonuniform magnetic dynamics which gives rises to many important phenomenons such as the \ndomain walls and skyrmions. Here we develop the two-dimensional atomistic model and \nprovide a torque analysis method to study the ferrimagnetic oscillation. Under the spin -transfer \ntorque, the magnetization oscillate s in the exchange mode or the flipped exchange mode . When \nthe Gd composition is increased, the exchange mode firstly disappears , and then appears again \nas the magneti zation compensation point is reached . We show that t hese results can only be \nexplained by analyzing the spatial distribution of magnetization and effective fields . In \nparticular, when the sample is small , a spatial nonuniform oscillation is also observed in the \nsquare film . Our work reveals the importance of spatial magnetic distributions in understanding \nthe ferrimagnetic dynamics. The method developed in this paper provides an important tool to gain a deeper understanding of ferrimagnets and antiferromagnets. The observed ultrafast \ndynamics can also stimulate the development of THz oscillators . \n \nIntroduction \nTerahertz (THz) frequency range s from microwave to infrared [1], which has wide \napplications in the fields of biomedicine [2], materials science [3] and communication [4]. High \nfrequencies can be produced by the current -induced oscillations in magnetic materials. In the \nmost widely used ferromagnet s (FMs), t he frequency ranges from Megahertz (MHz) to \nGigahertz (GHz) [5-7]. To generate and control higher frequency in the THz range, rece nt \nstudies turn to the antiferromagnet s (AFM s) [8-15], which consists of identical sublattices that \nare arranged antiparallelly through the strong exchange interaction . Theoretical studies have \nsuggested that it is possible to control the AFM moments by the spin transfer torque (STT) . \nThe application of spin curren t on AF M leads to a THz pr ecessing frequency. However, the \nmaterial grain structure and the magnetoelastic e ffects make it more complicated to control the \nAFM moments [16, 17] . \nSimilar to the AFM , the exi stence of strong exchange field in the ferr imagnet (FiM) allows \nit to generate high frequency in the THz range [18, 19] . However, the FiM is composed of \ndifferent sublattices , which results in a symmetry breaking in the dynamic equation of the Neel \nvector. In addition , it exhibits finite magnetization, allowing the easy detection using the tunnel \nmagnetoresistance effect (TMR) . Furthermore, t he ability to control the composition allows us \nto fabricate the FiM with different properties [20]. For example, the compositi on can be altered \nto reach the magnetization compensation (xMC) or the angular momentum compensation ( xAMC) [21, 22] . Previous studies have shown that the current induced magnetization oscillation in FiM \ncan be classified as the FM mode with GHz frequency and exchange mode with THz oscillation \n[7, 19] . These theoretical studi es describe the FiM using the two -sublattices macrospin model, \nwhere the magnetization dynamics is described by two coupled Landau -Lifshitz -Gilbert -\nSlonczewski (LLG S) equations [11, 19] . As a result , the two -sublattices macrospin model \ncannot capture the inhomogeneous magnetization dynamics such as the domain wall and the \nskyrmions , which can be significant as we have learned from the FM system [23]. The \nmacrospin model has made a great contribution in describing the dynamics of FM. However, \nas a simplified model , the two -sublattices model lacks the spatial description of the FM system . \nSpecifically, it is difficult to take into account the influence of neighboring atoms on the central \natom. The same is true for FiM. Therefore, the spatial description in the two-dimensional \natomistic model i s particularly important for a more realistic description of the magnetization \ndynamics. \nIn this paper, we have developed a two -dimensional (2D) atomistic model to study the STT \ndriven magnetization dynamics in the FiM, (FeCo) 1-xGdx, where x denotes the Gd composi tion \n[24, 25] . We find that the direction of the charge current Jc determines the chirality of \nmagnetization oscillation . We propose a torque analysis method to underst and this result. In \naddition, t he variation of x leads to different phase diagram s of magnetization oscillation. This \ncan only be understood after taking the spatial nonuniform distribution of magnetic properties \ninto consideration [26]. Furthermore, the size of system has a great influence on the stability \nof oscillation , which can be attribute d to the nonuniform oscillation dynamics induced by the \nedge effect . These new results presented here reveal the necessity of studying the nonuniform magnetic properties in order to correctly underst and the FiM dynamics. \n \nMethod ology \nThe 2D atomistic model is illustrated in Fig. 1(a), where the Gd atoms are randomly \ndistributed [27]. The FiM layer is then used as the free layer in the magnetic tunnel junction \n(MTJ) as shown in Fig. 1(b ). The Jc flows into the FiM layer and creates the STT acting on the \nmagnetization. The m agnetization dynamics in FiM is governed by the coupled LLG S \nequations [28], \n𝜕𝐦𝑖\n𝜕𝑡=−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼𝐦𝑖×𝜕𝐦𝑖\n𝜕𝑡−𝛾𝑖𝐵D,𝑖𝐦𝑖×(𝐦𝑖×𝐩) (1) \nwhere i denotes different sublattice s. p is defined as the polarization of the pinned layer. The \nthree terms on the right -hand side (RHS) represent the precession, the Gilbert damping and the \ndamping -like STT, respectively. The effect ive field (Heff) consists of the exchange interaction \nand crystalline anisotropy . It is obtained from the Hamiltonian ℋ=A∑𝐒𝑖∙𝐒𝑖+1− 𝑖\n𝐾∑(𝐒𝑖∙𝐳̂)2\n𝑖 with the exchange constant A and the anisotropy constant K. As shown in Fig. \n1(a), each atom is surrounded by four neighbors, resulting in three types of exchange \ninteraction, i.e., AGd-Gd = –1.26×10-21 J, AFeCo -FeCo = –2.83×10-21 J, AFeCo -Gd = 1.09×10-21 J. The \nHamiltonian expression of dipolar interaction is: ℋ𝑑𝑖𝑝𝑜𝑙𝑒 =−𝜇0\n4∑3(𝑅𝑖𝑗 ∙ 𝜇𝑖)(𝑅𝑖𝑗 ∙ 𝜇𝑗)\n𝑅𝑖𝑗5 − 𝑗≠𝑖\n𝜇𝑖 ∙ 𝜇𝑗\n𝑅𝑖𝑗3, where 𝑅𝑖𝑗 is the vector connecting spins 𝜇𝑖 and 𝜇𝑗. In the present sample with 100 \natoms, the dipolar field that each atom receives from all other atoms is 103 times s maller than \nthe exchange field. Therefore, we ignore the dipolar interaction. 𝐵D,𝑖=ℏ\n2𝐽c���\n𝑒𝑀s,𝑖𝑡FiM represents \nthe strength of STT, where tFiM is the thickness of the FiM layer, η is the spin transfer efficiency, \nMs is the saturation magnetization. The magnetization dynamics study is performed by using a home -made code that numerically integrates the LLG S equation s through the fourth –order \nRunge –Kutta methods (RKMs) [29]. The parameters are the same as that in [30], based on \nwhich we can determine xMC = 0.23 and xAMC = 0.21. \n \nResult and Discussion \nFig. 1(c) shows the phase diagram of the current driven magnetization dynamics in the \nsample with x = 0.1 . Under a negative Jc, the magn etization first switches (region 1), i.e., mFeCo \nchanges from + z to −z since p is opposite to the net magnetization. When Jc is further increased, \nthe magnetization of both atoms rotates in the counter -clockwise ( CCW ) direction at small \nangle (region 2), which is known as the exchange mode. For a n even larger Jc, the effect of \nspin current overcomes the exchange interaction, resulting in the rotation of mGd in the sphere \nwith mz,Gd < 0 (cf. region 3) , and we call it the flipped exchange mode . In this region, the atoms \nrotate in circles with different area s. However, since the atoms still experience strong exchange \ninteraction, the ir oscillation frequenc ies are identical , which indicates that the magnetization in \nthe larger circle has larger linear speed. Finally, when Jc is further increased , both mFeCo and \nmGd are aligned to the direction of p, i.e., −z direction. \nSimilarly, when the positive Jc is applied, both mFeCo and mGd rotate in the clockwise (CW) \ndirection that is opposite to the one under negative Jc. At larger positive Jc, the system enters \nthe flipped exchange mode and finally both mFeCo and mGd align along p in the + z direction . \nHowever, in the samples with a larger x, a different phase diagram is observed. As shown in \nFig. 1(d) for the sample with x = 0.15, a negative Jc first switches the magnetization , which is \nthe same as the x = 0.1 sample . However, when Jc is further increased, the system directly enters the flipped exchange mode. In this case, the exchange mode , where mFeCo and mGd rotate \nin the opposite direction, does not exist anymore . The disappearance of exchange mode as a \nfunction of x has not been reported before , and it ca nnot be explained using the two -sublattice s \nmacrospin model as discussed below . \nBefore study ing the reason for the different phase diagram s as a function of x, we firstly \nprovide a torque analysis method to understand the ferrimagnetic oscillation. The torque s \nexperienced by each atom under the cur rent can be understood more clearly by convert ing the \nLLG equation (Eq. 1) into the Landau -Lifshitz (LL) form as \n 𝜕𝐦𝑖\n𝜕𝑡(1+𝛼2)=−𝛾𝑖𝐦𝑖×𝐇eff,𝑖−𝛾𝑖𝛼𝐦𝑖×(𝐦𝑖×𝐇eff,𝑖)+𝛼𝛾𝑖𝐵𝐷𝐦𝑖×𝐩−𝛾𝑖𝐵𝐷𝐦𝑖×(𝐦𝑖×𝐩) \n(2) \nfrom this equation, we can see that the stable oscillation can be initiated when the Gilbert \ndamping (the second term on the RHS) is balanced by the damping -like STT (the last term) . \nAs shown in Fig. 2(a), when Jc is applied in the +z direction, the Gilbert damping \n[−m×(m×Heff)] and the damping -like STT [−m×(m×p)] acting on the Gd atom are pointing to \nthe opposite directions. In contrast, the se two torques on the FeCo atom are pointing to the \nsame direction. Therefore, the magnetization oscillation in this case is initiated by the Gd atom , \nand then the FeCo atom is dragged into oscillation via the exchange i nteraction. Since the \noscillation is initiated by the Gd atom, we can then determine the rotation direction by \nanalyzing the precession torque s experienced by Gd, i.e., the first and third terms on the RHS \nof Eq. (2) . As shown in Fig. 2(a), both −m×Heff and m×p are pointing to the same direction, \nresulting in the CW rotation when one looks from the top. This explains the magnetization \noscillation and the rotation direction for the positive Jc region in Fig. 1(c). Similarly, we can analyze the magnetization oscillation in the system where both mFeCo and Jc are pointing to the \n−z direction . As shown in Fig. 2(b), the oscillation is still initiated by the Gd atom , on which \nthe Gilbert damping and the damping -like STT are balance d. However, the atoms rotate in the \nCCW direction as a result of the precession torque . Therefor e, the torque analysis method \npresented here agrees with the numerical phase diagram presented in Fig. 1(c), and we can \nconclude that in the sample with a fixed x, the magneti c oscillation (i.e., balance of torque) and \nrotation direction are determined by the same atom (Gd in this case ) which is not related to the \ndirection of Jc or the state of magnetization. \nBased on the torque analysis method , we find that the steady oscillation only occurs when \nthe Gilbert damping is balanced by the damping -like STT . Therefore, the oscillation mode is \ndetermined by the magnitude of Heff and Jc. For example, i n the sample with a small x = 0.1 , \nthe Gilbert damping can be balanced by the damping -like S TT at Jc = +5×1011 A/m2, allow ing \nthe magneti c oscillation in the exchange mode [marked as the star in Fig. 1(c)]. In contrast, \nwhen x is increased, the amount of Gd atoms is increased, resulting in more Gd-Gd interactions. \nSince AGd-Gd is larger than AFeCo -Gd, Heff,Gd becomes larger . Therefore, when the current \nmaintains at Jc = +5×1011 A/m2, the Gd atom in the sample with increased x can no longer \nmaintain the torque balance required for the oscillation in the exchange mode [marked as the \nstar in Fig. 1(d)]. However, at this point, the oscillation still occurs, but in the flipped exchange \nmode. Now we need to figure out why the torque balance can be achieved in this mode. Since \nthe Gilbert damping is independent on the oscillation mode, it is the Heff,Gd that has to be \nreduced. This can be realized in several ways. Firstly, some FeCo atoms around Gd can be \nflipp ed to reduce Heff,Gd. Assume mz,Gd < 0 and the surrounding mz,FeCo > 0, the corresponding Hex,Gd points to −z direction , which combines with Han,Gd and the resulting Heff,Gd is too large \nto be balanced by the damping like STT. When some FeCo atoms are flipped to mz,FeCo < 0, the \nexchange field s produced by these atoms change to +z direction. This reduce s the Hex,Gd along \nthe −z direction , so that Heff,Gd and the damping like STT can be balanced to initiate the \noscillation. Although the oscillation condition can be satisfied under this picture, it cannot \nexplain the oscillation in the flipped exchange mode at Jc = +5×1011 A/m2, i.e., the average \nmz,Gd is larger than 0 in this mode. Therefore, in addition to the flipping of some FeCo atoms, \nwe can further suspect that some Gd atoms are also switched from mz,Gd < 0 hemisphere to \nmz,Gd > 0 hemisphere to assist the oscillation. For example, when mz,Gd < 0, both Han,Gd and \nHex,Gd point in the −z direction, the damping -like STT provided by Jc has to overcome both of \nthem to initiate the oscillation. In contrast, if Gd atoms are flipped to mz,Gd > 0, Han,Gd changes \nto +z direction, which assist s the damping -like STT to balance with Hex,Gd. Therefore, the \nresults shown Fig. 1(d), i.e., the system oscillates in the flipped exchange mode instead of the \nexchange mode when x is increased to x = 0.15 at Jc = +5×1011 A/m2, can be explained by \ncombining several mechanisms . Furthermore, these explanations point out that it is necessary \nto take the complicate d spatial magnetic information [31] into consideration, which cannot be \ncaptured by the macrospin model and one has to resort to the atomistic model. \nTo verify our explanation s, we then look into the effect of spatial distribution on \nmagnetization and effective fields. In Fig. 3(a), the Gd atoms are marked as the blue sphere s in \nthe sample with x = 0.15. The rest are the FeCo atoms. Under Jc = +5×1011 A/m2, all the atoms \noscillate and the average effect exhibits as the flipped exchange mode which corresponds to \nFig. 1(d). mz of each atom is denoted in the color bar with red and blue represents + z and −z, respectively. It can be clearly seen that the magnetization of some atoms has been flipped to \nthe opposite state , i.e., some FeCo and Gd atoms have been flipped to the mz < 0 and mz > 0 \nhemisphere , respectively. Furthermore, we plot the Hex,z experienced by each atom in Fig. 3(b). \nIt can be seen that the e xchange field nea r the Gd atom are generally small, which form s a \nboundary between the Gd and Fe Co atoms. The apparent drop of the e xchange field at the \nboundary separating Gd and Fe Co atoms supports our explanations that Hex,Gd is required to be \nreduced to maintain the oscillation , and t his can be realized by flipping the magnetization of \nsome Gd and FeCo atoms. In comparison, in the sample with x = 0.1 and Jc = +5 ×1011 A/m2, \nthe magnetization oscillates in the exchange mode as shown in Fig. 1(c). \nIn addition, some discontinuities appear at the boundary between the flipped exchange mode \n(region 3) and the region 4. At this boundary, we observed an unstable oscillation, which does \nnot occur in the sample with x larger than 0.15. This unstable oscillation is manif ested as the \nback and forth fluctuation of the angle between mGd and the + z axis. We attribute these \ndiscontinuities to the unstable oscillation. At this boundary, since most Gd atoms are pointing \nto the hemisphere with mGd > 0, the oscillation condition requires that Heff,Gd should align to \nthe –z direction to balance the torques. This can be achieved by either pulling mFeCo to the +z \naxis or moving mGd away from the +z axis. In the sample with smaller x, Heff,FeCo is larger, \nwhich makes all mFeCo already aligned to the +z axis. Therefore, only the latter option is feasible. \nHowever, in this case, STT pulls mGd to the + z direction. Their competition leads to the back \nand forth movement of mGd. \nFig. 3(c) shows the comparison of current range for the two oscillation modes as a function \nof x. The ratio in the y axis is calculated as the current range of the exchange mode over the entire oscillation range. When x is small, both modes exist, and the current range of the \nexchange mode is around half as small as the flipped exchange mode. As x increases, the ratio \nis gradually reduced. At x = 0.15, the exchange mode disappears and the ratio remains zero \nuntil x = xMC. As explained in Fig. 3(a) and 3(b), this is attributed to the change of spatial \nmagnetic properties when the amount of Gd is varied . Interestingly, a s x is further increased, \nthe exchange mode appears again and the corresponding current range expands as a function \nof x. Noticing that this transition happens at x = xMC, we then explain th is result based on the \nchange of the dominate magnetization when x exceeds xMC. When x is smaller than xMC, the \ndominate magnetization is mFeCo, and the positive Jc drives the magnetization into oscillation \n[see Fig. 1(c)]. Note that this is different from the scenario under negative Jc, where the \nmagnetization switching happens first followed by the oscillation. However, when x exceeds \nxMC, the dominate magnetization changes from mFeCo to mGd. In this case, the characteristics of \nthe positive and negative Jc swap s, i.e., under the positive Jc, the magnetization is firstly \nswitched and then oscillating, whereas it directly enters oscillation under the negative Jc. The \nphase diagram for x > xMC is schematically illustrat ed as the insert of Fig. 3(c). As a result , the \ncorresponding effective fields of FeCo and Gd atoms are also changed. For example, under Jc \n= +5×1011 A/m2, FeCo points to the −z direction whereas Gd points to the +z direction. Using \nthe torque analysis method presented in Fig. 2, we can find that the oscillation is now initiated \nby the FeCo atom , which is different from the sample with x < xMC. To initiate the oscillation, \nthe system resorts to the balance between Heff,FeCo and the damping like STT acting on FeCo. \nIn addition, in the samples with 0.15 < x < xMC, we attribute the disappearance of exchange \nmode to the increase of Heff,Gd as a function of x. However, since the oscillation is determined by FeCo in the samples with x > xMC, and t he exchange interaction between FeCo -FeCo is \nstronger than FeCo -Gd, Heff,FeCo decreases when the x is increased . The reduction of Heff,FeCo \nleads to the balance between the damping like STT and Heff,FeCo . Therefore, the system can \noscillate in the exchange mode , without entering the flipped exchange mode. This explains the \nreappearance of the exchange mode when x exceeds xMC. In addition, Heff,FeCo is further reduced \nas when x is increased, resulting in a larger current range for the oscillation in exchange mode \n[cf. Fig.3(c)] . \nIn the previous section, we discussed ferr imagnetic oscillation with a fixed sample size. As \nwe have seen the importance of the spatial distribution, we finally study the effect of sample \nsize on the ferrimagnetic oscillation . In this section, x is set to 0.5 to avoid “non-integer Gd \natoms ” in different samples. For example, if we want to study the magnetization d ynamics in \ndifferent samples with x fixed at 0.2, the number of Gd atoms will be 3.2 and 12.8 for the \nsamples of 16 and 64 atoms, respectively. However, we have to set them as integer numbers in \nthe code, e.g., 3 and 13. This variation in the number of ato m will lead to an unfair comparison \nfor samples with different size. This can be avoided by setting x to 0.5. As a result, the resulting \nphase diagram for the systems studied in Fig. 4 is the same as the sample with x = 0.1 which is \nillustrated in Fig. 1(c ). The relationship between frequency and Jc at different sizes which is \nshown in Fig. 4(a), for the sample with 16 or 36 atoms, f shows a step when Jc is larger than \n1.3×1012 A/m2. However, for larger samples, f is independent on the size [see Fig. 4(b)]. It is \nworth noting that the nature of discontinuity shown in Fig. 4(a) is different from that in Fig. \n1(c) which has been pointed out in the previous section . We have attributed the discontinuity \nin Fig. 1(c) to the back and forth oscillation of mGd,z. In contrast, as shown in Fig. 4(c) and 4(d), the stable oscillations are confined in the x-y plane with mz remains the same. In addition, the \nfrequency step occurs in the region of the flipped exchange mode (region 3) rather than at the \nboundary of regions 3 and 4. To understand these results, we study the oscillation trajectories \nof the sample with 16 atoms at Jc = 1×1012 A/m2, where a uniform oscillation is observed [see \nFig. 4(c)]. In contrast, for a larger Jc = 1.3×1012 A/m2, the oscillation becomes nonuniform as \nshown in Fig. 4(d) . The oscillation mode s of both Fig. 4(c) and 4(d) belong to the flipped \nexchange mode. This nonuniform oscillation can be understood as the edge effect. In the system \nstudied here, each center atom interacts with four neighboring atoms, where the edge atoms are \nonly affected by two or three nearby atoms. When the number of edge atoms is larger than the \ncenter atoms, the averaged oscillatio n trajectory becomes nonuniform, resulting in the \nfrequency step. For the systems studied here, this condition is only satisfied in samples with 16 \nand 36 atoms, whereas the number of center atoms will be dominat ing in samples with more \nthan 36 atoms . Thes e results also reveal that it is important to use the model that can capture \nthe spatial dynamics during the study of magnetization switching or oscillation in a large sized \nsample. \n \nConclusion \nIn conclusion, the spatial dependent ferrimagnetic oscillation is studied using the two-\ndimensional atomistic model. As the composition of Gd in the sample is increased, it is found \nthat the exchange mode firstly disappears, and then reappears after the magnetization \ncompensation point is reached. By studying the spatial distribution of the magnetization and \nexchange field, we conclude that the spatial nonuniform magnetic properties have to be taken into consideration to correctly understand the magnetic dynamics in ferrimagnets. Furthermore, \nthe oscillation dynamics is strongly affected by the sample size, which again emphasize the \nimportance of the spatial information, which can only be described by the atomistic model. We \nalso proposed a torque analysis method to gain a better understanding on the ferrimagnetic \noscillation. The methodologies and results presented in this paper can greatly stimulate the \nstudy of the ultrafast ferrimagnetic or antiferromagnetic dynamics. \nCorrespondi ng Authors : †zhangxue2@shanghaitech.edu.cn, †zhuzhf@shanghaitech.edu.cn \nAcknowledgemen ts: X.Z, J.R, Z.Y., Z.X. Y.Y and Z.Z. acknowledge the support from the \nNational Key R&D Program of China (Grant No. 2022YFB4401700), Shanghai Sailing \nProgram (Grant No. 20YF1430400) and National Natural Science Foundation of China \n(Grants No. 12104301 and No. 62074099). B.C. and G.L. would thank the support by \nMOE-2017-T 2-2-114, MOE-2019-T 2-2-215 and FRC-A-8000194-0 1-00. \nReference \n[1] P. H. Siegel, IEEE Trans. Microwave Theory Tech. 50, 910 (2002).\n[2] M. C. Beard, G. M. Turner, C. A. Schmuttenmaer, Phys. Med. Biol. 47, 3841 (2002).\n[3] T. -J. Yen, W. Padilla, N. Fang, D. Vier, D. Smith, J. Pendry, D. Basov, X. Zhang, Science. 303, 1494\n(2004). \n[4] M. Tonouchi, Nat. Photonics. 1, 97 (2007).\n[5] A. A. Kovalev, G. E. 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Red and \nblue arrow s denote the magnetization direction of FeCo and Gd, respectively. m is calculated \nby averaging the atoms of the same type. The point marked with the star represent s Jc = +5×1011 \nA/m2. \n \n \nFig. 2. Illustrations of the t orque s experienced by each atom when mFeCo and Jc are pointing in \nthe (a) + z and (b) −z directions. \n \nFig. 3. Spatial distribution of (a) mz and (b) Hex,z in the sample with x = 0.15 and Jc = +5×\n1011A/m2. The squares with blue balls represent Gd atoms and others represent FeCo atoms. (c) \nRatio of current range as a function of x. The insert illustrat es the phase diagram for x > xMC. \n \n \nFig. 4. (a) f as a function of Jc for sample s with different sizes. The number of atoms is shown \nin the legend. (b) f as a function of sample size with Jc = 1.5×1012 A/m2. The average o scillation \ntrajectory of the sample with 16 atoms at (c) Jc = 1×1012 A/m2 and (d) Jc = 1.3×1012 A/m2. \n \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. 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Nath1,\u0003\n1School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram-695551, India\n2Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India\n(Dated: October 16, 2020)\nStructural, electronic, and magnetic properties of modi\fed cubic spinel compound LiNi 0:5Mn1:5O4\nare studied via x-ray di\u000braction, resistivity, DC and AC magnetization, heat capacity, neutron\ndi\u000braction,7Li nuclear magnetic resonance, magnetocaloric e\u000bect, magnetic relaxation, and mag-\nnetic memory e\u000bect experiments. We stabilized this compound in a cubic structure with space group\nP4332. It exhibits semiconducting character with an electronic band gap of \u0001 =kB'0:4 eV. The\ninteraction within each Mn4+and Ni2+sub-lattice and between Mn4+and Ni2+sublattices is found\nto be ferromagnetic (FM) and antiferromagnetic (AFM), respectively. This leads to the onset of a\nferrimagnetic transition at TC'125 K. The reduced values of frustration parameter ( f) and ordered\nmoments re\rect magnetic frustration due to competing FM and AFM interactions. From the7Li\nNMR shift vs susceptibility plot, the average hyper\fne coupling between7Li nuclei and Ni2+and\nMn4+spins is calculated to be \u0018672:4 Oe/\u0016B. A detailed critical behaviour study is done in the\nvicinity ofTCusing modi\fed-Arrott plot, Kouvel-Fisher plot, and universal scaling of magnetization\nisotherms. The magnetic phase transition is found to be second order in nature and the estimated\ncritical exponents correspond to the 3D XY universality class. A large magneto-caloric e\u000bect is\nobserved with a maximum value of isothermal change in entropy \u0001 Sm'\u000011:3 J/Kg-K and a\nmaximum relative cooling power of RCP'604 J/Kg for 9 T magnetic \feld change. The imaginary\npart of the AC susceptibility depicts a strong frequency dependent hump at T=Tf2well below\nthe blocking temperature Tb'120 K. The Arrhenius behaviour of frequency dependent Tf2and\nthe absence of ZFC memory con\frm the existence of superparamagnetism in the ferrimagnetically\nordered state.\nI. INTRODUCTION\nGeometrically frustrated quantum magnets are long\nbeen a \feld of attraction since they provide unique\nopportunity to realize novel quantum phases at low\ntemperatures.[1] One of the most studied geometrically\nfrustrated systems in three dimension (3D) is the anti-\nferromagnetic (AFM) pyrochlore lattice which features a\nnetwork of corner sharing tetrahedras. Prominent exam-\nples in this category are compounds with general formula\nAB2O4(spinels),A2B2O7(pyrochlores), and AB2. In\nparticular, in spinel oxides AB2O4,Bsite ion forms a\nfrustrated 3D pyrochlore lattice. Owing to ground state\ndegeneracy, these compounds have witnessed various ex-\notic low temperature phenomena ranging from quantum\nspin-liquid, spin-glass, \feld induced transitions, magneti-\nzation plateaus to heavy fermionic behaviour.[2{4] In ad-\ndition, there exists another series of compounds AB2X6,\noften referred as cubic modi\fed pyrochlore lattice which\nmostly contains either mixed-valent or two kinds of tran-\nsition metal ions. The compounds (Rb,Cs)Cr 2F6,[5]\n(K,Rb)Os 2O6,[6] CsW 2O6,[7] and CsNiCrF 6[8] belong to\nthis category and exhibit various exotic ground states.\nAmong the large class of spinel oxides, cubic LiMn 2O4\n(space group: Fd\u00163m, Mn3+: Mn4+= 1: 1) is known\nto be a celebrated high-voltage cathode material for\nrechargeable Li-ion battery.[9] It is reported to have\n\u0003rnath@iisertvm.ac.incharge ordering accompanied by orbital ordering due\nto the Jahn-Teller distortion in Mn3+ions and under-\ngoes an antiferromagnetic (AFM) long-range-ordering\n(LRO) at low temperatures.[10] In certain reports, the\ncompound is found to show a spin-glass (SG) be-\nhaviour without any magnetic LRO which is attributed\nto the Mn3+/Mn4+super-lattice charge order as well\nas the e\u000bect of frustration.[11] These contradictory be-\nhaviours are believed to be originated from the Mn\nsite disorder.[12] Recently, coexistence of LRO and SG\nstate is found in LiMn 2O4nanorods.[12] The Ni doped\nLiNi 0:5Mn1:5O4(abbreviated as LNMO) crystallizes in\ntwo di\u000berent phases depending on the synthesis condi-\ntions. The stoichiometric LNMO has P4332 space group\nand exhibits a 1: 3 cation order of Ni2+and Mn4+ions,\nwhile non-stoichiometric LiNi 0:5Mn1:5O4\u0000\u000ehasFd\u00163m\nspace group.[13]\nIn theFd\u00163mstructure, Ni and Mn atoms randomly oc-\ncupy one crystallographic site while in the P4332 struc-\nture, they occupy two inequivalent sites independently.\nIn theP4332 structure which can also be referred as\nmodi\fed cubic spinel, the edge sharing of MnO 6and\nNiO 6octahedras and the corner shared LiO 4tetrahedra\nlead to a complex three-dimensional (3D) structure [see\nFig. 1(a)]. When only the interaction among the Mn4+\nions is considered, LNMO forms a 3D network of corner\nsharing Mn4+triangles which is found to be a frustrated\nhyper-kagome lattice [see Fig. 1 (b)]. Further, when the\ninteraction between Mn4+and Ni2+ions are taken into\naccount, a network of corner shared tetrahedras is formedarXiv:2010.07709v1 [cond-mat.mtrl-sci] 15 Oct 20202\n(a) Mn\nNiLi\nO\nb\na\nc\nabc(b)\na\nbc(d) (c)\nabc\nFIG. 1. (a) Crystal structure of LNMO in three dimension. (b) Hyperkagome lattice made of Mn4+ions. (c) Pyrochlore lattice\nmade of Ni2+and Mn4+ions. (d) A section of the pyrochlore lattice showing corner sharing tetrahedras and spin structure\ndeduced from the neutron di\u000braction data at T= 5 K.\nwhere each tetrahedra consists of three Mn4+and one\nNi2+ions. Hence, the hyper-kagome lattice transforms\ninto a 3D pyrochlore lattice [see Fig. 1(c)]. Based on the\npreliminary magnetic measurements, LNMO is reported\nto show a magnetic transition at TC'125 K.[14]\nIn this paper, we present a detailed study of the phys-\nical properties of stoichiometric modi\fed cubic spinel\ncompound LNMO ( P4332). A ferrimagnetic order is de-\ntected atTC'125 K. We found that the interaction\nwithin each Mn4+and Ni2+sub-lattice is ferromagnetic\n(FM), whereas the interaction between these two sub-\nlattices is antiferromagnetic (AFM), which results in a\nferrimagnetic behaviour below TC. Multiple magnetic\ntransitions are observed below TC, likely due to mag-\nnetic frustration. It exhibits magnetic relaxation and\nmagnetic memory e\u000bect below TC, typically expected for\nsuperparamagnetic systems. A large magnetocaloric ef-\nfect (MCE) is obtained across the magnetic transition.\nThe critical analysis of magnetization and MCE data es-\ntablish LNMO as a 3D XY type magnet. The paper\nis organized in the following manner. The experimental\ndetails concerning sample preparation and various mea-\nsurements are described in Sec. II. Section III contains\nthe experimental results which includes powder x-ray\ndi\u000braction, resistivity, DC magnetization, heat capacity,\nneutron di\u000braction,7Li NMR, magnetocaloric e\u000bect, AC\nsusceptibility, magnetic relaxation, and magnetic mem-\nory e\u000bect measurements, followed by discussions. Our\nexperimental \fndings are summarized in Sec. IV.\nII. METHODS\nTraditional sol-gel synthesis method was adopted to\nsynthesize LNMO in polycrystalline form. At \frst, sto-\nichiometric amount of lithium nitrate (LiNO 3, 99.99%),\nmanganese nitrate tetra-hydrate [Mn(NO 3)2.4H2O,\u0015\n97%], and nickel nitrate hexa-hydrate [Ni(NO 3)2.6H2O,\n99.999%] were taken and dissolved into ethanol. Themixture was continuously stirred at 800C until the whole\nsolvent is evaporated from the mixture and dark black\ncoloured paste was found. The resulting paste was then\ntransferred into a crucible and preheated at 5000C for\n2 hrs and then at 8000C for 8 hrs. Subsequently, the fur-\nnace was switched o\u000b and the sample was cooled natu-\nrally within the furnace. The resultant sample was found\nto be formed in the space group Fd\u00163m, con\frmed from\nthe powder x-ray di\u000braction. In the next step, the re-\nsultant sample was ground thoroughly and pressed into\npellets. The pellets were heated at 7000C for 2 days and\nthen cooled very slowly to room temperature at a rate of\n0.10C/min. This post \fring of the Fd\u00163mphase sample\nat 7000C was done to ensure the formation of the cation\norderedP4332 phase. This method is well established\nand already experimented previously.[15]\nPhase purity of the sample was checked from the high\nquality powder x-ray di\u000braction (XRD) data, collected\nusing a PANalytical x-ray di\u000bractometer (Cu K \u000bradi-\nation,\u0015av'1:5418 \u0017A). The temperature dependent\npower x-ray di\u000braction was performed over a wide tem-\nperature range (15 K \u0014T\u0014300 K). For going below\nroom temperature, an Oxford Phenix low-temperature\nattachment to the di\u000bractometer was used. To solve\nthe magnetic structure, temperature dependent neutron\npowder di\u000braction (NPD) experiment was performed us-\ning the neutron powder di\u000bractometer ( \u0015'1:094\u0017A)\nwith three linear position-sensitive detectors at Dhruva\nreactor, Bhabha Atomic Research Center, India. Ri-\netveld re\fnement of the powder XRD data and NPD data\nwas performed using FullPROF software package.[16]\nThe DC magnetization ( M) was measured using a\nvibrating sample magnetometer (VSM) attachment to\na commercial Physical Property Measurement System\n(PPMS, Quantum Design) as a function of temperature\n(2 K\u0014T\u0014600 K) and magnetic \feld (0 to 9 T).\nFor the high temperature measurements ( T\u0015380 K),\na high-Toven was attached to the VSM. Similarly, AC\nsusceptibility was measured as a function of temperature\n(2 K\u0014T\u0014200 K) and frequency (50 Hz \u0014\u0017\u001410 kHz)3\nin an AC \feld of 5 Oe using ACMS option of the PPMS.\nFor the temperature dependent heat capacity ( Cp) mea-\nsurement, the relaxation technique was adopted and the\nmeasurement was carried out on a pressed pellet using\nheat capacity option of the PPMS. Electrical resistivity\n(\u001a) as a function of temperature was measured on a rect-\nangular pellet using the four probe technique in PPMS.\nThe NMR measurements were performed by employing\npulsed NMR technique on7Li (nuclear spin I= 3=2 and\ngyromagnetic ratio \rN=2\u0019= 16:546 MHz/T) nuclei over\na wide temperature range (4 K \u0014T\u0014290 K). For this\npurpose, we have used a liquid helium cryostat (Janis,\nUSA) with a \feld sweep superconducting magnet and a\nTecmag (Redstone) spectrometer. The spectral measure-\nments at di\u000berent temperatures were carried out either\nby Fourier transform of the NMR echo signal at a \fxed\n\feld ofH= 1:5462 T or by sweeping the \feld at a cor-\nresponding \fxed frequency of 25.58 MHz. Traditional\nsaturation recovery pulse sequence was used to measure\nthe7Li spin-lattice relaxation time ( T1).\nIII. RESULTS AND DISCUSSION\nA. X-ray Di\u000braction\nFigure 2 presents the XRD pattern of LNMO at two\nend temperatures 300 K and 15 K. To evaluate the unit\ncell parameters and atomic positions, Rietveld re\fnement\nwas performed on the powder XRD data. The initial\nstructural parameters for this purpose were taken from\nRef. [15]. All the peaks could be successfully indexed\nwith cubic non-centrosymmetric space group P4332. The\nre\fned unit cell parameters, volume of the unit cell, and\nthe atomic co-ordinates at room temperature are listed\nin Table I which are in close agreement with the previ-\nous report. As already described in Sec. I, LNMO ex-\nhibits 1:3 cation order, resulting in a superstructure cu-\nbicP4332 space group. The cation ordering in LNMO\ncan be visualized by the emergence of several low an-\ngle and low intensity Bragg peaks, such as (110), (210),\n(322), (410) etc.[17] These peaks are not allowed in the\nBragg re\rections in the normal face-centered cubic spinel\n(Fd\u00163m) structure, because of the re\rection conditions\n(h+k) = 2n, (h+l) = 2n, and (k+l) = 2n. Here,nis\nthe integer and ( h;k;l ) are the Miller indices. These low\nintensity peaks are highlighted in the inset of the upper\npanel of Fig. 2. It can be seen that all these small Bragg\nre\rections could be perfectly indexed by using P4332\nspace group. We have also tried to do the re\fnement\nwithFd\u00163mspace group, which could not index these\nsmall Bragg peaks, thus con\frming the phase purity of\nthe sample with P4332 space group.\nIt is reported that the material synthesis following sol-\ngel method leads to the formation of nano-crystalline\nform of LNMO.[18] To estimate the crystallite size, we\n\ftted the XRD peaks at room temperature by a Gaus-\nsian function and evaluated the full width at half maxima\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s84 /s32/s61/s32/s51/s48/s48/s32/s75/s32\n/s32 /s32/s49/s46/s56/s54\n/s32/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s52/s56 /s53/s52 /s54/s48/s32/s40/s51/s50/s50/s41\n/s40/s52/s49/s48/s41/s40/s51/s51/s48/s41\n/s40/s52/s49/s49/s41\n/s40/s52/s50/s49/s41/s40/s51/s51/s50/s41 /s40/s52/s51/s48/s41/s40/s52/s51/s49/s41\n/s40/s53/s49/s48/s41/s40/s52/s51/s50/s41\n/s40/s53/s50/s48/s41\n/s49/s54 /s50/s48 /s50/s52/s40/s50/s49/s48/s41\n/s40/s49/s49/s48/s41\n/s32/s32/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s32/s32/s73\n/s111/s98/s115\n/s32/s73\n/s99/s97/s108\n/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110\n/s32/s73\n/s111/s98/s115/s32/s45/s32/s73\n/s99/s97/s108\n/s32/s32\n/s84 /s32/s61/s32/s49/s53/s32/s75/s32\n/s32 /s32/s49/s46/s56/s48FIG. 2. Upper panel: X-ray powder di\u000braction pattern (open\ncircles) of LNMO at room temperature. The red solid line\nis the Rietveld re\fnement \ft with P4332 space group. The\nexpected Bragg positions are indicated by green vertical ticks.\nBlue solid line at the bottom denotes the di\u000berence between\nobserved and calculated intensities. Inset: Enlarged portion\nof the XRD pattern to visualize small Bragg peaks. Lower\npanel: Rietveld re\fnement of the XRD pattern of LNMO at\n15 K.\n(FWHM) of the individual peak. Subsequently, by using\nthe Scherrer equation, \u001c=K\u0015=\f cos\u0012(where,\u001cis the\ncrystallite size, Kis the dimensionless shape factor which\nhas a typical value of 0.9, \u0015is the x-ray wavelength, and \f\nis the line broadening at FWHM), the average crystallite\nsize is calculated to be \u001860 nm.[19] Further, the analysis\nof the Scanning-Electron-Microscopy (SEM) data also re-\nveals the nano-crystalline nature of LNMO sample with\naverage particle size 100-150 nm.\nAs shown in lower panel of Fig. 2, no extra peaks could\nbe detected down to 15 K. Figure 3 depicts the tempera-\nture variation of lattice constant ( a) and unit cell volume\n[Vcell(T)] obtained from the re\fnement. Both the quan-\ntities are found to decrease systematically during cooling\nand neither any structural transition nor any lattice dis-\ntortion is observed in the entire measured temperature\nrange (15 K\u0014T\u0014300 K). Following the method de-\nscribed in Ref. [20], Vcell(T) is \ftted by\nVcell(T) =\rU(T)=K0+V0; (1)\nwhereV0is the unit cell volume of the crystal structure at\nT= 0 K,K0is the bulk modulus, and \ris the Gr uneisen4\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s53/s52/s50/s46/s55/s53/s52/s51/s46/s54/s53/s52/s52/s46/s53/s32/s86\n/s99/s101/s108/s108/s32/s40/s197/s51\n/s41\n/s84 /s32/s40/s75/s41/s32/s86 /s32\n/s99/s101/s108/s108\n/s32/s32/s32 /s32/s102/s105/s116\n/s56/s46/s49/s53/s48/s56/s46/s49/s53/s53/s56/s46/s49/s54/s48/s56/s46/s49/s54/s53\n/s32/s97\n/s97 /s32/s40/s197/s41\nFIG. 3. Variation of lattice constant ( a) and unit cell volume\n(Vcell) with temperature. The solid line represents the \ft of\nVcell(T) by Eq. (1).\nTABLE I. Crystal structure data for LNMO at room tem-\nperature (cubic, space group: P4332). The obtained lattice\nparameters from the re\fnement are a=b=c= 8:1643(1) \u0017A\nandVcell'544:2\u0017A3. Our \ft yields quality factors Rp'10:1,\nRwp'6:66,Re'4:88, and goodness of \ft \u001f2= [Rwp\nRe]2'\n1:87. Listed are the Wycko\u000b positions and the re\fned atomic\ncoordinates ( x,y, andz) for each atom.\nAtom Site x y z\nLi 8c\u00000:0002(4)\u00000:0002(4)\u00000:0002(4)\nNi 4b 0:625 0 :625 0 :625\nMn 12 d 0:125 0 :3768(3) 0 :8732(3)\nO1 8c 0:3816(7) 0 :3816(7) 0 :3816(7)\nO2 24e 0:1509(6)\u00000:1395(9) 0 :1238(7)\nparameter. U(T) is the internal energy and it can be\nexpressed in terms of the Debye approximation as\nU(T) = 9pkBT\u0012T\n\u0012D\u00133Z\u0012D=T\n0x3\nex\u00001dx: (2)\nHere,pis the number of atoms in the unit cell and kBis\nthe Boltzmann constant. The parameters evaluated from\nthe \ftting are Debye temperature \u0012D'965 K,\r=K 0'\n1:24\u000210\u00004Pa\u00001, andV0'542:48\u0017A3.\nB. Resistivity\nTemperature dependent electrical resistivity [ \u001a(T)]\nmeasured in zero \feld is shown in Fig. 4. It increases\nrapidly with decreasing temperature which indicates that\nthe ground state is insulating in nature. Below 218 K,\n\u001a(T) could not be measured since it exceeded the mea-\nsurable range of the instrument. To evaluate the acti-\nvation energy, the temperature dependent conductivity\n/s50/s50/s53 /s50/s53/s48 /s50/s55/s53 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48\n/s48/s46/s48/s48/s51/s54 /s48/s46/s48/s48/s52/s50/s45/s49/s56/s45/s49/s53/s45/s49/s50\n/s32\n/s84 /s32/s40/s75/s41/s32 /s40 /s109 /s41/s32/s32/s102/s105/s116\n/s32/s32/s32\n/s32\n/s49/s47 /s84 /s32/s40/s75/s45/s49\n/s41/s108/s110/s32/s91 /s40 /s32/s109/s45/s49\n/s41/s93FIG. 4. The electrical resistivity \u001a(T) of LNMO in zero \feld.\nInset: ln(\u001b) vs 1=T. The solid line is the \ft using Eq. (3).\n(\u001b= 1=\u001a) data were \ftted by Arrhenius equation\n\u001b(T) =Aexp\u0012\n\u0000\u0001\nkBT\u0013\n; (3)\nwhere,Ais the proportionality constant and \u0001 is\nthe activation energy. In the inset of Fig. 4, ln( \u001b)\nis plotted against 1 =Tto highlight the activated be-\nhaviour. Our \ft in the whole measured temperature\nrange (218 K\u0014T\u0014300 K) yields \u0001 =kB\u00190:4 eV. This\nvalue of \u0001=kBcategorizes LNMO as a semiconductor.\nC. DC Magnetization\nThe upper panel of Fig. 5 presents the temperature\ndependent DC magnetic susceptibility \u001f(T) (\u0011M=H )\nmeasured in an applied \feld of H= 0:5 T and 1.5 T.\nIn the high temperature regime, \u001f(T) shows a grad-\nual increase with decreasing temperature. Below about\n140 K,\u001f(T) increases rapidly, indicating the onset of a\nferrimagnetic/ferromagnetic ordering. From the d\u001f=dT\nvsTplot, the ordering temperature is found to be\nTC'125 K. As depicted in the inset of the upper panel\nof Fig. 5, the zero-\feld cooled (ZFC) and \feld cooled\n(FC) susceptibility data at H= 50 Oe show a signi\f-\ncant bifurcation below TC. Such an irreversibility is a\ncharacteristic behaviour of ferrimagnetic/ferromagnetic\ncompounds[21] and is also observed for various SG[22, 23]\nand superparamagnetic[24] systems. Moreover, the ZFC\n\u001f(T) shows a well de\fned maxima at Tb'120 K, which\ncorresponds to the blocking temperature, typically ex-\npected for a superparamagnet. A signi\fcant di\u000berence\nin the behaviour of FC \u001f(T) is expected between a su-\nperparamagnet and a SG system. For instance, FC\nmagnetization always increases for a superparamagnet\nwhereas for a SG system, it either remains \rat or de-\ncreases with decreasing temperature, below Tb.[25, 26]\nAs noticed from Fig. 5, the FC \u001f(T) belowTbincreases\nmonotonously with decreasing T, which is a primary indi-\ncation of the superparamagnetic blocking process.[25, 27]5\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s50/s51/s52\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s49/s53/s51/s48\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s50/s48/s50/s32 /s32/s40/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s32/s84 /s32/s40/s75/s41/s32\n/s48/s72 /s32/s61/s32/s48/s46/s53/s32/s84\n/s32\n/s48/s72 /s32/s61/s32/s49/s46/s53/s32/s84/s32/s70/s67\n/s32/s90/s70/s67/s32/s32\n/s32 /s84 /s32/s40/s75/s41/s32/s40/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s32/s32\n/s72 /s32/s61/s32/s53/s48/s32/s79/s101/s84\n/s98\n/s32/s32/s49/s47 /s32/s40/s99/s109/s51\n/s47/s109/s111/s108/s41/s45/s49\n/s32/s84 /s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32\n/s48/s72 /s32/s61/s32/s48/s46/s53/s32/s84\n/s32/s102/s105/s116/s48/s72 /s32/s40/s84/s41\n/s32/s32/s77 /s32/s40 /s47/s102/s46/s117/s46/s41/s32\n/s84 /s32/s61/s32/s50/s32/s75\nFIG. 5. Upper panel: Temperature dependent DC magnetic\nsusceptibility \u001f(T) at two di\u000berent applied \felds. Inset: \u001f(T)\nmeasured at 50 Oe in ZFC and FC conditions. Lower panel:\n1=\u001fvsTin\u00160H= 0:5 T. Inset: Magnetization isotherm at\nT= 2 K.\nIn order to con\frm this behaviour, we have performed a\ndetailed ac susceptibility and magnetic memory e\u000bect ex-\nperiments which are discussed later.\nThe lower panel of Fig. 5 shows the inverse magnetic\nsusceptibility (1 =\u001f) for\u00160H= 0:5 T. In the paramag-\nnetic regime ( T > T C), 1=\u001ftypically shows a linear be-\nhaviour with temperature, due to uncorrelated moments.\nIn contrast, the observed high temperature non-linear be-\nhaviour with a strong positive curvature is a possible sig-\nnature of ferrimagnetic nature of LNMO.[21, 28] To ex-\ntract the magnetic parameters, we \ftted the \u001f(T) data\nby the modi\fed Curie-Weiss law\n\u001f(T) =\u001f0+C\nT\u0000\u0012CW: (4)\nHere,\u001f0is the temperature-independent susceptibility,\nCis the Curie constant, and \u0012CWis Curie-Weiss tem-\nperature. Our \ft in the high-temperature regime ( T\u0015\n450 K) (see the lower panel of Fig. 5) yields the param-\neters:\u001f0'0:0015 cm3/mol,C'1:604 cm3K/mol,\nand\u0012CW'144:4 K. From the value of C, the e\u000bec-\ntive moment is calculated to be \u0016e\u000b=p\n3kBC=N A\u00162\nB'\n3:58\u0016B, whereNAis the Avogadro's number. This\nis close to the expected spin-only value of \u0016e\u000b=s\u00120:5\u0002[\u0016e\u000b(Ni2+)]2+ 1:5\u0002[\u0016e\u000b(Mn4+)]2\n0:5 + 1:5\u0013\n\u0016B=\n3:64\u0016B, taking\u0016e\u000b= 2:83\u0016Band 3.87\u0016Bfor Ni2+(S=\n1) and Mn4+(S= 3=2), respectively.[14] The posi-\ntive value of \u0012CWimplies that the dominant interaction\namong the magnetic ions is ferromagnetic (FM) in na-\nture. The inset of the lower panel of Fig. 5 shows a com-\nplete magnetization isotherm ( MvsH) atT= 2 K. It\nshows a very weak hysteresis and the magnetization sat-\nurates quickly at \u00160H= 0:5 T, typically expected for a\nferrimagnet. The saturation magnetization ( Ms) is found\nto be\u00183:2\u0016B. Assuming a two sub-lattice model of\nmagnetic species Ni2+and Mn4+with antiferromagnetic\n(AFM) coupling between them and using molecular-\feld\napproximation, the saturation magnetization for LNMO\ncan be written as Ms= [S(Mn4+)\u0002g\u00021:5\u0000S(Ni2+)\u0002\ng\u00020:5]\u0016B.[21, 28] Taking S(Mn4+) = 3=2,S(Ni2+) = 1,\nandg= 2,Msis calculated to be 3 :5\u0016B. Usually, the\nvalue ofgfor Mn4+and Ni2+is always more than 2 which\nshould produce Mslarger than 3 :5\u0016B.[29] Clearly, our\nexperimental value of Msis smaller than the expected\nspin-only value.\nThe extent of frustration in a spin system can be\nquanti\fed by the frustration ratio f=j\u0012CWj\nTC.[1] Ac-\ncording to the mean \feld theory \u0012CWis nothing but\nthe sum of all exchange couplings present in the sys-\ntem i.e.\u0012CW=P\niJi, whereiis the number of nearest\nneighbor spins.[30] Typically, for a non-frustrated AFM\nsystem\u0012CW\u0018TC(orTN) andfhas a value close to\n1. However, for a highly frustrated antiferromagnet, f\nvalue is much larger than 1 ( >10).[31] On the other\nhand, for a system having FM and AFM interactions,\nthe value of \u0012CWis reduced due to opposite sign of the\nexchange couplings. This results in a decreased value of\nf. For LNMO, the frustration ratio is calculated to be\nf'144:4=125'1:16. Since LNMO is having highly\nfrustrated pyrochlore geometry, a reduced value of f\nclearly implies co-existence of AFM and FM interactions\nin the system. Indeed, our neutron powder di\u000braction\nexperiments (discussed later) con\frm this proposition.\nD. Heat Capacity\nThe temperature dependent heat capacity [ Cp(T)]\nmeasured in zero \feld is shown in the upper panel of\nFig. 6. A sharp and distinct peak is observed at TC'\n124 K, indicating the magnetic transition. In order to\nanalyze the low temperature Cp(T) data, we \frst used\nthe relation Cp(T) =\fT3, which did not \ft the data\ne\u000bectively. However, the low temperature Cp(T) data\ncould be \ftted nicely by adding an extra term \u000eT3=2to\nthe above relation, i.e., Cp(T) =\fT3+\u000eT3=2. Here,\nthe \frst term ( \fT3) represents the lattice contribution\nand the second term ( \u000eT3=2) is typical for ferromag-\nnetic/ferrimagnetic and glassy systems.[32, 33] The in-\nset of the upper panel of Fig. 6 depicts the enlarged6\n/s48/s52/s48/s56/s48/s49/s50/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48 /s56 /s49/s54/s48/s49\n/s32\n/s32/s32\n/s32/s67\n/s112\n/s32/s67/s32\n/s112/s104/s32\n/s32/s67/s32\n/s109/s97/s103/s67\n/s112/s32/s40/s74/s47/s109/s111/s108/s45/s75/s41\n/s32/s67\n/s109/s97/s103/s47/s84 /s32/s40/s74/s47/s109/s111/s108/s45/s75/s50\n/s41\n/s84 /s32/s40/s75/s41/s48/s49/s48/s50/s48\n/s83\n/s109/s97/s103/s32/s40/s74/s47/s109/s111/s108/s45/s75/s41\n/s48/s49/s48/s50/s48/s32/s102/s105/s116\n/s32/s67\n/s112/s32/s40/s74/s47/s109/s111/s108/s45/s75/s41/s32\n/s84 /s32/s40/s75/s41\nFIG. 6. Upper panel: Cp(T) measured in zero magnetic \feld.\nThe dashed and solid lines represent the phonon ( Cph) and\nmagnetic (Cmag) contributions, respectively. Inset: Enlarged\nview of low temperature portion of Cp(T) data. The solid\nline is the \ft as described in text. Lower panel: Cmag=Tand\nmagnetic entropy ( Smag) vsTalong the left and right y-axes,\nrespectively.\nview of the low temperature portion of Cp(T). The solid\nline is the \ft using the above relation, in the temper-\nature range 2-16 K. The resulting \fand\u000evalues are\n\u00181:07\u000210\u00004J mol\u00001K\u00004and\u00180:0081 J mol\u00001K\u00005=2,\nrespectively. The electronic contribution \rTis not con-\nsidered in the \ftting procedure, since LNMO is an insu-\nlator at low temperatures.\nFor an estimation of the phonon contribution Cph(T),\nwe \ftted the experimental Cp(T) data in the high tem-\nperature regime ( T\u0015160 K) by the Debye function\nCph(T) = 9R\u0012T\n\u0012D\u00133Z\u0012D=T\n0x4ex\n(ex\u00001)2dx: (5)\nHere,Ris the universal gas constant. The best \ft was\nobtained with \u0012D'735 K which is close to the value ob-\ntained from the Vcell(T) analysis. The high temperature\nCph(T) \ft was extrapolated down to low temperatures\nand subtracted from the experimental Cp(T) data to ob-\ntain the magnetic contribution Cmag(T). The obtained\nCmag(T) is shown as a solid line in the upper panel of\nFig. 6.Cmag(T)=TvsTis presented in the left y-axis of\nthe lower panel of Fig. 6. The magnetic entropy Smag(T)\nFIG. 7. Temperature evolution of the neutron powder\ndi\u000braction pattern of LNMO shown for the low angle regime.\nAn enhancement in the intensity of fundamental nuclear\nBragg peaks is evident below TC. These Bragg peaks at\n2\u0012'10:81\u000e, 13:31\u000e, 17:17\u000e, and 18:84\u000ecorrespond to (1,1,0),\n(1,1,1), (2,1,0), and (2,1,1) planes, respectively.\nis calculated by integrating the Cmag(T)=Tin the whole\nmeasured temperature range as\nSmag(T) =ZT\n2:1KCmag(T)\nTdT: (6)\nAs shown in the lower panel of Fig. 6, the value of\nSmagis found to be\u001820:6 J/mol-K at 250 K, which is\nvery close to the theoretically expected value of Smag=\n0:5\u0002Rln[2S(Ni2+) + 1] + 1:5\u0002Rln[2S(Mn4+) + 1] =\n21:8 J/mol-K. One broad hump is observed in Cmag(T)\nat aroundT\u001850 K which cannot be attributed to any\nmagnetic transition as \u001f(T) does not show any feature at\nthis temperature. Similar feature is reported earlier for\nBaMn 2As2and BiMn 2PO6where it is proposed that this\nbroad hump is associated with the temperature depen-\ndent change in the population of the Zeeman levels below\ntransition, arising due to the temperature dependent ex-\nchange \feld.[34, 35] Using a Weiss molecular \feld theory\non a Heisenberg model, it is predicted that the hump\nlike feature is pronounced for the systems with higher\nspin values.[34, 35] Thus, the observed hump in Cmag(T)\nis obvious since LNMO has two high spins: S= 1 and\nS= 3=2.\nE. Neutron Di\u000braction\nFigure 7 depicts a series of neutron powder di\u000braction\n(NPD) patterns for LNMO, over the temperature range\n5\u0000300 K. An enhancement in the intensity of funda-\nmental nuclear Bragg peaks at \u001810:81\u000e, 13:31\u000e, 17:17\u000e,\nand 18:84\u000eare observed below TC. The observation of\nadditional intensity at the positions of fundamental nu-\nclear Bragg re\rections (with no additional peaks) indi-\ncate the presence of a ferro- or ferrimagnetic ordering.\nAs shown in the upper panel of Fig. 8, all the nuclear7\n/s48/s49/s48/s48/s48/s50/s48/s48/s48/s51/s48/s48/s48\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s49/s48/s48/s48/s50/s48/s48/s48/s51/s48/s48/s48/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s32/s32\n/s32/s32\n/s32/s32\n/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s84 /s32/s61/s32/s49/s52/s48/s32/s75/s78/s101/s117/s116/s114/s111/s110/s32/s67/s111/s117/s110/s116/s115/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s84 /s32/s61/s32/s53/s32/s75/s32\n/s32/s78/s105\n/s32/s77/s110/s77/s111/s109/s101/s110/s116/s32/s40\n/s66 /s47/s105/s111/s110/s41\n/s84 /s32/s40/s75/s41\nFIG. 8. Rietveld re\fned neutron powder di\u000braction pat-\nterns atT= 140 K (upper panel) and T= 5 K (lower panel).\nOpen circles represent the experimental data, solid line rep-\nresents the calculated curve, and di\u000berence between them is\nshown as a solid line at the bottom. Vertical marks corre-\nspond to the position of all allowed Bragg re\rections for the\nnuclear (top row) and magnetic (bottom row) re\rections. In-\nset: Temperature variation of ordered magnetic moment for\nNi2+and Mn4+ions.\npeaks atT= 140 K could be re\fned using cubic crystal\nstructure with space group P4332. The re\fned struc-\ntural parameters are listed in Table II. These values are\nin close agreement with the re\fned values obtained from\nthe powder XRD data. The lower panel of Fig. 8 shows\nthe Rietveld re\fnement of NPD pattern at T= 5 K. All\nthe low angle peaks with additional intensity could be in-\ndexed with a propagation vector k= (0;0;0) and space\ngroupP4332. The symmetry analysis shows that the ob-\nserved NPD patterns can be \ftted assuming a collinear\nferrimagnetic structure with magnetic moments aligned\nalong the [110] direction. The magnetic spin structure\ndetermined from Rietveld re\fnement of the NPD pat-\ntern atT= 5 K is shown in Fig. 1(d). Within each Ni2+\nor Mn4+sublattice, the moments are arranged parallel\nto each other providing a FM intra-sublattice interac-\ntion whereas between the Ni2+and Mn4+sublattices the\nalignment is found to be antiparallel which provides an\nAFM inter-sublattice interaction. From Fig. 1(d) it is\nclearly evident that the system is still frustrated due toTABLE II. Structural parameters of LNMO re\fned from the\nneutron di\u000braction data at T= 140 K (Structure: cubic\nand Space group: P4332). The obtained lattice parame-\nters from the re\fnement are a=b=c= 8:2000(8) \u0017Aand\nVcell'551:36(9) \u0017A3. The \ft yields quality factors Rp'21:5,\nRwp'18:2,Re'15:8, and goodness of \ft \u001f2= [Rwp\nRe]2'\n1:327. Listed are the Wycko\u000b positions and the re\fned atomic\ncoordinates ( x,y, andz) for each atom.\nAtom Site x y z\nLi 8c 0:0002(19) 0 :0002(19) 0 :0002(19)\nNi 4b 0:625 0 :625 0 :625\nMn 12 d 0:125 0 :3787(11) 0 :8713(11)\nO1 8c 0:3809(5) 0 :3809(5) 0 :3809(5)\nO2 24 e 0:1503(4)\u00000:1417(5) 0 :1298(5)\ncompeting FM and AFM interactions. However, the ex-\ntent of frustration is de\fnitely less than a conventional\nAFM pyrochlore lattice. This indeed fall in line with the\n\u001f(T) analysis where the reduced value of fis attributed\nto the co-existence of AFM and FM interactions.\nOne can also understand the exchange interactions by\nlooking at the bond angles. From the re\fnement of NPD\ndata atT= 5 K, we found the angles 6Mn\u0000O1\u0000Mn'\n93:8\u000e,6Mn\u0000O2\u0000Mn'98:4\u000e, and6Mn\u0000O2\u0000Ni'\n96:80\u000eand 95:3\u000e. According to Goodenough-Kanamori\nrule,[36] the superexchange interaction between Mn4+\n(3d3) ions through the O2\u0000(2p2) ion is expected to\nbe AFM when the angle 6Mn\u0000O\u0000Mn is linear\n(\u0018180\u000e) and it crosses over to FM interaction for the\nangle<135\u000e.[37] On the contrary, for the interaction\nbetween Mn4+(3d3) and Ni2+(3d8) ions via O2\u0000(2p2)\nion, it is reported that FM interaction occurs for angle\n6Mn\u0000O\u0000Niclose to 180\u000eand AFM interaction for\nthe angle close to 90\u000e.[36] Thus, the obtained FM inter-\naction between Mn4+ions and AFM interaction between\nMn4+and Ni2+ions are consistent with the bond angle\nanalysis using NPD data.\nAtT= 5 K, the re\fned values of the ordered mo-\nment at 4b(Ni2+) and 12d(Mn4+) sites are found to\nbe\u0016= 1:60(1)\u0016Band 2.80(1) \u0016B, respectively, which\nare smaller as compared to the expected spin only values\n(2\u0016B=Ni2+forS= 1 and 3 \u0016B=Mn4+forS= 3=2),\nassumingg= 2. Such a reduced moment is typically\nobserved in low-dimensional and frustrated spin systems\nwhich is attributed to the e\u000bect of quantum \ructuations\nand magnetic frustration, respectively.[38] As shown in\nthe inset of Fig. 8 the value of the ordered moment for\nboth Ni2+and Mn4+decreases as the temperature ap-\nproachesTC, typical behaviour of sub-lattice magneti-\nzation in the ordered state. Using these values of the\nordered moments at T= 5 K, one expects a saturation\nmagnetization of Ms'(\u0016Mn4+\u00021:5\u0000\u0016Ni2+\u00020:5)\u0016B'\n3:4\u0016B, assuming a ferrimagnetic spin structure which is\nindeed consistent with Ms'3:2\u0016Bobtained from the\nMvsHcurve atT= 2 K.8\n/s49/s46/s50 /s49/s46/s51 /s49/s46/s52 /s49/s46/s53 /s49/s46/s54\n/s50/s48/s48/s32/s75\n/s48/s72/s32 /s40/s84/s41/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s50/s57/s48/s32/s75\n/s50/s54/s48/s32/s75\n/s50/s52/s48/s32/s75\n/s50/s50/s48/s32/s75\n/s49/s55/s53/s32/s75\n/s49/s53/s48/s32/s75\n/s49/s52/s48/s32/s75\n/s49/s51/s48/s32/s75\n/s49/s50/s48/s32/s75\n/s49/s49/s48/s32/s75\n/s49/s48/s48/s32/s75\n/s56/s48/s32/s75\n/s52/s32/s75\n/s32\n/s52/s48/s32/s75/s72\n/s114/s101/s102/s55\n/s76/s105/s32/s78/s77/s82\n/s50/s53/s46/s53/s56/s32/s77/s72/s122\n/s76/s105\nFIG. 9. Field-sweep7Li NMR spectra at di\u000berent tem-\nperatures for the polycrystalline LNMO sample measured at\n25.58 MHz. The vertical dashed line corresponds to the7Li\nresonance frequency of the non-magnetic reference. Inset:\nCoupling of Li nucleus with four cubic units made of Ni2+\nand Mn4+ions.\nF.7Li NMR\nSince Li is coupled strongly with the magnetic Mn4+\nand Ni2+ions, (see the inset of Fig. 9) it is possible to get\nthe information about the static and dynamic properties\nof the spins by performing7Li NMR.\n1.7Li NMR Spectra\nFor a quadrupolar7Li (I= 3=2) nucleus, one would\nexpect two satellite peaks along with the central line\ndue to three allowed transitions. Our7Li NMR spec-\ntra, however, display a single spectral line in the whole\ntemperature range as shown in Fig. 9 which corresponds\nto the central transition (+1 =2$\u0000 1=2). The absence\nof satellite peaks could be either due to low quadrupolar\nfrequency or the distribution of the satellite peak inten-\nsity over a wide frequency/\feld range. A single spectral\nline in7Li NMR is typically observed in low-dimensional\noxides.[39] The line shape is found to be symmetric in\nthe entire measured temperature range suggesting the\nabsence of magnetic anisotropy in the compound. The\n/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s54/s49/s50\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s52/s56/s49/s50/s55\n/s76/s105/s32/s78/s77/s82\n/s50/s53/s46/s53/s56/s32/s77/s72/s122\n/s32/s32/s70/s87/s72/s77/s32/s40/s84/s41/s32 /s32/s102/s105/s116\n/s32/s32/s70/s87/s72/s77/s32/s40/s84/s41\n/s49/s46/s53/s32/s84/s32/s40/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s32/s32/s75/s32/s40 /s37 /s41\n/s84 /s32/s40/s75/s41/s32 /s32/s102/s105/s116\n/s32/s32/s75 /s32/s40/s37/s41\n/s49/s46/s53/s32/s84/s32/s40/s99/s109/s51\n/s47/s109/s111/s108/s41FIG. 10. Upper panel: Fullwidth at half maximum (FWHM)\nof7Li NMR spectra plotted as a function temperature. Inset:\nFWHM vs \u001fmeasured at 1.5 T is plotted with temperature\nas an implicit parameter. The solid line is a linear \ft. Lower\npanel: Temperature-dependent7Li NMR shift ( K) vs tem-\nperature. Inset:7Li NMR shift vs \u001fmeasured at 1.5 T is\nplotted with temperature as an implicit parameter. The solid\nline is a linear \ft.\nNMR line broadens drastically below TCwhich re\rects\nthat Li nucleus senses the static internal \feld in the or-\ndered state. With decreasing temperature, the peak po-\nsition of the spectral line is found to be shifted. The\nupper panel of Fig. 10 depicts the fullwidth at half maxi-\nmum (FWHM) of7Li NMR spectra plotted as a function\ntemperature. At high temperatures ( T\u0015200 K), it is al-\nmost temperature independent, increases abruptly below\nabout 150 K (near TC), and then levels o\u000b to a constant\nvalue with lowering temperature. The over all temper-\nature dependent behaviour of FWHM is similar to that\nof the bulk \u001f(T) data. The FWHM is plotted against \u001f\nwith temperature as an implicit parameter in the inset of\nthe upper panel of Fig. 10. It indeed gives a straight line\nbehaviour over the whole measured temperature range\nimplying that the linewidth traces the bulk \u001f(T).\nThe7Li NMR shift ( K) was extracted from the central\npeak positions of the spectra in Fig. 9 by using the rela-\ntionK(T) = [Href\u0000H(T)]=H(T), whereH(T) andHref\nare the resonance \feld of the sample and the nonmagnetic\nreference sample, respectively. The temperature depen-9\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53/s48/s46/s49/s49\n/s32/s32/s49/s47 /s84\n/s49/s32/s40/s109/s115/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s55\n/s76/s105/s32/s78/s77/s82\n/s50/s53/s46/s53/s56/s32/s77/s72/s122\n/s32/s32/s49/s45 /s77 /s40/s116/s41/s47 /s77 /s40 /s41\n/s116/s32/s40/s115/s41/s32/s49/s48/s48/s32/s75\n/s32/s49/s50/s48/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s32/s32/s32 /s32/s102/s105/s116\nFIG. 11. Spin-lattice relaxation rate 1 =T1vs temperature\nmeasured at 25.58 MHz. Inset: The longitudinal magnetiza-\ntion recovery curves at various temperatures. The solid lines\nare the \fts, as described in the text.\ndent NMR shift [ K(T)] is presented in the lower panel\nof Fig. 10. It shows a constant behaviour at high tem-\nperatures and then increases abruptly below 150 K sug-\ngesting the occurrence of a ferrimagnetic ordering. The\noverall behaviour of K(T) is again similar to the that\nobserved for \u001f(T) and FWHM( T). SinceK(T) measures\nthe intrinsic spin susceptibility \u001fspin(T), one can write\nthe relation\nK(T) =Kchem+Ahf\nNA\u0016B\u001fspin(T); (7)\nwhereKchem is the temperature-independent chemical\nshift andAhfis the hyper\fne coupling constant between\nthe Li nuclei and electronic (Ni2+, Mn4+) spins. The\nKvs\u001fplot with temperature as an implicit parameter\nis presented in the inset of the lower panel of Fig. 10.\nIt produces a linear behaviour in the entire measured\ntemperature range. From the slope of a straight line\n\ft, the transfer hyper\fne coupling is evaluated to be\nAhf'672:4 Oe/\u0016B.\n2. Spin-lattice relaxation rate 1/ T1\nIn order to probe the low energy spin dynamics,7Li\nNMR spin-lattice relaxation rate (1 =T1) is measured at\nthe \feld corresponding to the central peak position at\ndi\u000berent temperatures. The measurements are carried\nout in the temperature range 4 K to 290 K. Recovery\nof the longitudinal magnetization is monitored after a\nsaturation pulse sequence. Some of the representative\nrecovery curves in the low temperature regime are shown\nin the inset of Fig. 11. These recovery curves are \ftted\nby single exponential function[39]\n1\u0000M(t)\nM0=Ae\u0000t=T1; (8)whereM(t) is the nuclear magnetization at a time tafter\nthe saturation pulse and M0is the equilibrium magneti-\nzation. The 1 =T1extracted from the \ft is plotted with\nrespect to the temperature in Fig. 11.\nAt high temperatures ( T\u0015160 K), 1=T1is al-\nmost temperature independent. In the paramagnetic\nregime, where the temperature is higher than the ex-\nchange energy between the spins, the temperature inde-\npendent behaviour of 1 =T1is obvious due to uncorrelated\nmoments.[40] Typically, when one approaches the mag-\nnetic transition from high temperatures, 1 =T1is antici-\npated to show a sharp peak or divergence with temper-\nature due to the critical slowing down of the \ructuating\nmoments. Our 1 =T1does show a weak anomaly at TCand\nthen decreases rapidly below TC. The decrease below TC\nindicates the relaxation due to scattering of magnons by\nthe nuclear spins.[39]\nG. Critical Scaling of Magnetization\nFor a better understanding of the magnetic proper-\nties, we have performed the critical behaviour study of\nmagnetization in the vicinity of critical temperature TC.\nScaling hypothesis suggests that the second order phase\ntransition near TCcan be characterized by a set of crit-\nical exponents ( \f,\r, and\u000e) and magnetic equations of\nstate.[41] Spontaneous magnetization MS(T) atT T C, and\nisothermal magnetization ( MvsH) atT=TCare re-\nlated to the critical exponents by the following equations:\nMS(T) =M0(\u0000\")\f;\"< 0; (9)\n\u001f\u00001\n0(T) =\u0000(\")\r;\"> 0; (10)\nM=XH1=\u000e;\"= 0; (11)\nwhere\"= (T\u0000TC)=TCis the reduced temperature and\nM0,\u0000, andXare the critical amplitudes.\nUsually, the Arrott-Noakes equation of state[42] is em-\nployed for a reliable estimation of critical exponents and\nTCusing magnetic isotherms, which can be written as\n(H=M )1=\r=A\"+BM1=\f: (12)\nAccording to this equation, for a particular set of \fand\n\rvalues,M1=\fvs (H=M )1=\rplots should produce a\nset of parallel straight lines in the high \feld region for\ndi\u000berent temperatures around TCand the isotherm at\nT=TCshould pass through origin. The plot of M1=\fvs\n(H=M )1=\ris often called the modi\fed Arrott plot (MAP)\nand the exponents re\rect the universality class of the spin\nsystem. The mean \feld exponents ( \f= 0:5 and\r= 1)\nlead to conventional Arrott plot ( H=M vsM2), which is\ntraditionally used for critical behaviour analysis.[43] We\nhave measured several magnetic isotherms around the TC10\n/s48 /s49 /s50/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s49/s53/s51/s48/s52/s53/s54/s48/s49/s49/s53 /s49/s50/s48 /s49/s50/s53 /s49/s51/s48 /s49/s51/s53/s49/s50/s51\n/s48/s50/s52/s54/s56\n/s49/s49/s50 /s49/s50/s48 /s49/s50/s56 /s49/s51/s54/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s49/s53/s32/s75\n/s49/s51/s53/s32/s75\n/s32/s32/s40/s77 /s41/s50\n/s32/s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41/s50\n/s40\n/s48/s72/s47/s77 /s41/s32/s40/s84/s45/s102 /s46/s117/s46/s47\n/s66/s41\n/s32/s32/s40/s77 /s41/s49/s47\n/s32/s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41/s49/s47\n/s40\n/s48/s72/s47/s77 /s41/s49/s47\n/s32/s40/s84/s45/s102 /s46/s117/s46/s47\n/s66/s41/s49/s47/s32/s61/s32/s48/s46/s51/s53/s48\n/s61/s32/s49/s46/s51/s51/s50\n/s49/s51/s53/s32/s75/s49/s49/s53/s32/s75/s40/s98/s41/s40/s99/s41 /s40/s97/s41\n/s77\n/s83/s32/s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41\n/s84\n/s67/s32/s61/s32/s49/s50/s53/s46/s56/s55/s40/s51/s41/s32/s75\n/s32/s61/s32/s48/s46/s51/s53/s48/s32/s40/s50/s41/s32/s32\n/s84 /s32/s40/s75/s41/s84\n/s67/s32/s61/s32/s49/s50/s53/s46/s56/s51/s40/s51/s41\n/s32/s61/s32/s49/s46/s51/s50/s48/s40/s50/s41\n/s48/s46/s48/s48/s48/s46/s51/s53/s48/s46/s55/s48\n/s32/s40/s84/s45/s102/s46/s117/s46/s47\n/s66/s41\n/s40/s100/s41\n/s84 /s32/s40/s75/s41/s77\n/s83/s40/s100/s77\n/s115/s47/s100/s84/s41/s45/s49\n/s32/s40/s75/s41/s32\n/s48/s45/s49\n/s40/s100\n/s48/s45/s49\n/s47/s100/s84/s41/s45/s49\n/s32/s40/s75/s41\n/s84/s67/s32/s61/s32/s49/s50/s53/s46/s56/s51/s40/s54/s41/s32/s75\n/s32/s61/s32/s48/s46/s51/s53/s50/s32/s40/s51/s41\n/s84/s67/s32/s61/s32/s49/s50/s53/s46/s56/s52/s32/s40/s56/s41\n/s32/s61/s32/s49/s46/s51/s49/s52/s32/s40/s50/s41\n/s32/s32\nFIG. 12. (a) The Arrott plots ( M2vsH=M ) for LNMO at di\u000berent temperatures, above and below TC. (b) The modi\fed\nArrott plots ( M1=\fvs (H=M )1=\r) for LNMO at di\u000berent temperatures. The solid lines are the linear \fts to the data in the\nhigh \feld regime ( H\u00152:5 T) and are extrapolated to H=M = 0. (c) Spontaneous magnetization MSand zero \feld inverse\nsusceptibility \u001f\u00001\n0as a function of temperature in the left and right y-axes, respectively, obtained from the intercepts of the\nmodi\fed Arrott plots in the vicinity of TC. The solid lines are the \fts, as described in the text. (d) The Kouvel-Fisher plots\nforMSand\u001f\u00001\n0. The solid lines are the linear \fts.\nin close temperature steps. To avoid residual magnetiza-\ntion, at each temperature the measurements are done\nafter cooling the sample in zero \feld from high temper-\natures (above TC). Figure 12(a) presents the standard\nArrott plots for LNMO. All the curves show a non-linear\nbehaviour and a downward curvature in the high \feld re-\ngion. This indicates that the transition is non-mean \feld\ntype and standard Arrott plot (associated with mean\n\feld theory) may not be an appropriate way to analyze\nthe critical behaviour of this system. Further, the order\nof phase transition can be determined from the slope of\nArrott plots. According to the Banerjee criterion, the\npositive slope corresponds to a second order phase tran-\nsition whereas the negative slope indicates a \frst order\nphase transition.[44] Thus, the positive slope found in\nFig. 12(a) con\frms second order nature of the transition.\nOn the other hand, the MAPs constructed using the\ncritical exponents of 3D Heisenberg model resulted morelinear behaviour in the high \feld region compared the\nmean-\feld one. Therefore, we took the theoretical val-\nues of\fand\rfor 3D Heisenberg model as the starting\ntrial values to construct MAPs. The linear \fts to the\nMAPs in the high \feld regime were extrapolated down\nto zero \feld and the values of MS(T) and\u001f\u00001\n0(T) were\nobtained from the intercepts on the M1=\fand (H=M )1=\r\naxes, respectively. The temperature dependent MSand\n\u001f\u00001\n0were \ftted by Eqs. (9) and (10), respectively and\nthe values of \fand\rwere estimated. These set of \fand\n\rvalues were again used to construct a new set of MAPs.\nThis whole process was repeated several times, until we\ngot a set of parallel straight lines in the high \feld region\nwith a set of stable \f,\r, andTCvalues. The \fnal MAPs\nare shown in Fig. 12(b). The obtained MSand\u001f\u00001\n0val-\nues are plotted as a function of temperature in Fig. 12(c).\nThe solid lines are the \fts using Eqs. (9) and (10), re-\nspectively. The \ft using Eq. (9) yields \f= 0:350(2)11\n/s48 /s51 /s54 /s57/s48/s50/s52\n/s45/s50 /s45/s49 /s48 /s49/s48/s46/s48/s48/s46/s51/s48/s46/s54/s84 /s32/s61/s32/s49/s50/s53/s46/s54/s32/s75\n/s32/s32\n/s48/s72 /s32/s40/s84/s41/s32/s77 /s32/s40 /s47/s102/s46/s117/s46/s41\n/s32/s76/s111/s103/s32 /s77\n/s32/s32\n/s76/s111 /s103/s32 /s72/s32/s61/s32/s52/s46/s55/s55/s32/s40/s51/s41\nFIG. 13. Magnetization isotherm ( MvsH) atT'TC.\nInset: logMvs logHplot atT'TC.\nwithTC= 125:87(3) K and the \ft using Eq. (10) yields\n\r= 1:320(2) with TC= 125:83(3) K. These values are\nalmost equal to the values obtained from the \fnal MAPs\n(\f= 0:350 and\r= 1:332). As seen from Fig. 12(b),\nthough straight lines are obtained in the high \feld re-\ngion but signi\fcant deviation from linearity was found in\nthe low \feld region. This is likely due to the averaging\nover magnetic domains which are mutually misaligned.\nIn the next step, Kouvel-Fisher (KF) method is used\nto determine \f,\r, andTCmore accurately.[45] The equa-\ntions involved in the KF method are\nMS(T)\u0014dMS(T)\ndT\u0015\u00001\n= (1=\f)(T\u0000TC); (13)\nand\n\u001f\u00001\n0(T)\u0014d\u001f\u00001\n0(T)\ndT\u0015\u00001\n= (1=\r)(T\u0000TC): (14)\nFrom these two equations, it is apparent that\nMS(T)[dMS(T)=dT]\u00001vsTand\u001f\u00001\n0(T)[d\u001f\u00001\n0(T)=dT]\u00001\nvsTplots should yield straight lines with slopes 1 =\fand\n1=\r, respectively. The beauty of this method is that,\nwithout any previous knowledge about TC, one can es-\ntimate it from the intercept of the straight line \fts on\nthe temperature axis. The KF plots for LNMO are pre-\nsented in Fig. 12(d). The critical exponent values eval-\nuated from the KF plots are \f= 0:352(3) with TC=\n125:83(6) K and \r= 1:314(2) with TC= 125:84(8) K.\nThese values of critical exponents match nicely with the\nones obtained from the MAPs, indicating the reliability\nof the values of critical exponents.\nFigure 13 presents the critical isotherm at T'TC=\n125:6 K. According to Eq. (11), log Mvs logHplot at\nthe critical temperature should produce a straight line\nwith slope 1 =\u000e. We have plotted log Mvs logHin the\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s49/s48/s48/s50/s48/s48/s48 /s54/s48/s48 /s49/s50/s48/s48 /s49/s56/s48/s48/s48/s52/s56/s49/s50\n/s32/s49/s50/s49/s32/s75\n/s32/s49/s50/s48/s32/s75\n/s32/s49/s49/s57/s32/s75\n/s32/s49/s49/s56/s32/s75\n/s32/s49/s51/s49/s32/s75\n/s32/s49/s51/s50/s32/s75\n/s32/s49/s51/s51/s32/s75\n/s32/s49/s51/s52/s32/s75/s84 /s32/s62/s32 /s84\n/s67\n/s32/s32/s109/s32/s50\n/s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41/s50\n/s104/s47/s109 /s32/s40/s84/s45/s102/s46/s117/s46/s47\n/s66/s41/s84 /s32/s60/s32 /s84\n/s67/s84 /s32/s62/s32 /s84\n/s67\n/s32/s32/s109/s32 /s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41\n/s104 /s32/s40/s84/s41/s32/s49/s50/s49/s32/s75\n/s32/s49/s50/s48/s32/s75\n/s32/s49/s49/s57/s32/s75\n/s32/s49/s49/s56/s32/s75\n/s32/s49/s51/s49/s32/s75\n/s32/s49/s51/s50/s32/s75\n/s32/s49/s51/s51/s32/s75\n/s32/s49/s51/s52/s32/s75/s84 /s32/s60/s32 /s84\n/s67\n/s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48/s48/s46/s49/s49/s49/s48/s109/s32 /s40\n/s66/s32/s47/s32/s102/s46/s117/s46/s41\n/s104 /s32/s40/s84/s41/s84 /s32/s60/s32 /s84\n/s67\n/s84 /s32/s62/s32 /s84\n/s67FIG. 14. Upper panel: The reduced magnetization m=\nj\"j\u0000\fM(H;\") vs reduced magnetic \feld h=Hj\"j\u0000(\f+\r).\nThe re-normalized curves in di\u000berent temperatures, just\nabove and below TCare collapsing into two separate branches.\nInset: The log-log plot of mvshto magnify the data in the\nlow \feld region. Lower panel: m2vsh=m plots where the\ncurves just above and below TCare collapsing into two sepa-\nrate branches.\ninset of Fig. 13 and a straight line \ft results \u000e= 4:77(3).\nOne can also estimate \u000eby using the Widom relation[47]\n\u000e= 1 +\r\n\f: (15)\nBy using the \fand\rvalues from the KF method and\nMAPs, the corresponding \u000evalue is estimated to be \u000e=\n4:733(4) and \u000e= 4:771(2), respectively. It is found that\nthe\u000evalue obtained from the Widom relation and the\ncritical isotherm analysis are very close to each other,\nre\recting the self-consistency of our \fand\restimations.\nAccording to scaling hypothesis,[46] the obtained criti-\ncal exponents should follow the universal scaling function\nM(H;\") =\"\ff\u0006 \nH\nj\"j\f+\r!\n; (16)12\nTABLE III. Experimentally evaluated critical exponents ( \f,\r, and\u000e) andTCfor LNMO obtained from MAP, KF plot, Critical\nIsotherm analysis, and MCE. For a comparison, the theoretically predicted values for di\u000berent universality classes (mean-\feld,\n3D Heisenberg, 3D Ising, and 3D XY) are also listed (taken from Ref. [46]).\nParameters MAP KV plot Critical\nIsotherm\nAnalysisMCE Mean\nField\nModel3D\nHeisen-\nberg\nModel3D XY\nModel3D\nIsing\nModel\n\f 0.350(2) 0.352(3) { { 0.5 0.365 0.345 0.325\n\r 1.320(2) 1.314(2) { { 1 1.386 1.316 1.241\n\u000e 4.771(2) 4.733(4) 4.77(3) 4.72(9) 3 4.80 4.80 4.82\nn 0.61(2) 0.61(3) { 0.61(2) { { { {\nTC(K) 125.83(3) 125.84(8) 125.6 { { { { {\nwheref+andf\u0000are the regular functions for T < T C\nandT > T C, respectively. Equation (16) can further\nbe simpli\fed in terms of reduced magnetization ( m) and\nreduced \feld ( h) as\nm=f\u0006(h); (17)\nwherem=j\"j\u0000\fM(H;\") andh=Hj\"j\u0000(\f+\r). Equa-\ntions (16) and (17) suggest that for appropriate choice\nof critical exponents, all the curves in the mvshand\nm2vsh=m plots should fall into two separate branches:\nf+forT > T Candf\u0000forT < T C. This behaviour can\nbe clearly visualized in Fig. 14. In the inset of the up-\nper panel of Fig. 14 we have plotted mvshin log-log\nscale which magnify the data in the low \feld regime and\ncon\frms no dispersion among the curves in two branches.\nThis further con\frms the reliability of the estimated crit-\nical exponent values.\nFor a comparison, all the values of critical exponents\nevaluated by the above methods along with the theoret-\nical values corresponding to various universality classes\n(mean-\feld, 3D Heisenberg, 3D Ising, and 3D XY) are\nlisted in Table III. One can see that our estimated crit-\nical exponents ( \f\u00180:35,\r\u00181:32, and\u000e\u00184:772)\nare very close to the 3D XY-Model. This implies that\nLNMO belongs to the 3D-XY universality class. This\nis quite consistent with the magnetic structure deduced\nfrom the NPD data where the spin alignments are re-\nstricted only to the ab-plane. There are certain com-\npounds known to show the evidence of magnetic LRO be-\nlonging to the 3D-XY universality class e.g.CuGeO 3[48],\nGd2IFe2, Gd 2ICo2, and Gd 2BrFe 2[49].\nH. Magnetocaloric E\u000bect\nThe MCE is de\fned as the reversible change in tem-\nperature of a magnetic material while magnetic \feld is\napplied or removed in an adiabatic condition. Generally,\nisothermal entropy change (\u0001 Sm) and adiabatic tem-\nperature change (\u0001 Tad) with respect to the change in\nmagnetic \feld quantify the MCE of a system. From\nthe Maxwell's thermodynamic relation, ( @S=@H )T=(@M=@T )H, \u0001Smcan be estimated using the magneti-\nzation isotherm ( MvsH) data as\n\u0001Sm(H;T) =ZHf\nHidM\ndTdH: (18)\nFigure 15(a) presents the 3D plot of \u0001 Smwith the change\nin \feld (H) and temperature ( T). \u0001Smchanges gradu-\nally withHand shows a maximum entropy change across\ntheTC, with a highest value of \u0001 Sm'11:2 J/kg-K for\n9 T \feld change. Furthermore, \u0001 Tadis calculated from\nthe zero \feld heat capacity and magnetization isotherms\nusing the relation\n\u0001Tad=\u0000ZHf\nHiT\nCpdM\ndTdH: (19)\nThe 3D plot of \u0001 Tadas a function of HandTis shown in\nFig. 15(b). \u0001 Tadshows a gradual temperature evolution\nwith respect to Hshowing a maximum value \u00184 K for\n9 T \feld change near TC. The shape of both \u0001 Sm(T) and\n\u0001Tad(T) peaks appear to be asymmetric and expanded\nover a wide temperature range around TC.\nAnother important parameter which is very useful to\nevaluate the performance of a magneto-caloric material\nis relative cooling power ( RCP ). It de\fnes the amount of\nheat transfer between it's hot and cold reservoirs which\ncan be written mathematically as\nRCP =ZThot\nTcold\u0001Sm(T;H)dT: (20)\nHere,TcoldandThotare the temperatures corresponding\nto the cold and hot reservoirs, respectively. Thus, RCP\ncan be evaluated approximately as\njRCPjapprox = \u0001Speak\nm\u0002\u000eTFWHM; (21)\nwhere \u0001Speak\nmis the peak value and \u000eTFWHM is the full-\nwidth at half maximum (FWHM) of the \u0001 Sm(T) curves\nin Fig. 15(a). The highest value RCP is found to be\n\u0018604 J/Kg for 9 T \feld change. This value of RCP is\nquite high and comparable to other well known magneto-\ncaloric materials, having TCaround this temperature.13\n[d]\nFIG. 15. (a) The isothermal change in magnetic entropy (\u0001 Sm) as function of temperature ( T) and \feld ( H). (b) The\nadiabatic change in temperature (\u0001 Tad) as a function of temperature ( T) and \feld ( H). (c) Universal curve plot of normalized\nentropy change as a function of \u0012with the application of 1 T to 9 T magnetic \feld change. (d) Absolute value of entropy\nchange at the peak position \u0001 Speak\nmand relative cooling power ( RCP = \u0001Speak\nm\u0002\u000eTFWHM ) as a function of magnetic \feld in\nthe left and right y-axes, respectively. Solid lines are the \fts as described in the text.\nIn Table IV, we have made a comparison of the RCP\nand \u0001Speak\nm values of LNMO with some other magneto-\ncaloric materials with TC= 110\u0000140 K. In most of the\nmaterials with second order phase transition, the shape\nof the \u0001Smcurves is broad and asymmetric which is one\nof the main reasons for the large value of RCP . Though,\nmaterials with \frst order phase transition show giant\nMCE and large \u0001 Smand \u0001Tadvalues but the width\nof these peaks are narrow, which restricts the usability\nof these materials for a cyclic operation. Another draw-\nback is that the materials with \frst order phase tran-\nsition show hysteretic behaviour which leads to energy\nloss.[50] Therefore, materials with second order phase\ntransition are more preferred for the magnetic refriger-\nation purpose than the ones with \frst order phase tran-\nsition. It has been theoretically predicted that geometri-\ncally frustrated magnets could show enhanced MCE com-pared to the ordinary non-frustrated magnets.[51] In par-\nticular, pyrochlore lattices are predicted to show highest\nMCE among the geometrically frustrated lattices. Subse-\nquently this idea has been utilized to design several new\nmagnetocaloric materials with strong spin \ructuations\nand/or frustration.[52, 53] Indeed, LNMO is a frustrated\npyrochlore magnet and hence the asymmetric behaviour\ncan be attributed to the e\u000bect of magnetic frustration.\nThus, the obtained large \u0001 SmandRCP values make\nLNMO a promising compound for magnetic refrigeration\npurpose.\nFurther, MCE can also be utilized to gain more insight\nabout the nature of the magnetic phase transition and the\ncritical exponents. The universal scaling curve construc-\ntion of \u0001Smproposed by Franco et. al. [60] is generally\nused for this purpose. This method is tested on vari-\nety of materials and found to be a very e\u000ecient way of14\nTABLE IV. Comparison of experimental \u0001 Speak\nm andRCP values for LNMO with some reported magneto-caloric materials\nhavingTC= 110\u0000140 K for a \feld change of 5 T.\nsystem TC(K) Nature of transition \u0001 Speak\nm(J/Kg-K) RCP (J/Kg) Ref.\nTb2NiMnO 6 110 Second order 5.2 - [54]\nDyGa 113 Second order 5.8 381.9 [55]\nLiNi 0:5Mn1:5O4 125.8 Second order 7.76 302 This work\nGdCo 0:2Mn1:8 140 Second order 4.11 320 [56]\nTb5Ge2\u0000xSi2\u0000xMn2x, 2x= 0:1 123 First order 20.84 330.43 [57]\nDy(Co 0:98Ni0:02)2 126 First order 11 304 [58]\nYFe 2H4:2 132 First order 7.11 263 [59]\ninvestigating the nature of the phase transition.[61] To\nconstruct the universal scaling curves, \frst we normal-\nized all the \u0001 Sm(T) curves with their respective peak\nvalues (i. e. \u0001 Sm/\u0001Speak\nm) for each \feld change and then\nplotted as a function of \u0012in Fig. 15(c). Here, \u0012is the\nre-scaled temperature, which is given by\n\u0012=\u0000(T\u0000TC)=(Tr1\u0000TC) forT\u0014TC (22)\nand\n\u0012= (T\u0000TC)=(Tr2\u0000TC) forT >T C: (23)\nIn the above, Tr1andTr2are the two reference temper-\natures corresponding to 0 :5\u0002\u0001Speak\nm values and TC'\n125:8 K [obtained from the Kouvel-Fisher plot]. As it is\nseen from Fig. 15(c), all the normalized curves for dif-\nferent \feld changes collapse into a single curve similar\nto the other reported compounds showing second order\nphase transition.[61, 62]\nThe \u0001Speak\nmandRCP values are plotted in Fig. 15(d),\nas a function of \feld in the left and right y-axes, respec-\ntively. For the purpose of critical analysis, we have \ftted\nthe \feld dependent \u0001 Speak\nm andRCP curves by the fol-\nlowing power laws[60]\nj\u0001Speak\nmj/Hn(24)\nand\njRCPj/H(1+1=\u000e): (25)\nThe \ft of Eq. (24) to the \u0001 Speak\nm vsTdata yields\nn= 0:61(2). The exponent nis related to the critical\nexponents\fand\rby the relation\nn= 1 +\u0012\f\u00001\n\f+\r\u0013\n: (26)\nUsing\fand\rvalues obtained from the modi\fed Arrot\nplot and Kouvel-Fisher plots, the value of nis estimated\nto be 0.61(2) and 0.61(3), respectively. These values of\nnobtained from various analysis methods are consistent\nwith each other and con\frms the reliability of our critical\nanalysis technique. Similarly, the \ft of RCP vsHdata\nusing Eq. (25) yields \u000e= 4:72(9), which is consistent with\nFIG. 16. Field and temperature dependence of the exponent\nnusing Eq. (27).\nthe values obtained from modi\fed Arrrott plot, Kovel-\nFisher plot, and critical isotherm analysis techniques.\nA more quantitative analysis of the phase transition\ncan be done by using the method proposed recently by\nLaw et. al. [63]. The exponent nin Eq. (24) is normally\n\feld and temperature dependent and can also be calcu-\nlated locally as\nn(T;H) =d lnj\u0001Smj\nd lnH: (27)\nWe used Eq. (27) to quantify the local HandTvaria-\ntion of exponent nand plotted as a 3D plot in Fig. 16.\nSince in the low \feld range, the system has multi-domain\nstates, we have shown only the values corresponding to\nhigh \felds ( H > 2 T) in Fig. 16. At low temperatures\n(T 27 K) region, the data could\nbe \ftted well by Eq. (29) resulting Ea=kB'61:7 K\nand\u001c0'3:7\u000210\u00006s. Such an Arrhenius behaviour\nis often considered to be the characteristic feature of\nsuperparamagnetism.[69] The value of \u001c0is also seems\nto be within range expected for superparamagnets (10\u00006\nto 10\u00009s)[70].\nIn the low temperature ( T < 27 K) region, the data\nshow a signi\fcant deviation from the Arrhenius law as\nevident in the upper panel of Fig. 19. This indicates that\nthere are two di\u000berent relaxation mechanisms involved.\nWe \ftted the low- Tdata by the dynamical scaling law\nor power law, predicted for SG systems[71]\n\u001c=\u001c0exp\u0012Tf2\nTg\u00001\u0013\u0000z\u00170\n: (30)\nHere,\u001c0has the same physical meaning as \u001c0,Tgis the\nfreezing temperature as \u0017approaches zero, and \u001c0=\u0018z\nwithzbeing the dynamical critical exponent. The cor-\nrelation length has the form \u0018= (Tf2=Tg\u00001)\u0000\u00170with\n/s48/s46/s48/s50/s56 /s48/s46/s48/s51/s53 /s48/s46/s48/s52/s50/s45/s49/s48/s46/s53/s45/s57/s46/s48/s45/s55/s46/s53\n/s45/s48/s46/s54 /s45/s48/s46/s51 /s48/s46/s48/s45/s52/s46/s56/s45/s52/s46/s48/s45/s51/s46/s50\n/s50/s46/s53 /s51/s46/s48 /s51/s46/s53/s50/s52/s46/s51/s50/s53/s46/s50/s50/s54/s46/s49\n/s32/s32/s108/s110/s32/s40 /s41\n/s49/s47 /s84\n/s102/s50\n/s32/s32/s108/s111/s103\n/s49/s48/s40 /s41\n/s108/s111/s103\n/s49/s48/s40/s84\n/s102/s50/s47/s84\n/s103/s45/s49/s41\n/s32/s32/s84\n/s102/s50/s32/s40/s75/s41\n/s108/s111/s103\n/s49/s48/s40 /s41FIG. 19. Upper panel: ln( \u001c) vs 1/Tf2plot. The solid line is\nthe \ft using Eq. (29) in the high temperature region. Lower\npanel: log 10(\u001c) vs log 10(Tf2=Tg\u00001) plot. The solid line is\nthe \ft using Eq. (31). Lower inset: Tf2vs log 10\u0017plot in the\nlow temperature region. The solid line is the \ft as described\nin the text.\ncritical exponent \u00170. Equation (30) can be rewritten in a\nsimpli\fed form as\nlog10\u001c= log 10\u001c0\u0000z\u00170log10\u0012Tf2\nTg\u00001\u0013\n: (31)\nAs shown in the lower panel of Fig. 19, Eq. (31) \fts\nwell to the low- Tdata giving \u001c0'5:08\u000210\u00009s and\nz\u00170'(6:6\u00060:2). In the \ftting process we have \fxed\nTg'(21:03\u00060:01) K, obtained from the y-intercept of\nthe linear \ft of Tf2vs log 10\u0017plot (see the inset of lower\npanel of Fig. 19). The larger value of \u001c0clearly indicates\nthat the spin dynamics is slower than conventional SG\nsystems (\u001810\u000013s).[66] Such a high value of \u001c0is previ-\nously reported for various reentrant-SG systems.[70, 72]\nFurther, the value of z \u00170also falls in the range ex-\npected for typical SG systems ( z\u00170'4{12) and com-\nparable to the values reported for various reentrant-SG\nsystems.[70, 72]\nThus, the frequency dependence of Tf2follows an Ar-\nrhenius behaviour at higher temperatures which could\nsuggest that the system is superparamagnet far above17\n/s48 /s54/s48 /s49/s50/s48 /s49/s56/s48/s49/s56/s55/s50/s49/s56/s55/s54/s49/s56/s56/s48/s72 /s32/s61/s32/s49/s48/s48/s32/s79/s101\n/s116\n/s119/s32/s61/s32/s54/s48/s32/s115\n/s32/s102/s105/s116\n/s84 /s32/s61/s32/s50/s32/s75/s32\n/s116/s32/s40/s109/s105/s110/s41/s32/s32/s54/s53/s55/s48/s32/s115/s44/s32/s32/s32/s48/s46/s52/s56\n/s32/s32/s49/s50/s48/s48/s54/s32/s115/s44/s32/s32/s32/s48/s46/s51/s56\n/s49/s55/s55/s48/s49/s55/s55/s51/s84 /s32/s61/s32/s49/s48/s32/s75/s32\n/s32/s77 /s32/s40/s71/s32/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s32/s32/s50/s53/s56/s49/s53/s32/s115/s44/s32/s32/s32/s48/s46/s51/s55/s49/s54/s56/s52/s49/s54/s56/s56/s49/s54/s57/s50\n/s32/s84 /s32/s61/s32/s55/s48/s32/s75\nFIG. 20. Time dependence of ZFC magnetization at di\u000berent\ntemperatures ( T= 2;10;and 70 K) in H= 200 Oe and after a\nwaiting time of 60 s. The solid lines are the \fts using Eq. (32).\nthe critical region ( T\u001dTg). As we move closer to\nTg, Arrhenius behaviour breaks down and power law be-\nhaviour takes over which evidences the existence of SG\ntransition at Tg'21:03 K, similar to the dilute magnet\nLiHoxY1\u0000xF4(x= 0:045).[73] In order to resolve this\nambiguity, memory e\u000bect measurements are discussed\nlater.\nJ. Nonequilibrium Dynamics\n1. Magnetic Relaxation\nTo explore the low temperature spin dynamics in\nLNMO, we have performed the magnetic relaxation time\nmeasurement at di\u000berent temperatures (2 K, 10 K, and\n70 K), below Tb. The sample was cooled in ZFC con-\ndition from high temperature paramagnetic state ( T\u0015\n200 K) to the desired temperature. After waiting for\ntw= 60 s at the desired temperature, a small magnetic\n\feld ofH'100 Oe is applied. Thereafter, the growth\nof the magnetization with time [ M(t)] was recorded.\nThe resulting M(t) curves are plotted in Fig. 20. The\ntime evolution of magnetization curves are found to fol-\nlow a stretched exponential behaviour speci\fed by theKohlrausch-Williams-Watts relation[74, 75]\nM(t) =M0\u0000Mgexp[\u0000(t=\u001c)\f]: (32)\nHere,M0is the intrinsic magnetization at t= 0,Mg\nis associated with the glassy component of the magneti-\nzation,\u001cis the characteristic relaxation time, and \fis\nthe stretching exponent. Typically, the value of \fvaries\nbetween 0 and 1 which decides the dynamics of a spin sys-\ntem.\fis also a function of temperature and is strongly\ndependent on the nature of energy barrier involved in\nthe relaxation process. Following Eq. (32), when \f= 0,\nM(t) = constant; means no relaxation. Similarly, \f= 1\nimplies relaxation of a spin system with a single time con-\nstant due to the presence of uniform energy barrier. On\nthe other hand, \f <1 implies the presence of the distri-\nbution of non-uniform energy barriers, typically observed\nfor SG and superparamagnetic systems.[22, 24, 74, 76]\nTheM(t) curves for T= 70 K, 10 K, and 2 K are\nwell \ftted by Eq. (32), yielding \f'0:48, 0.38, and\n0.37, respectively. These values are less than 1 sug-\ngesting the evolution of magnetization through a num-\nber of intermediate metastable states. The value of \u001c\nincreases with decreasing temperature as expected for\nglassy systems.[22, 77] The relaxation behaviour observed\natT= 10 K and T= 2 K is quite natural because both\nof these temperatures are well below Tf2. However, the\nslow relaxation behaviour observed at high temperatures\n(T= 70 K> T f2) is unusual and suggests that the per-\nsistence of SG/superparamagnetism beyond Tf2in the\nferrimagnetically ordered state.\n2. Magnetic Memory E\u000bect\nSince the bifurcation of the DC ZFC and FC \u001f(T) data\noccurs for both SG and superparamagnetic systems, the\nhistory-dependent measurements are required in order to\ndelineate the microscopic character of the spin system.\nTherefore, in the following, we have described the mag-\nnetic memory e\u000bect measurements in both FC and ZFC\nprotocols.\nIn the FC protocol, the sample was cooled in a small\nmagnetic \feld of H= 100 Oe from high temperature\nparamagnetic state ( T\u0015200 K) to the lowest measured\ntemperature T= 2 K at a constant rate of 0.5 K/min\nwith intermediate stops at three di\u000berent temperatures\n(Tw\n1= 70 K,Tw\n2= 12 K, and Tw\n3= 5 K). At each stop,\nthe magnetic \feld was switched o\u000b and a waiting time of\ntw= 3 hours was given for the magnetization to relax.\nAfter each tw, the same magnetic \feld was applied and\n\feld cooled cooling (FCC) process was resumed. In this\nway, the recorded magnetization ( Mw\nFCC) is plotted in\nthe upper panel of Fig. 21 which shows step like features\nat each stopping temperature. Once it reached 2 K, the\nmagnetization was recorded by heating the sample at the\nsame rate (0.5 K/min) back to 200 K in the same mag-\nnetic \feld without any intermediate stop. The resulting\n\feld cooled warming magnetization data are referred as18\n/s50/s48/s48/s48/s50/s53/s48/s48\n/s48 /s53/s48 /s49/s48/s48/s49/s55/s48/s48/s49/s56/s48/s48/s49/s57/s48/s48/s84/s119\n/s51/s32/s61/s32/s53/s32/s75/s84/s119\n/s50/s32/s61/s32/s49/s50/s32/s75\n/s32/s77 /s32/s40/s71/s99/s109/s51\n/s47/s109/s111/s108/s41/s32/s77/s119\n/s70/s67/s67\n/s32/s77/s109/s101/s109\n/s70/s67/s87 \n/s32/s77/s114/s101/s102\n/s70/s67/s67/s84/s119\n/s49/s32/s61/s32/s55/s48/s32/s75\n/s32/s84/s119 /s50\n/s32/s61/s32/s49/s50/s32/s75/s84/s119 /s51\n/s32/s61/s32/s53/s32/s75\n/s84/s119\n/s49/s32/s61/s32/s55/s48/s32/s75/s84/s119 /s50/s32/s61/s32/s49/s50/s32/s75\n/s84/s119 /s51/s32/s61/s32/s53/s32/s75\n/s32/s32/s77 /s32/s40/s71/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s84 /s32/s40/s75/s41/s32/s77/s114/s101/s102\n/s90/s70/s67/s87 \n/s32/s77/s109/s101/s109\n/s90/s70/s67/s87 \n/s84/s119 /s49\n/s32/s61/s32/s55/s48/s32/s75\n/s50/s53 /s53/s48 /s55/s53/s48/s49/s50/s51/s77 /s32/s61/s32 /s77/s109/s101/s109 /s90/s70/s67/s87/s32/s45/s32 /s77/s114/s101/s102 /s90/s70/s67/s87\n/s84 /s32/s40/s75/s41\nFIG. 21. Memory e\u000bect measured as a function of temper-\nature in FC (upper panel) and ZFC (lower panel) protocols\ninH= 100 Oe, as described in the text. Inset of the up-\nper panel magni\fes the features at the interruption points:\nT= 5 K, 12 K, and 70 K. The di\u000berence in magnetization,\n\u0001M(=Mmem\nZFCW\u0000Mref\nZFCW ) vsTis plotted in the inset of the\nlower panel.\nMmem\nFCW in the upper panel of Fig. 21. As depicted in the\ninset of the upper panel of Fig. 21, Mmem\nFCW also exhibits\nchange of slope at each stopping temperature ( Tw\n2= 12 K\nandTw\n3= 5 K) as in Mw\nFCC. These characteristic fea-\ntures clearly imply that the system tries to remember the\nthermal history of magnetization during cooling, thus,\nshowing the magnetic memory. A weak change of slope\nwas also seen at the stopping temperature, Tw\n1= 70 K\nwhich is well above Tf2. Similar behaviour is also re-\nported for reentrant SG compound Lu 2MnNiO 6and it is\nmentioned that the unusual behaviour where memory ex-\nists aboveTfand belowTCcan be attributed to the e\u000bect\nof magnetic frustration due to competing FM and AFM\ninteractions.[38] A FCC curve, Mref\nFCC in the same \feld\nwithout any interruption is also measured as a reference.\nIn the ZFC protocol, the sample was cooled in zero\nmagnetic \feld form T\u0015200 K to 2 K at a constant rate\nof 0.5 K/min with three intermediate stops at Tw\n1= 70 K,\nTw\n2= 12 K, and Tw\n3= 5 K. At each stop, the sam-\nple was allowed to relax for a waiting time of 3 hours.\nAfter reaching 2 K, the magnetization Mmem\nZFCW was col-\nlected by warming the sample upto 200 K, after applyinga small magnetic \feld of 100 Oe. The reference data,\nMref\nZFCW were also taken by measuring the magnetization\nduring warming in the same magnetic \feld 100 Oe, af-\nter the sample was cooled in zero magnetic \feld without\nany intermediate stops. From the data presented in the\nlower panel of Fig. 21, it can be seen that there is nei-\nther any clear dip nor any change in slope in the Mmem\nZFCW\ndata at the stopping temperatures, implying the absence\nof ZFC memory e\u000bect. The di\u000berence in magnetization,\n\u0001M(=Mmem\nZFCW\u0000Mref\nZFCW ) vsTis also plotted in the\ninset of the lower panel of Fig. 21 to highlight no mem-\nory e\u000bect. In order to make sure that there is no ZFC\nmemory, we have also studied memory e\u000bect by measur-\ning AC susceptibility in the ZFC protocol, following the\nsame procedure as discussed above. Similar to the DC\n\u001f(T), the real ( \u001f0) and imaginary ( \u001f00) parts of the AC\nsusceptibility data (not shown) don't show any change of\nslope at the stopping temperatures. This further proves\nthat no memory is imprinted by aging under zero \feld.\nThe above memory e\u000bect can be understood from the\nsimple two-state model proposed by Sasaki et:al: [25] and\nTsoiet: al: [24] for non-interacting magnetic nano parti-\ncles (superparamagnets). In this model, it is assumed\nthat a superspin associated with the dipole magnetic\nmoment of a nano particle can occupy one of the two\nstates with energies \u0000KV\u0006HM sV, whereKis the bulk\nanisotropy constant, Vis the volume of the nanoparticle,\nandHis the applied \feld. Therefore, a broad distribu-\ntion of particle volumes results in a broad distribution of\nanisotropic energy barriers. The occupation probability\np1(t) of one of the two states, in which the superspin is\nantiparallel or parallel to the applied \feld is p1(t) = 0:5\n[i.e.M(t) = 0] at any time t, ifp1(t= 0) = 0:5 and\nH= 0. Thus, in a ZFC process, which starts from a\ninitial demagnetized state [ M(t= 0) = 0], p1(t) and\nhence the total magnetization is independent of the wait-\ning time (tw), whereas for a FC process which starts from\na initial magnetized state, p1(t) and hence the total mag-\nnetization is dependent on tw. In the light of this model,\na superparamagnet should not show ZFC memory but\nit can show FC memory simply because of the blocked\n(frozen) superspins. On the other hand, a SG system\ncan show both FC and ZFC memories which are well ex-\nplained by Sasaki et: al: [25] considering the random en-\nergy model[78] as well as the droplet theory[79] proposed\nfor SG systems. The above models have been employed\nto describe the experimental data of various superpara-\nmagnets and SG systems.[24{27] Thus, the absence of\nZFC memory discriminates the dynamics of LNMO from\nthe behaviour of a SG and establishes the superpara-\nmagnetic nature at low temperatures.[24, 76] It is quite\nsurprising that despite having large average particle sizes\n(\u0018100 nm), the compound still behaves like a superpara-\nmagnet. It is to be noted that the unconventional super-\nparamagnetism is also reported in several compounds in\npolycrystalline form.[80]19\n/s49/s55/s55/s54/s49/s55/s56/s52\n/s49/s55/s52/s48/s49/s56/s48/s48\n/s48 /s49/s48/s48/s48/s48 /s50/s48/s48/s48/s48/s56/s54/s52/s56/s55/s51\n/s48 /s49/s48/s48/s48/s48 /s50/s48/s48/s48/s48/s56/s52/s53/s57/s49/s48/s48 /s53/s48/s48/s48 /s49/s48/s48/s48/s48/s49/s55/s55/s54/s49/s55/s56/s48/s49/s55/s56/s52\n/s48 /s53/s48/s48/s48 /s49/s48/s48/s48/s48/s56/s54/s53/s56/s55/s48/s49/s50/s32/s75 /s53/s32/s75/s49/s50/s32/s75/s116\n/s51/s116\n/s50\n/s32/s32/s77 /s32/s40/s71/s32/s99/s109/s51\n/s47/s109/s111/s108/s41/s116\n/s49\n/s116\n/s49/s116\n/s50/s116\n/s51\n/s49/s50/s32/s75 /s53/s32/s75 /s49/s50/s32/s75/s40/s97/s41\n/s32/s77 /s32/s40/s71/s32/s99/s109/s51\n/s47/s109/s111/s108/s41/s32\n/s53/s32/s75 /s53/s32/s75 /s49/s50/s32/s75/s116\n/s51/s116\n/s50/s116\n/s49\n/s40/s98/s41\n/s32/s32/s77 /s32/s40/s71/s32/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s116/s32/s40/s115/s41/s40/s99/s41\n/s32/s77 /s32/s40/s71/s32/s99/s109/s51\n/s47/s109/s111/s108/s41/s32\n/s116/s32/s40/s115/s41/s116\n/s49\n/s53/s32/s75/s116\n/s50\n/s49/s50/s32/s75/s116\n/s51\n/s53/s32/s75/s40/s100/s41/s116\n/s51\n/s32/s32/s77 /s32/s40/s71/s32/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s116/s32/s40/s115/s41/s116\n/s49\n/s32/s77 /s32/s40/s71/s32/s99/s109/s51\n/s47/s109/s111/s108/s41\n/s116/s32/s40/s115/s41/s116\n/s49/s116\n/s51\nFIG. 22. Magnetic relaxation measured in negative T-cycle\nfor (a) ZFC and (b) FC methods, as described in the text.\nInsets: Magnetic relaxation at 12 K measured during t1and\nt3for negative T-cycle and in ZFC and FC methods. The\nsolid lines are the \fts using Eq. (32). Magnetic relaxation\ndata measured in positive T-cycle for ZFC and FC methods\nare shown in (c) and (d), respectively.\n3. Memory E\u000bect using Magnetic Relaxation\nTo investigate the memory e\u000bect in further detail, we\nhave performed the magnetic relaxation measurements\nfollowing the protocol reported by Sun et:al: [81] for both\nnegative and positive- Tcycles.\nNegative T\u0000cycle : In a negative temperature cy-\ncle, we have measured the magnetic relaxation in both\nZFC and FC protocols and are plotted in Fig. 22(a) and\n(b), respectively. In the ZFC procedure, the sample was\n\frst cooled from 200 K down to 12 K in zero \feld. At\n12 K, a small \feld of 100 Oe was applied and M(t) was\nrecorded for t1= 2 hours, which is found to grow expo-\nnentially with t. The sample was again cooled down to\n5 K in the same magnetic \feld and M(t) was measured\nfort2= 2 hours. The nature of M(t) curve was found to\nbe almost constant with t. Thereafter, the temperature\nwas restored back to 12 K and M(t) was recorded for\nt3= 2 hours in the same \feld. At this temperature, the\nM(t) curve was again found to grow exponentially with t.\nAs shown in the inset of Fig. 22(a), the M(T) data mea-\nsured during t1andt3follow a continuous growth curve.\nIn the FC process, the sample was cooled in a small mag-\nnetic \feld of 100 Oe down to 12 K. Once it reached 12 K,\nthe magnetic \feld was switched o\u000b and the decay of mag-\nnetization with twas measured for t1= 2 hours. Thesample was further cooled down to 5 K in zero \feld and\nM(t) was recorded for t2= 2 hours at 5 K. This M(t)\ncurve was found to be almost constant with t. Subse-\nquently, the sample was heated back to 12 K in zero \feld\nandM(t) was recorded for t3= 2 hours at 12 K. The\nM(t) curve measured during t3was found to decay expo-\nnentially with tas a continuation of M(t) curve recorded\nduringt1[see the inset of Fig. 22(b)].\nThus, the continuous growth and decay of magnetiza-\ntion during t1andt3obtained for the ZFC and FC pro-\ncesses, respectively indicate that the state of the system\nat 12 K is recovered after a temporary cooling. This a\nclear demonstration of the memory e\u000bect where the sys-\ntem tries to remember the initial state even after going\nthrough a change of magnetization. These continuous\ncurves could be \ftted well by the stretched exponential\nfunction [Eq. (32)] with \f'0:37 and 0.48 for the ZFC\nand FC processes, respectively which are consistent with\nthe magnetic relaxation measurements.\nPositive T\u0000cycle : Similar to the negative T-cycle,\nwe have also done magnetic relaxation measurements in\npositiveT-cycle in both ZFC and FC protocols. For ZFC\nprocedure, the sample was \frst cooled from 200 K down\nto 5 K in zero magnetic \feld and M(t) was recorded for\nt1= 2 hours, after applying a small magnetic \feld of\n100 Oe. Then, the sample was heated up to 12 K in\nthe same \feld and M(t) was measured for t2= 2 hours.\nThereafter, the sample was again cooled back to 5 K in\nthe same \feld and M(t) was recorded for t3= 2 hours.\nIn the FC procedure, the sample was \frst cooled down to\n5 K in a small magnetic \feld of 100 Oe. Once the tem-\nperature reached 5 K, the magnetic \feld was switched\no\u000b andM(t) was recorded for t1= 2 hours. By keeping\nthe \feld zero, the sample was heated to 12 K and M(t)\nwas recorded for t2= 2 hours. In zero \feld, the sample\nwas further brought back to 5 K and M(t) was measured\nfort3= 2 hours. The measured ZFC and FC data are\npresented in Fig. 22(c) and (d), respectively. Unlike the\nnegativeT-cycle there is no continuity found in the M(t)\ndata measured during t1andt3for both ZFC and FC\nmeasurements in the positive T-cycle. This clearly sug-\ngests that positive T-cycling or temporary heating erases\nthe memory and re-initializes the relaxation process in\nboth ZFC and FC methods. Thus, no memory e\u000bect is\nobserved when the temperature is restored back to 5 K.\nThe asymmetric response of magnetic relaxation with\nrespect to both negative and positive T-cycles is typically\nobserved for both SG and superparamagnetic systems. In\nSG systems, this behaviour can be explained on the basis\nof the hierarchical model.[81, 82] Similarly, for superpara-\nmagnetic systems, this behaviour is very well explained\nby Tsoiet: al: [24] and Bandyopadhyay et: al: [26] in the\nlight of simple two-state superparamagnetic model[25].\nThus, the observed memory e\u000bects in ZFC and FC pro-\ntocols for the negative T-cycle further justi\fes the super-\nparamagnetic nature of the system under investigation.20\nIV. SUMMARY\nWe have successfully synthesized the polycrystalline\nsample of a new modi\fed cubic spinel compound LNMO.\nIt is found to crystallize in a cubic structure with a\nnon-centrosymmetric space group P4332 and exhibits 1:3\ncation order of Ni2+and Mn4+ions. Physical property\nmeasurements suggest semiconducting nature of the com-\npound which exhibits a long-range ferrimagnetic ordering\natTC'125 K. The analysis of the neutron di\u000braction\ndata reveals a collinear ferrimagnetic spin structure, with\nmagnetic moments aligned along [110] direction. The\nmoments of each Ni2+or Mn4+sublattice are coupled\nferromagnetically whereas the inter-sublattice interaction\nis antiferromagnetic. The compound is still frustrated\ndue to competing AFM and FM interactions which is\nalso the reason for the reduction of frustration ratio ( f),\ndespite a highly frustrated pyrochlore lattice geometry.\nThe reduced ordered moment of Ni2+or Mn4+ions also\nindicates the presence of signi\fcant frustration.\nThe critical exponents obtained from the analysis of\nmagnetization data near TCvia modi\fed Arrott and\nKouvel-Fisher plots fall in the category of 3D XY uni-\nversality class. The reliability of these critical exponents\nand the value of TCare further con\frmed from the scal-\ning of magnetization isotherms. A reversible MCE with\na large value of \u0001 SmandRCP has been observed over a\nwide temperature range across TCwhich can be ascribed\nto the e\u000bect of magnetic frustration. Even the critical\nanalysis of \feld dependent j\u0001Speak\nmjandRCP curves pro-\nduce exponents close to the ones obtained from the crit-\nical analysis of magnetization. The second order natureof the phase transition has also been con\frmed from the\nuniversal scaling of \u0001 Smand the nature of n(H;T) curve.\nOur results demonstrate that, LNMO is a promising re-\nfrigerant material, where frustration associated with the\ncompeting exchange interactions drives MCE.\nThe magnetic relaxation below Tbfollows stretched ex-\nponential function demonstrating that the system evolves\nthrough a number of metastable states. \u001f00(T) below\nTCdepicts multiple anomalies, likely due to the e\u000bect\nof magnetic frustration. The hump at T=Tf2shows\nstrong frequency dependency with unusual dynamics. It\nfollows a Arrhenius behaviour in the high temperature\nregime (T > 27 K), suggesting superparamagnetic be-\nhaviour and a power law behaviour in the low temper-\nature regime ( T < 27 K), suggesting SG dynamics at\nlow temperatures. However, the absence of ZFC mem-\nory e\u000bect below Tb'120 K rules out the possibility of\nSG transition and con\frms the superparamagnetic be-\nhaviour down to the lowest measured temperature. In-\ndeed, the behaviour of DC \u001f(T) measured in FC and ZFC\nconditions also seems to substantiate the superparamag-\nnetic nature of the compound with blocking temperature\nTb'120 K. Nevertheless, the critical slowing down be-\nhaviour following a power law at low temperature and\nthe absence of ZFC memory contradict each other and\nwarrants further investigations.\nV. 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Liu2 \n1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, \nand Institute for Advanced Materials, South China Academy of Advanced Optoelectronics , \nSouth China Normal University, Guangzhou 510006, China \n2Laboratory of Solid State Microstructures and Innovative Center for Advanced \nMicrostructures, Nanjing University, Nanjing 210093, China \n \n[Abstract] The ultrafast magnetic dynamics in compensated ferrimagnets not only provides \ninformation similar to antiferromagnetic dynamics, but more importantly opens new \nopportunities for future spintronic devices [Kim et al., Nat. Mater. 16, 1187 (2017) ]. One of \nthe most essential issues for device design is s earching for low-power -consuming and \nhigh-efficient methods of controlling domain wall. In this work, we propose to use the \nvoltage -controlled magnetic anisotropy gradient as an excit ation source to drive the domain \nwall motion in ferrimagnet s. The ultrafast wall motion under the anisotropy gradient is \npredicted theoretically based on the collective coordinate theory , which is also confirmed by \nthe atomistic micromagnetic simulations. The antiferromagnetic spin dynamics is realized at \nthe angular momentum comp ensation point , and t he wall shift ing has a constant speed under \nsmall gradient and can be slightly accelerated under large gradient due to the broadened wall \nwidth during the motion. For nonzero net angular momentum, the Walker breakdown occurs \nat a critical anisotropy gradient significantly depend ing on the second anisotropy and \ninterfacial Dzyaloshinkii -Moriya interaction , which is highly appreciated for further \nexperiments including the materials selection and device geometry design. More importantly , \nthis work unveils a low -power -consuming and high-efficient method of controlling the \ndomain wall in ferrimagnets, benefiting to future spintronic applications. \n \nKeywords: magnetic dynamic s, domain wall , magnetic anisotropic gradient , ferrimagnets \n \n \nEmail: qinmh@scnu.edu.cn I. Introduction \nAntiferromagnetic materials show fast magnetic dynamics and produce non -perturbin g \nstray fields , attributing to their zero magnetization and ultralow susceptibility . These \nadvantages make them promising candidates for next generation of high-density and \nhigh-speed spintronic devices .1-5 However, the magnetic field immunity of antiferromagnetic \nmaterials also hinders the detection and manipulation of magnetic states .6-8 Thus, it is still \nchallenging to experimental ly study the antiferromagnetic spin dynamics , although several \nstimuli have been predicted to drive the fast domain wall motion in the earlier theoretical \nworks .9-18 Therefore, a reliable and direct detection of the magnetic states remains to be a \ncommon issue for antiferromagnetic spintronic researches. \nTo overcome this deficiency, an immediate alternative strategy is to consider \nferrimagnetic (FiM) systems where the fast magnetic dynamics in the vicinity of angular \nmomentum compensation temperature TA can be achieved,19 at which the net momentum \nvanishes while the net magnetic moment is nonzero . It has been theoretically predicted and \nexperimentally confirmed that the FiM dynamics at TA is similar to the antiferromagnetic \ndynamics . More importantly, the magnetic states of a FiM system at TA can be effectively \ndetected and addressed through the magnetoelectric20-22 and magneto -optical23 responses, \nbenefiting from their nonzero magnetic moment , and thus highly appreciated . \nIn fact , the magnetic field - and electrical current -driven fast domain wall motion s in \nangular momentum compensated ferrimagnets have been experimentally reported, \nrespectively.19,24-26 Also , the Walker breakdown field, under which the domain wall begins to \nprecess and reaches to a threshold speed, is significantly increased and the domain wall \nmobility is extensively enhanced when the net angular momentum approaches to zero . At TA, \nthe field diverg es, and the domain wall speed keeps increasing linearly with field due to the \nexcluded Walker breakdown, exactly the same as in antiferromagnets. For example, the \ndomain wall speed as high as ~ 20 km s-1T-1 was reported at TA in rare earth 3d transition \nmetal ferrimagnets.19 Thus, the magnetic dynamics in ferrimagnets at TA not only provides \nequivalent information for antiferromagnetic spin dynamics, but more importantly opens new \nopportunities for future spintronic devices. \nOn the other hand , searching for well-controlled and low -power -consum ed method s to \nmodulat e FiM domain w all is one of the most important issues for spintronic device operation , \nnoting that the shortcomings of these proposed schemes may be detrimental for future \napplications . For instance, the dispersion characteristic of magnetic field generally limits the \ndensity of ferrimagnetic elements and hinders the further optimization of device dimension. Moreover, some of the electrical current related schemes normally generate Joul e heating and \nunnecessary energy loss , significantly affecting the data transportation process where a stable \noperating temperature is benefiting . Along this line, electric field control could be highly \npreferred,27 to be explained in detail below. \nFirst, numerous experiments have revealed the voltage control of magnetism. For \nexample, the voltage induced magnetic anisotropy gradient has been experimentally reported \nin magnetic heterostructures through elaborate structure design.28-30 Under such a gr adient, \nthe magnetic domain wall tends to move towards the low anisotropy side in order to save free \nenergy. As a matter of fact, the anisotropy gradient has been proven to efficiently drive the \nskyrmions motion and antiferromagnetic domain wall motion,31-33 and this scheme could be \nalso utilized to control the FiM domain wall motion. More importantly, this alternative \nscheme is promising for future spintronic applications considering the low-energy cost and \nthe high operating efficiency. However, as far as we know, few works on this subject have \nbeen reported , while the dynamics of FiM domain wall under anisotropy gradient is certainly \nan urgent topic to be understood , in order to provide instruction for future experiments and \npromot e the application proces s for spintronics. \nIn this work, we study the domain wall dynamics of ferrimagnets under an anisotropy \ngradient, using the collective coordinate theory and atomistic Landau -Lifshitz -Gilbert (LLG) \nsimulations. It is demonstrated that the wall speed and precession direction depend closely on \nthe net angular momentum. At the angular momentum compensation point, the Walker \nbreakdown vanishes and the wall moves at a maximal speed, similar to the case of \nantiferromagnetic dynamics. It will be shown that the wall remain s to shift at a const ant speed \nunder small gradient, while the motion can be slightly accelerated under large gradient due to \nthe broadened wall width during the motion. Furthermore, for a nonzero angular momentum, \nthe Wa lker breakdown gradient could be modulated by utilizing a second anisotropy and the \ninterfacial Dzyaloshinkii -Moriya (DM) interaction . These results provid e useful information \nfor future material design and spintronic applications. \n \nII. Analytical analysis and numerical simulation \nWe investigate theoretically the domain wall motion for ferrimagnet s such as rare earth \nand transition metal compounds , whose magnetic structure is depicted in Fig. 1 (a) where the \nspins of two inequivalent sublattices ar e coupled antiferromagnetically .34 We set n1,2(r, t) (n1 \n= -n2), M1,2 (M1,2 = M1,2·n1,2), 1,2, g1,2, and 1,2 to be the local unit vector at time t and position r, magnetization moment , gyromagnetic ration, Landé -g factor, and Gilbert damping \nconstant of the two sublattices. Thus, the spin density of the sublattice i is given by si = Mi/i \nwith i = giB/ћ, where µB is the Bohr magneton. It is noted that the net magnetization M = \nM1 + M2 is nonzero at TA where the net angular momentum s = s1 s2 = 0, because of the \ndifferent Landé -g factors between the two sublattices. \n \n2.1. Analytical treatment \nFollowing the collective coordinate approach , the low -temperature magnetic dynamics of \nFiM model is described by the Lagrangian density L = LB U with the spin Berry phase LB \nand the potential -energy density U.19,35 In detail, the Berry phase is associated with the \nstaggered spin density s = (s1 + s2)/2 and the net spin density s, which can be described \nby:18,19,35 \n( ) ( ) ,BsLs n n m a n n\n (1) \nwhere n (n1 – n2)/2, and m (n1 + n2)/2, ṅ represents the derivative with respect to time, \na(n) is the vector potential generated by a magnetic monopole of unit charge satisfying n a \n= n. The potential -energy density is given by \n2\n2 2 2 ()( ) ( ).2 2 2 2 2ex\nz x y zA K z k DU n n mn e n n\n (2) \nHere, the first and second terms are the inhomogeneous and homogeneous exchange energ ies \nwhere Aex > 0 is the exchange stiffness and is the magnetic susceptibility . The third term is \nthe easy -axis anisotropy along the z axis (nanowire axis) with positive K which changes \nlinearly with the z-coordinate K(z) = K0 z·dK/dz. The fourth term is the so-called second \nanisotropy or intermediate anisotropy defined along the x axis with k > 0, and this anisotropy \nshould be weaker than the easy -axis anisotropy along the z-axis. The last term is the \ninterfacial DM interaction with D > 0 and ey is the unit vector in the y direction . To obtain an \nmore explicit expression of the Lagrangian density, w e replace m with m = s ṅ n,36,37 and \nobtain \n2 2 2 2( ) ( ) ( ),2 2 2 2 2ex\ns z x y zA K k DL n n n a n n n e n n\n (3) \nwhere s2 parametrizes the inertia of dynamics. The dissipative dynamics can be described by introducing the Rayleigh function density R = sṅ2/2 with s 1s1 2s2 \naccounting for the energy and spin loss due to the magnetic dynamics.38 \nNow we discuss the low -energy dynamics of FiM domain wall. Following the earlier \nwork, we introduce two collective coordinates, the position q(t) and azimuthal angle (t) in Eq. \n(3) to characterize the FiM domain wall under an anisotropy gradient . We consider the Walker \nansatz39 for the domain wall profile: n(z, t) = (sech(( z-q)/)cos, sech(( z-q)/)sin, \ntanh(( z-q)/)) where is the domain wall width. After a pplying the Euler -Lagrange equation, \nwe obtain t he equations of motion for the two coordinates : \n/, Mq G Mq F \n (4) \n00 / sin cos sin , I Gq I k D \n (5) \nwhere M = 2/ is the mass with the cross -sectional area of the domain wall , I = 2 is \nthe moment of inertia, G = 2s is the gyrotropic coefficient, = s is the relaxation time, F \n= 4·dK/dz is the force exerted by an anisotropy gradient , k0 = 2k, and D0 = D/2. \nA specific solution to Eq. ( 4) and Eq. (5) for k = D = 0 gives the domain wall velocity v \nand domain wall plane precession speed : \n2\n22,\nsdKvs s dz\n\n (6) \n.svs\n (7) \nEq. (6) shows that velocity v increases linearly with dK/dz and reaches the maximum at \nthe angular momentum compensation point TA where s vanishes (s ~ 0). To illustrate that \nthis velocity can be high in real materials, one gives a crude estimation of v by taking the \nwell-known FiM compound GdFeCo as an example .19,24,26 Setting the internal parameters \nexchange stiffness Aex = 50 pJ/m, anisotropy constant at high anisotropy end K0 = 0.5 MJ/m3, \nM1 = 440 kA/m, M2 = 400 kA/m, 1 = 2 = 0.01, g1 = 2.2, and g2 = 2.0 , one obtains a wall \nmotion velocity v ~ 1.2 km/s at the compensation point under an anisotropy gradient dK/dz = \n300 GJ/m4, comparable to the current - and the field -driven motions for antiferromagnetic \ndomain wall motions . Furthermore, as shown in Eq. (7), the domain wall plane rotates with \nthe domain wall propagation without any favored orientation due to k = 0, which is closely dependent of s. \n \n2.2. Numerical calculation \nIn order to check the validity of the above analytical treatment , we also perform the \nnumerical simulations based on the atomistic LLG equation. Here, the corresponding \none-dimensional discrete Hamiltonian is given by :40 \n22\n11 ( ) ( ) ( ),zx\ni i i i x i i i i\ni i i iH=J K S K S S S D S S\n (8) \nwhere t he first term is the exchange interaction with J = 1, Si is the normalized spin moment \nvector at lattice site i. The second term is the anisotropy energy with the easy axis along the \nz-direction, and the anisotropy constant at site i is described by Ki = K 0 – ia·K where K \ndescribes the anisotropy gradient magnitude, a is the lattice constant . The third term is the \nsecond anisotropy Kx along the x-axis, and the last term is the DM interaction with Di = (0, Dy, \n0). \nThen, t he dynamics is investigated by solving the stochastic LLG equation,41-43 \n 2( ) ,(1 )ii\ni i i i i\nii tM SS H S H\n (9) \nwhere Hi = − H/Si is the effective field. Without loss of generality, we set the damping \nconstants 1 = 2 = 0.01, the gyromagnetic ratio s 1 = 1.1 and 2 = 1.0 corresponding to the \nLandé g-factors g1 = 2.2 and g2 = 2.0 for the two sublattices .44 \nTo investigate the dynamics in the vicinity of the momentum compensation point, several \nsets of (M1, M2) are employed, as listed in Table I. Unless stated elsewhere, the LLG \nsimulations are performed on a 1 1 400 lattices with open boundary conditions using the \nfourth -order Runge -Kutta method with a time step t = 1.0 10−4 s/Jeff where s is the \nsaturation moment and eff = (1 + 2)/2. After a sufficient relaxation of the domain structure , \nthe anisotropy gradient is applied to drive the domain wall motion , as schematically depicted \nin Fig. 1 (a). \nAs a matter of fact, a comparison between the analytical treatment and the atomic model \ncan be useful in qualitative sense . It is seen from the atomistic model that various torques act \non the wall spins .16 The two spins neighboring the central wall spin deviate differently from the ea sy-axis with 1 > 2, resulting in the net damping torque d from the exchange \ninteraction on the central spin, as depicted in Fig. 1(b). The damping torques d ~ − S (S \nH) point in an opposite direction on the two sublattices and drive the wall motion. Moreover , \nthe precession torques p ~ − S H pointing into the same direction on the two sublattices are \nunequal in magnitude in the case of s 0, resulting in the precess ion of the wall plane with \nthe wall propagation , in agreement with Eq. (7). For s = 0, torques p on the two sublattices \nare equal and the domain wall plane is fixed. \nThus, the fast domain wall motion and the precession of the wall plane in ferrimagnets \nare theoretically revealed and qualitatively confirmed by the atomic model simulation s. \nSubsequently, we present the analytically derived and numerically calculated results to \ndemonstrate the quantitative consistence between the analytical derivation and atomistic \nsimulation on one hand, and more importantly to unveil the FiM dynamics in details. \n \nIII. Results and discussion \n3.1. Domain wall dynamics \nWe first present the domain wall dynamics by discussing the wall velocity and precession \nspeed as a function of the anisotropy gradient respectively. Fig. 2(a) shows the numerically \nsimulated (empty points) and Eq. (6) -based calculated (solid lines) wall v elocit y v as a \nfunction of K for various δs and K0 = 0.01 J, Kx = 0 and Dy = 0. It is seen that the simulated \ndata fit the calculations perfectly , confirming the validity of the analytical treatment . Here, \ntwo issues deserve highlighting. First, the driving torque increases with the increasing K, \nwhich significantly enhances the wall motion speed. Specifically, v increases linearly with K, \nnoting that here only low anisotropy gradient is considered and the domain wall width is \nhardly changed during the motion. Second, for a fixed K, v increases with decreasing \nmagnitude of s, and reaches to the maximum at the angular momentum compensation point \ns = 0, as clearly shown in Fig. 2(b) w here v(s) curves for various K are presented. \nWe then discuss the domain wall plane precession which appears for a nonzero s in \naccompanying with the wall motion, as shown in Fig. 2(c) where the angular velocity (d/dt) \nof the plane as a function of K is plotted . Also two issues are highlighted . First, t he angular \nvelocity increases linearly with K or the wall speed v. In comparison with the dynamics for s = 0 where the domain wall plane is fixed, the wall plane precession leads to additional \nenergy dissipation, resulting in the low wall mobility for nonzero s under the same K. \nSecond and more interestingly, the precession direction of the wall plane depends on the sign \nof s. Specificall y, the wall plane precesses clockwise around the easy -axis for s > 0, while \ndoes counter clockwise for s < 0, as clearly shown in Fig. 2(d) where the x and y components \nof local quantity n, nx and ny, are presented at various times for s > 0 (top half) and s < 0 \n(bottom half) . With the wall motion, opposite precession directions are clearly observed in the \ntwo cases , in consistent with the theoretical predict ion in Eq. (7). \nSo far, the anisotropy gradient driven domain wall motion in the vicinity of the angular \nmomentum compens ation point of ferrimagnets has been clearly uncovered in our theoretical \nanalysis and LLG simulations. In experiments, anisotropy gradient could be induced by \ntuning electric field on particular heterostructures , and efficiently drives the domain wall \nmotion generating Joule heat much less than those electrical current related methods . Thus, \nthe proposed method in the work is expected to be both low power -consuming and \nhigh-efficient, which is essential for future spintronic applications. \n \n3.2. Roles of internal parameters \nBased on the good consistency between the analytical analysis and numerical calculations, \none is able to discuss the roles of various internal parameters. An unveiling of these roles \nwould be highly appreciated for practical applicatio ns including the materials selection, \ndevice geometry design, and performance optimization. \nFirst, the anisotropy constant K0 determines the wall width which can be estimated by \na(J/2K0)1/2, and in turn affects the wall speed which increases with as demonstrated in Eq. \n(6). Thus, contra ry to K, a large K0 results in a small and makes the wall motion slow, as \nconfirmed in our simulations. In Fig. 3(a), the simulated and calculated speeds as functions of \nK0 for various K at the angular momentum compensation point are plotted, not only showing \nthe excellent consistence between the simulation and analytical derivation but also clearly \nrevealing that the anisotropy magnitude enables an decelerated wall motion. In addition, an \nenhanced damping term a lways reduces the wall mobility,45,46 and v decreases with the \nincrease of the damping constant α. As a matter of fact, v linearly increases with 1/ α, as shown in Fig. 3(b) where presents the simulated and calculated v as functions of 1/ α at δs = 0 for \nfixed K0 and K. \nIn the above analysis, the wall width is simply set to be unchanged during the wall \nmotion, which well describes the case of very low anisotropy gradient. However, when the \nwall shifts under a high gradient, the wall is considerably enlarged, resulting in an accelerated \ndomain wall motion. This phenomenon has been also observed in our simulations (dashed \nlines) and calculations (solid lines) in Fig. 4 which give the evolution of the wall position (Fig. \n4(a)) and the local wall velocity (Fig. 4(b)) for various K at δs = 0. In this case, the wall \nwidth could be updated to = a(J/2Kc)1/2 with Kc the anisotropy on the wall central spin. A \nconstant velocity is obtained under a low gradient K ~ 0.5 10-5J/a, while an acceleration of \nthe wall motion under a high gradient K ~ 2 10-5J/a is clearly observed. \nSecond, the intermediate anisotropy Kx could be non -negligible in some FiM materials , \nand affects the wall motion. Subsequently, we check the effect of Kx on the dynamics of \ndomain wall in ferrimagnets. The time evolutions of the wall position for various Kx for δs = \n0.022 are presented in Fig. 5(a) which exhibits three types of wall motion .16 As discussed \nabove, the wall has no favored orientation for Kx = 0 and rotates continuously and moves \nconstantly with a reduced speed. The consideration of the intermediate anisotropy suppresses \nthe rotation of the wall plane, and in turn significantly affects the wall motion . In the case of \nsmall anisotropy of Kx = 2.5 10-5 J, the Wal ker breakdown occurs under the anisotropy \ngradient K larger than the threshold value KWB. Here , the Walker breakdown gradient \nKWB can be estimated by Kxs/4s.47,48 In the case of high anisotropy Kx = 10 10-5 J, the \nprecession of the domain wall is completely suppressed for the considered K, resulting in the \nwall motion with a maximal velocity. On the other hand, the wall motion at the angular \nmomentum compensation point δs = 0 where no precession of the wall is available is \nindependent of the anisotropy Kx, as clearly shown in Fig. 5(b) where presents the mean \nvelocity of the domain wall as a function of Kx for δs = 0 and δs = 0.0 22. Moreover, f or a fixed \ngradient K below the Walker breakdown KWB, δs = 0.022 is with a magnetization smaller \nthan δs = 0, and the domain wall motion for δs = 0.0 22 is slightly faster than δs = 0. \nThird, a DM interaction could be induced at interface between heavy metal and \nferrimagnet and modulated efficiently through elaborate heterostructure design. Similarly, the interfacial DM interaction Dy(0, 1, 0) also suppresses the precession of the wall plane and \nspeed s up the wall motion below KWB. In Fig. 5(c), the simulated velocit ies as a function of \nDy for δs = 0 and δs = 0.0 22 for K = 0.5 10-5J/a is plotted, revealing the critical DM \ninteractions Dc which can be given by |Dc| = 8δs2·KWB / πs for nonzero δs.26,47,48 Under a \nfixed K, the Walker b reakdown occurs for | Dy| < |Dc|, while vanishes for | Dy| > |Dc|. The \nsimulated | Dc| (empty points) as a function of δs for various K is presented in Fig. 5(d), well \nin consistent with the theoretical derivation (solid lines) . Thus, this prediction could be used \nto improve the Walker breakdown field and to enhance the domain wall mobility, which is \nvery meaningful for spintronic applications. \n Thus, the domain wall motion depending on the internal parameters has been clearly \nunveiled, which definitely provides usefu l information for future material selection and device \ndesign. For example, high domain wall mobility could be available in ferrimagnet with not \ntoo large K0 and considerable second anisotropy, as suggested in our calculations. Moreover, a \nlarge DM interac tion generated in interface between heavy metal and ferrimagnet \nsignificantly improves the Walker breakdown field and ensures the fast domain wall motion. \nOf cause, these predictions given here deserve to be checked in further experiments. \n \nIV. Conclusion \nTo summarize , we have studied analytical ly and numerically the dynamics of the domain \nwall in ferrimagnet s driven by the magnetic anisotropy gradient. The wall moves towards the \nlow anisotropy side to release the free energy and reaches to a maxim al velocity at the angular \nmomentum compensation point where exhibits the antiferromagnetic dynamics . Moreover, \nthe net spin angular momentum determines not only the wall velocity but also the precession \ndirection of the domain wall plane . Furthermore, for nonzero net angular momentum, Walker \nbreakdown occurs under a critical anisotropy gradient which significantly depends on the \nintermediate anisotropy and interfacial DM interaction. 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Mougin, M. Cormier, J. Adam, P. Metaxas, and J. Ferr, Europhys. Lett. 78, 57007 \n(2007). \n \n Table I. Parameters chosen for the simulation. \n \nParameter 1 2 3 4 5 6 7 \nM1 1.13 1.12 1.11 1.1 1.09 1.08 1.07 \nM2 1.0 1.04 1.02 1.0 0.98 0.96 0.94 \ns -0.03273 -0.0218 -0.0109 0 0.0109 0.0218 0.03273 \n \n \n \n \n \n \n \nFIG.1. (color online) (a) Illustration of a domain wall in ferrimagnetic nanowire under an \nanisotropy gradient. Here the asymmetry of the domain wall center is exaggerated. (b) A \nschematic depiction of torques acting on the central spins of the domain wall. \n \n \n \nFIG.2. (color online) The simulated (empty points) and calculated (solid lines) velocities as \nfunctions of (a) K for various s, and (b) s for various K for K0 = 0.01. (c) The simulated \n(empty points) and calculated (solid lines) angular velocit ies of the wall plane as function s of \nK for various s, and (d) the evolutions of the local nx and ny for s > 0 (top half) and s < 0 \n(bottom half) . The rotations of the wall plane are shown in the insert of (d). \n \n \n \nFIG.3. (color online) The simulated (empty points) and calculated (solid lines) velocities at \nthe momentum compensation point as functions of (a) K0 for various value of K for = 0.01, \nand (b) 1/ for K0 = 0.01 J and K = 0.5 10-5 J/a. \n \n \n \nFIG.4. (color online) The simulated (dashed lines) and calculated (solid lines) (a) evolutions \nof the wall positions for various K, and (b) instantaneous speed for various K. \n \n \n \n \nFIG.5. (color online) For K = 0.5 × 10-5 J/a, the simulated (a) evolutions of the wall position \nfor various Kx, and mean velocities as functions of (b) Kx and (c) Dy for δs = 0 and δs = 0.022. \n(d) The simulated (empty points) and calculated (solid lines) | Dc| as a function of δs for \nvarious K. \n \n" }, { "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). 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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": "0707.1953v1.Effect_of_antiferromagnetic_exchange_interactions_on_the_Glauber_dynamics_of_one_dimensional_Ising_models.pdf", "content": "arXiv:0707.1953v1 [cond-mat.mtrl-sci] 13 Jul 2007Effect of antiferromagnetic exchange interactions on the Gl auber dynamics of\none-dimensional Ising models\nM. G. Pini1∗and A. Rettori2,3†\n1Istituto dei Sistemi Complessi, CNR, Sezione di Firenze,\nVia Madonna del Piano, I-50019 Sesto Fiorentino (FI), Italy\n2Dipartimento di Fisica, Universit` a di Firenze, Via G. Sans one 1, I-50019 Sesto Fiorentino (FI), Italy\n3INFM-CNR National Research Center S3, Via Campi 213/A, I-41 100 Modena, Italy\n(Dated: October 26, 2018)\nWe study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of\ntwo different one-dimensional (1D) Ising models with spin ±1. The first model is an Ising chain\nwith antiferromagnetic exchange interaction limited to ne arest neighbors and subject to an oscil-\nlating magnetic field. The system of master equations descri bing the time evolution of sublattice\nmagnetizations can easily be solved within a linear field app roximation and a long time limit. Res-\nonant behavior of the magnetization as a function of tempera ture (stochastic resonance) is found,\nat low frequency, only when spins on opposite sublattices ar e uncompensated owing to different\ngyromagnetic factors ( i.e., in the presence of a ferrimagnetic short range order). The s econd model\nis the axial next-nearest neighbor Ising (ANNNI) chain, whe re an antiferromagnetic exchange be-\ntween next-nearest neighbors (nnn) is assumed to compete wi th a nearest-neighbor (nn) exchange\ninteraction of either sign. The long time response of the mod el to a weak, oscillating magnetic field\nis investigated in the framework of a decoupling approximat ion for three-spin correlation functions,\nwhich is required to close the system of master equations. Th e calculation, within such an approx-\nimate theoretical scheme, of the dynamic critical exponent z, defined as 1 /τ≈(1/ξ)z(whereτis\nthe longest relaxation time and ξis the correlation length of the chain), suggests that the T= 0\nsingle spin-flip Glauber dynamics of the ANNNI chain is in a di fferent universality class than that\nof the unfrustrated Ising chain.\nPACS numbers: 75.10.-b, 75.10.Pq, 75.50.Ee, 75.50.Gg\nI. INTRODUCTION\nAfter the publication of fundamental papers1,2on stochastic resonance (SR), it was realized that the response\namplitude of a nonlinear dynamic system to an external periodic signa l is greatly enhanced as a function of noise\nstrength, in the presence of a matching between the frequency o f the external force and the escape rate across an\nintrinsic energy barrier. Most of the SR research3was pursued on dynamic systems with a double well potential,\nsubject to both periodic and random forces, while only a few investig ations of SR in extended or coupled systems\nhave yet been conducted.4\nThe Ising model with Glauber dynamics5can be viewed as a set of coupled two-state oscillators, where the c oherent\nsignal is provided by an external oscillating magnetic field and therma l fluctuations are the only source of random\nnoise. Each spin is assumed to be in interaction with a heat reservoir o f some sort, which causes it to flip between\nthe values σ= +1 and σ=−1 randomly with time. In the presence of magnetic coupling between t he spins, the\ntransition probability for one spin to flip is assumed to depend on the c onfiguration of the neighboring spins. The\ntime evolution of the system is described by a master equation where the transition rates verify the detailed-balance\ncondition. Solvingthe masterequation, the time dependence ofthe magnetizationand ofthe spincorrelationfunctions\ncan be obtained. For exchange interaction limited to nearest neighb or (nn) spins, the response of the Ising model with\nGlauber dynamics to an oscillating magnetic field was investigated in one (1D),5,6two (2D),7and three (3D)8,9spatial\ndimensions. For the 1D nn Ising ferromagnet, Brey and Prados6obtained an analytic expression, within the linear\nfield approximation, for the amplitude and the phase of the induced m agnetization. The amplitude always presents\na maximum as a function of temperature, with a genuine resonant be havior only for low frequencies. The Glauber\ndynamics of the 1D Ising model with antiferromagnetic next-neare st neighbor (nnn) exchange interaction competing\nwith the nn one was investigated by Yang,10who employed a decoupling approximation to solve the master equatio n\nand get an analytical expression for the time-dependent magnetiz ation. He also found, by heuristic arguments, the\ndynamic critical exponent z, defined as 1 /τ≈(1/ξ)z(whereτis the longest relaxation time and ξis the correlation\nlength of the chain)11, to bez= 2, the same as that of the unfrustrated 1D nn Ising model.\nIn this paper, we study - at finite temperature T >0 - the effect of antiferromagnetic (AF) exchange interactions on\nthe single spin-flip Glauber dynamics of two different one-dimensional Ising models. Our interest in kinetic 1D Ising\nmodels with AF interactions is motivated by recently sinthesized coba lt-based12,13and rare-earth-based14,15single\nchain magnets, showing slow relaxation of the magnetization at low te mperature. The magnetic properties of the2\nformer chain compound, [Co(hfac) 2NITPhOMe], can be described in terms of a 1D Ising model with AF nn ex change\ncoupling.13,16However, the resulting short rangeorderis ferrimagnetic, owing t o the alternationalong the chain oftwo\ndifferent kinds of magnetic centers (a metal ion, Co2+, and a nitronyl-nitroxide radical, PhOMe), both with S= 1/2\nbut with different gyromagnetic factors. In spite of further comp lications due to non-collinearity of the spins,16this\nsystem was shown to be the first experimental realization of a 1D nn Ising model with Glauber dynamics.13The single\nchainmagnetsbelongingtothe latter classofrare-earth-basedc ompounds, ofgeneralformula[M(hfac) 3(NiTPhOPh)],\nwhere M=Eu, Gd, Tb, Dy, Ho, Er, orYb, and PhOPhis anitronyl-nitro xideradical, arecharacterizedby strongIsing-\ntype anisotropy and by the simultaneous presence of both nn and n nn exchange interactions between the magnetic\ncenters, with the last ones being antiferromagnetic in nature.14,15\nThe paper is organized as follows. In Section II we investigate the Gla uber dynamics in a collinear Ising chain,\nwith antiferromagnetic exchange interaction limited to nearest neig hbors and different gyromagnetic factors on the\ntwo opposite sublattices, subject to an oscillating magnetic field. Th e system of master equations describing the\ntime evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time\nlimit. Resonant behavior of the magnetization as a function of tempe rature (stochastic resonance) is found, at\nlow frequency, only when spins on opposite sublattices are uncompe nsated owing to different gyromagnetic factors\n(i.e., in the presence of a ferrimagnetic short range order). In Sectio n III we investigate the 1D axial next-nearest\nneighbor Ising (ANNNI) model, where an antiferromagnetic exchan ge between next-nearest neighbor spins is assumed\nto compete with a nearest-neighbor exchange interaction of eithe r sign. The long time response of the model to a\nweak, oscillating magnetic field is investigated in the framework of a de coupling approximation (required in order to\nclose the system of master equations) for three-spin correlation functions, which in principle is more accurate than the\none reported in Ref. 10. As a consequence, our approximate calcu lation of the dynamic critical exponent zsuggests\nthat the T= 0 single spin-flip Glauber dynamics of the ANNNI chain is in a different un iversality class than that of\nthe unfrustrated Ising chain. Finally, the conclusions are drawn in S ection IV.\nII. GLAUBER DYNAMICS IN THE NEAREST-NEIGHBOR FERRIMAGNETI C ISING CHAIN\nWe consider a one-dimensional Ising model with a nearest-neighbor antiferromagnetic exchange interaction, J <0,\nin the presence of a time-dependent external field. The Hamiltonian of the system is\nH=−JN/summationdisplay\nj=1σz\njσz\nj+1−µ0H(t)N/2/summationdisplay\nj=1(gAσz\n2j−1+gBσz\n2j) (1)\nwhereµ0is the Bohr magneton, and H(t) =H0e−iωtis an external magnetic field applied along the zdirection and\noscillating in time with frequency ω. Spins on opposite sublattices are allowed to take possibly different g yromagnetic\nfactors (gA/ne}ationslash=gB), while we assume σz\nj=±1∀j. Hereafter, the zindex will be dropped for ease of notation. In the\nabsence ofa magnetic field, if gA/ne}ationslash=gBthe ground state is ferrimagnetic, with opposite uncompensated m agnetizations\non the two sublattices; if gA=gBthe ground state is antiferromagnetic, with compensated sublatt ice magnetizations.\nWhen the system is endowed with single spin-flip Glauber dynamics,5its time evolution is described by the master\nequation\n∂\n∂tp(σ,t) =/summationdisplay\nj/bracketleftig\nWj(Rjσ)p(Rjσ,t)−Wj(σ)p(σ,t)/bracketrightig\n(2)\nwherep(σ,t) is the probability for the system to assume the configuration σ={σ1,···,σj,···,σN}at timet,Rjσ\nis the configuration obtained from σby flipping spin j, andWj(σ),Wj(Rjσ) are the transition rates between such\nconfigurations.\nFor a 1D Ising model of spins ( σj=±1) withferromagnetic nn exchange interaction J >0 and gyromagnetic factor\ng, Brey and Prados6showed that, for low frequency, a stochastic resonance phenom enon occurs: i.e., the induced\nmagnetization M(t) =gµ0/summationtextN\nj=1/an}bracketle{tσj;t/an}bracketri}htoscillates at the same frequency as the magnetic field, and the amplit ude of\nM(t) presents a sharp maximum as a function of temperature T. The resonance temperature, Tr, is determined by\nthe matching between the frequency, ω, of the external field and the inverse of the statistical time scale, 1/τ(Tr),\nassociated to the spontaneous ( i.e., in zero field) decay of the magnetization. In zero field, the magnet ization of the\n1D nn Ising ferromagnet was found5,6to relax to its equilibrium value, M= 0, with the asymptotic t→ ∞behavior\nM(t)≈e−t/τ(T)/√\nt. The relaxation time τwas found to be exponentially divergent for T→0,τ(T)≈e4J\nkT(where\nkdenotes Boltzmann’s constant), and to become of the order of th e inverse of the transition rate of an isolated spin\nforT→ ∞,τ≈1/α.5,63\nFor a 1D Ising model with antiferromagnetic nn exchange interaction ( J <0), the master equation (2) is still the\nstarting point for the study of the chain dynamics. In this case, if gA/ne}ationslash=gB, the transition rates in the presence of a\nfield are assumed to be different for even ( A) and odd ( B) lattice sites j\nWj(σ) =W(0)\nj(σ)/bracketleftig\n1−σjtanh(βA,B)/bracketrightig\n=1\n2α/bracketleftig\n1−1\n2γσj(σj−1+σj+1)/bracketrightig/bracketleftig\n1��σjtanh(βA,B)/bracketrightig\n(3)\nwhereW(0)\nj(σ) denote the transition rates in zero field. The transition rate of an isolated spin,1\n2α, is considered as\ntemperature independent and sets the time scale. In the case of in teracting spins, the probability per unit time of the\nj-th spin to flip depends on the orientation of its nearest neighbors. The magnetic field favors one orientation with\nrespect to the other. A correspondence between the paramete rsγ,βA,Bof the stochastic model and the parameters\nJ,gA,Bµ0H(t) of the statistical Ising model can be obtained5,6observing that at equilibrium∂\n∂tp(σ,t) = 0, so that\n/summationdisplay\nj/bracketleftig\nWj(Rjσ)peq(Rjσ,t)/bracketrightig\n=/summationdisplay\nj/bracketleftig\nWj(σ)peq(σ,t)/bracketrightig\n. (4)\nNext, requiring the detailed balance ( i.e., the microscopic reversibility) condition to be satisfied\nWj(Rjσ)\nWj(σ)=peq(σ,t)\npeq(Rjσ,t), (5)\nwithpeq(σ,t) =e−H(σ)\nkTandpeq(Rjσ,t) =e−H(Rjσ)\nkT,one readily obtains\nγ= tanh/parenleftig2J\nkT/parenrightig\n, βA,B= tanh/parenleftiggA,Bµ0H(t)\nkT/parenrightig\n. (6)\nThe evolution equation for the spin expectation value /an}bracketle{tσj;t/an}bracketri}ht=/summationtext\nσσjp(σ,t) is directly obtained from the mas-\nter equation to be∂\n∂t/an}bracketle{tσj;t/an}bracketri}ht=−2/an}bracketle{tσjWj(σ);t/an}bracketri}ht.5,6Considering that for model (1) the spins belong to two opposite\nsublattices, the system of evolution equations in the presence of a n oscillating field is\n∂\n∂(αt)/an}bracketle{tσ2j−1;t/an}bracketri}ht=−/an}bracketle{tσ2j−1;t/an}bracketri}ht+1\n2γ/parenleftig\n/an}bracketle{tσ2j−2;t/an}bracketri}ht+/an}bracketle{tσ2j;t/an}bracketri}ht/parenrightig\n+βA/bracketleftig\n1−1\n2γ/parenleftig\n/an}bracketle{tσ2j−2σ2j−1;t/an}bracketri}ht+/an}bracketle{tσ2j−1σ2j;t/an}bracketri}ht/parenrightig/bracketrightig\n∂\n∂(αt)/an}bracketle{tσ2j;t/an}bracketri}ht=−/an}bracketle{tσ2j;t/an}bracketri}ht+1\n2γ/parenleftig\n/an}bracketle{tσ2j−1;t/an}bracketri}ht+/an}bracketle{tσ2j+1;t/an}bracketri}ht/parenrightig\n+βB/bracketleftig\n1−1\n2γ/parenleftig\n/an}bracketle{tσ2j−1σ2j;t/an}bracketri}ht+/an}bracketle{tσ2jσ2j+1;t/an}bracketri}ht/parenrightig/bracketrightig\n(7)\nThe system is not closed owing to the presence of two-spin, time-de pendent correlation functions on the right\nhand sides. In order to solve it, a linear field approximation is made5,6so that tanh[( gA,Bµ0H0)/(kT)] can be\nexpanded for small values of the argument and two-spin correlatio ns can be evaluated in the absence of a field.\nMoreover, if in the long time limit t→ ∞the nn correlation functions are assumed5,6to take their equilibrium value\n/an}bracketle{tσjσj+1;t/an}bracketri}ht →η= tanh[J/(kT)], the system of two coupled equations of motion for the two sublat tice magnetizations\nM1(t) =gAµ0N/2/summationdisplay\nj=1/an}bracketle{tσ2j−1;t/an}bracketri}ht, M2(t) =gBµ0N/2/summationdisplay\nj=1/an}bracketle{tσ2j;t/an}bracketri}ht (8)\ncan be written in matrix form\n/parenleftigg∂\n∂(αt)M1(t)\n∂\n∂(αt)M2(t)/parenrightigg\n+/parenleftbigg1−gA\ngBγ\n−gB\ngAγ1/parenrightbigg/parenleftbigg\nM1(t)\nM2(t)/parenrightbigg\n=N(T)/parenleftbigg\ng2\nA\ng2\nB/parenrightbigg\ne−iωt\n(9)\nTaking into account that γ=2η\n1+η2, the temperature dependent coefficient N(T) can be expressed as\nN(T) =N\n2µ2\n0H0\nkT(1−γη) =N\n2µ2\n0H0\nkT1−η2\n1+η2. (10)\nThe above system can be decoupled diagonalizing the 2 ×2 non-symmetric matrix on the l.h.s. of Eq. (9). Denoting\nbyM1(t) andM2(t) the normal modes, one obtains\n/parenleftigg∂\n∂(αt)M1(t)\n∂\n∂(αt)M2(t)/parenrightigg\n+/parenleftbigg\nλ10\n0λ2/parenrightbigg/parenleftbigg\nM1(t)\nM2(t)/parenrightbigg\n=N(T)/parenleftbigg\nf1\nf2/parenrightbigg\ne−iωt4\nwhere the eigenvalues λn(n= 1,2) turn out to be independent of the gyromagnetic factors gAandgB\nλ1= 1−γ, λ 2= 1+γ, (11)\nand thefn(n= 1,2) coefficients are\nf1=gB\n2(gB+gA)f2=gB\n2(gB−gA). (12)\nThe relationships between the normal modes Mn(t) and the sublattice magnetizations Mn(t) (n= 1,2) are\nM1(t) =1\n2/bracketleftig\nM2(t)+gB\ngAM1(t)/bracketrightig\n,M2(t) =1\n2/bracketleftig\nM2(t)−gB\ngAM1(t)/bracketrightig\n(13)\n(i.e.,M1(t) andM2(t) are related to the net and the staggered magnetization, respec tively). Conversely, one has\nM1(t) =gA\ngB/bracketleftig\nM1(t)−M2(t)/bracketrightig\n, M2(t) =M1(t)+M2(t). (14)\nThe general solution for the normal modes is ( n= 1,2)\nMn(t) =Mn(t0)e−t−t0\nτn+N(T)fn/integraldisplayt\nt0dt′et′−t\nτne−iωt′(15)\nwhere the relaxation times τnare expressed, in terms of the eigenvalues λnof the non-symmetric 2 ×2 matrix, as\nτn= 1/(αλn), so that\nτ1=1\nα(1−γ), τ2=1\nα(1+γ). (16)\nIn the absence of an external magnetic field, N(T) = 0, the normal modes Mn(t) are found to relax exponentially.\nIn the low temperature limit, T→0, one has γ= tanh/parenleftig\n2J\nkT/parenrightig\n≈J\n|J|/parenleftig\n1−2e−4|J|\nkT/parenrightig\n, so that for antiferromagnetic\nnn exchange ( J <0), the first relaxation time is simply τ1≈1\n2α, while the second relaxation time is exponentially\ndiverging with decreasing T,τ2≈1\n2αe4|J|\nkT. For high temperatures, kT≫ |J|, both relaxation times become of the\norder of the inverse of the transition rate of an isolated spin, τ1≈τ2≈1/α.\nFor non vanishing magnetic field, the time dependence of the normal modes is obtained letting t0→ −∞in Eq. (15)\nMn(t) =N(T)fn\nλn1\n1−iωτne−iωt(n= 1,2) (17)\nThe total magnetization is\nMtot(t) =M1(t)+M2(t) =gB+gA\ngBM1(t)+gB−gA\ngBM2(t) =χ(ω,T)H0e−iωt, (18)\nwhere the complex susceptibility χ(ω,T) is given by\nχ(ω,T) =Nµ2\n0\nkT/bracketleftigg/parenleftiggB+gA\n2/parenrightig21+η\n1−η1\n1−iωτ1+/parenleftiggB−gA\n2/parenrightig21−η\n1+η1\n1−iωτ2/bracketrightigg\n. (19)\nIn the limit ω→0, the static susceptibility of the Ising ferrimagnetic chain in zero fie ld is correctly recovered:\nsee Appendix A.1, Eq. (A6), for details. As regards the dynamic res ponse of the system to a weak, oscillating\nmagnetic field, from Eq. (19) it is apparent that, for antiferromag netic nn exchange ( J <0) andT→0, the first\nterm on the r.h.s. is associated with a fast relaxation, while the secon d term with an exponentially slow relaxation.\nThus, a resonant behavior, similar to the one observed in the ferro magnetic nn Ising chain endowed with single spin-\nflip Glauber dynamics,6is possible only when spins on opposite sublattices are uncompensate d owing to different\ngyromagnetic factors ( i.e., in the presence of ferrimagnetic short range order). See Fig. 1, where the temperature\ndependence of the amplitude of χ(ω,T) is reported, for selected values of the frequency, both in the co mpensated\n(J <0 andgA=gB) and uncompensated ( J <0 andgA/ne}ationslash=gB) case.\nThe resonant behavior shown by the ferrimagnetic chain at low freq uency (see Fig. 1c) is a manifestation of the\nstochastic resonance phenomenon:3i.e., the response of a set of coupled bistable systems to a periodic driv e is5\nFIG. 1: (color online) Temperature dependence of the amplit ude of the complex susceptibility |χ(ω,T)|for an Ising chain\nwith antiferromagnetic nearest neighbor interaction J=−1, subject to a weak external magnetic field oscillating at fr equency\nω. Figures (a), (b) refer to the compensated case ( gA=gB= 2), while Figures (c) and (d) to the uncompensated case\n(gA= 2, gB= 3), for selected values of the frequency ( ω/α= 0.001 and 10). In Figs. (c,d) the thin (color) lines represent t he\ncontributions to the amplitude of the two terms on the r.h.s. of Eq. (19), while the thick (black) line is their sum. A reson ant\nbehavior (similar to the one predicted for the nn Ising ferro magnetic chain endowed with single spin-flip Glauber dynami cs)5,6\nis observed only in the uncompensated case for low frequency (notice the enhanced vertical scale in Fig. c).\nFIG. 2: (color online) Frequency dependence of the peak temp erature of the amplitude of the complex susceptibility |χ(ω,T)|\nof an Ising chain with nearest neighbor exchange interactio n. Triangles: compensated antiferromagnet ( J=−1,gA=gB= 2);\ncircles: uncompensated ferrimagnet ( J=−1,gA= 2,gB= 3); squares: ferromagnet ( J= +1,gA=gB= 2).6The dashed\nlines are guides to the eye.6\nenhanced in the presence of a stochastic noise when a matching occ urs between the fluctuation induced switching\nrate of the system and the forcing frequency. In the ferrimagne tic chain, the role of stochastic noise is played by\nthermal fluctuations and the resonance peak occurs when the de terministic time scale of the external magnetic field\nmatches with the statistical time scale associated to the spontane ous decay of the net magnetization Mtot(t). For low\nfrequency ω≪α(i.e., low temperature), the resonance condition for the uncompensa ted case is\nω−1≈τ2(Tpeak), (20)\nwhile for the compensated case only the mode with fast relaxation τ1≈O(α−1) contributes, providing a broad peak\nrather than a genuine resonance. For high frequency ω≫α(i.e., high temperature) a broad peak is found, both for\nthe uncompensated and the compensated case, since the two rela xation times τ1andτ2become of the order of 1 /α,\nso that the resonance condition cannot be fulfilled.6\nThe frequency dependence of the peak temperature Tpeakis reported in Fig. 2 both for the compensated (anti-\nferromagnetic) and the uncompensated (ferrimagnetic) chain, a nd compared with the ferromagnetic counterpart.6\nIn the compensated case, the frequency dependence of the pea k is very smooth, owing to the smooth temperature\ndependence of the relaxation time τ1, ranging between 1 /(2α) at low Tand 1/αat highT. In the uncompensated\ncase, a behavior very similar to the ferromagnetic one is observed f or low frequency: the reason is that for low ωthe\ndominant contribution to χ(ω) is provided by the second term on the r.h.s. of Eq. (19). At interme diate frequency,\na maximum is observed owing to the coming into play of the first term on the r.h.s. of Eq. (19). Finally, for ω≫α,\nthe amplitude of χ(ω) becomes\n|χ(ω,T)| ≈Nµ2\n0\nkTα\nω/bracketleftigg/parenleftiggB+gA\n2/parenrightig21+η\n1−η(1−γ)+/parenleftiggB−gA\n2/parenrightig21−η\n1+η(1+γ)/bracketrightigg\n(21)\nwhere both terms in square brackets on the r.h.s. of Eq. (21) pres ent a maximum at the same temperature, which is\nnumerically determined to be Tpeak≈1.66711|J|.\nIII. GLAUBER DYNAMICS IN THE AXIAL-NEXT-NEAREST-NEIGHBOR -ISING (ANNNI) CHAIN\nWe consider a 1D axial-next-nearest-neighbor Ising (ANNNI) mode l with spins alternating on two interlacing\nsublattices (denoted by AandB), with Hamiltonian\nH=−J1N/2/summationdisplay\ni=1/parenleftig\nσz\n2i−1σz\n2i+σz\n2iσz\n2i+1/parenrightig\n−J2N/2/summationdisplay\ni=1/parenleftig\nσz\n2i−1σz\n2i+1+σz\n2iσz\n2i+2/parenrightig\n−µ0H(t)N/2/summationdisplay\ni=1/parenleftig\ngAσz\n2i−1+gBσz\n2i/parenrightig\n.(22)\nThe intra-sublattice antiferromagnetic next-nearest neighbor c ouplingJ2<0 competes with the inter-sublattice near-\nest neighbor coupling J1, which may be of either sign. In what follows, we shall assume J1>0 (ferromagnetic\ncoupling). H(t) =H0eiωtis an external magnetic field applied along the zdirection and oscillating in time with fre-\nquencyω,µ0denotes the Bohrmagneton and the spins σz\ni=±1 areallowedto assume possibly different gyromagnetic\nfactors on odd and even sites ( gA/ne}ationslash=gB); thezindex shall be dropped for ease of notation.\nIn the limiting case gA=gB=g, Eq. (1) reduces to the well-known ANNNI (axial next-nearest-n eighbor Ising)\nmodel.17Depending on the competition ratio r=−J2/J1, this model in zero field is known to admit a ferromagnetic\nground state for r <1/2, and a (2 ,2) antiphase structure (two spins up, two spins down), with zero m agnetization,\nforr >1/2; forr= 1/2 the ground state is degenerate and disordered.18At finite temperatures, the model cannot\nsupport long range order; however, a strong short range order is present in the paramagnetic phase. For zero applied\nfield, as far as the thermodynamic propertiesare concerned,19the 1D ANNNI model can be mapped into an equivalent\n1D Ising model with only nearest neigbor interaction in an effective fie ld, and analytic results (see Appendix A.2) can\nbe obtained for the partition function and the spin correlationfunc tions.21,22In the presence of a static magnetic field,\nthe ground state of the generalized ANNNI model, i.e.a chain of alternating spins with different quantum numbers\nand different nnn exchangeinteractions on the two sublattices, wa sthoroughlyinvestigated,20and the thermodynamic\nproperties were exactly calculated (though numerically) by the tra nsfer matrix method.23,24\nHere we aim at investigating the long-time dynamic response of the AN NNI chain, Eq. (22), to a weak, external\nmagnetic field oscillating in time. The time evolution of the system is still d escribed by the master equation (2), but\nwith respect to the case of the nn Ising chain, the transition rates in zero field, W(0)\nj(σ), are now assumed to take the\nform\nW(0)\nj(σ) =1\n2α/bracketleftig\n1−1\n2γ1σj(σj−1+σj+1)/bracketrightig/bracketleftig\n1−1\n2γ2σj(σj−2+σj+2)/bracketrightig\n(23)7\nmeaning that the probability per unit time of the j-th spin to flip depends on the status of both its nearest neighbors\nand next nearest neighbors;1\n2α, the transition rate of an isolated spin, is arbitrary and sets the tim e scale. In the\npresence of a field applied along the zaxis, the transition rates Wj(σ) are given by\nWj(σ) =W(0)\nj(σ)/bracketleftig\n1−σjtanh(βA,B)/bracketrightig\n. (24)\nAs usual, a correspondence between the parameters γ1,γ2,βA,Bof the stochastic model and the parameters J1,\nJ2,gA,Bµ0H(t) of the statistical ANNNI model can be obtained requiring the deta iled balance ( i.e., the microscopic\nreversibility) condition, Eq. (5) to be satisfied at equilibrium. One find s10\nγ1= tanh/parenleftig2J1\nkT/parenrightig\n, γ2= tanh/parenleftig2J2\nkT/parenrightig\n, βA,B= tanh/parenleftiggA,Bµ0H(t)\nkT/parenrightig\n. (25)\nThe stochastic equation of motion for the spin expectation value /an}bracketle{tσj;t/an}bracketri}ht=/summationtext\nσσjp(σ,t) in the presence of an\noscillating field is then obtained, from the master equation, to be∂\n∂t/an}bracketle{tσj;t/an}bracketri}ht=−2/an}bracketle{tσjWj(σ);t/an}bracketri}ht, giving10\n∂\n∂t/an}bracketle{tσj;t/an}bracketri}ht=−/an}bracketle{tσj;t/an}bracketri}ht+1\n2γ1/parenleftig\n/an}bracketle{tσj−1;t/an}bracketri}ht+/an}bracketle{tσj+1;t/an}bracketri}ht/parenrightig\n+1\n2γ2/parenleftig\n/an}bracketle{tσj−2;t/an}bracketri}ht+/an}bracketle{tσj+2;t/an}bracketri}ht/parenrightig\n−1\n4γ1γ2/parenleftig\n/an}bracketle{tσjσj−1σj−2;t/an}bracketri}ht+/an}bracketle{tσjσj−1σj+2;t/an}bracketri}ht+/an}bracketle{tσjσj+1σj−2;t/an}bracketri}ht+/an}bracketle{tσjσj+1σj+2;t/an}bracketri}ht/parenrightig\n+ tanh/parenleftiggA,Bµ0H(t)\nkT/parenrightig\n×/bracketleftig\n1−1\n2γ1/parenleftig\n/an}bracketle{tσjσj−1;t/an}bracketri}ht+/an}bracketle{tσjσj+1;t/an}bracketri}ht/parenrightig\n−1\n2γ2/parenleftig\n/an}bracketle{tσjσj−2;t/an}bracketri}ht+/an}bracketle{tσjσj+2;t/an}bracketri}ht/parenrightig\n+1\n4γ1γ2/parenleftig\n/an}bracketle{tσj−1σj−2;t/an}bracketri}ht+/an}bracketle{tσj−1σj+2;t/an}bracketri}ht+/an}bracketle{tσj+1σj−2;t/an}bracketri}ht+/an}bracketle{tσj+1σj+2;t/an}bracketri}ht/parenrightig/bracketrightig\n. (26)\nwhere we remind that the subscripts AandBrefer to the case of jodd and jeven, respectively. This set of equations\nis not closed, owing to the time-dependent two-spin and three-spin correlation functions on the r.h.s. In order to solve\nit, we make the following approximations.\n•For sufficiently weak fields ( x= (gA,Bµ0H0)/(kT)≪1), the hyperbolic tangent on the r.h.s. of Eq. (26) is\nexpanded for low values of the argument (tanh x≈x) and two-spin correlation functions are calculated in the\nabsence of a field.\n•Three-spin correlation functions are decoupled, in all possible ways , into products of a single-spin expectation\nvalue and a two-spin correlation function\n/an}bracketle{tσjσj+mσj+n;t/an}bracketri}ht ≈ /an}bracketle{tσj;t/an}bracketri}ht/an}bracketle{tσj+mσj+n;t/an}bracketri}ht+/an}bracketle{tσj+m;t/an}bracketri}ht/an}bracketle{tσjσj+n;t/an}bracketri}ht+/an}bracketle{tσj+n;t/an}bracketri}ht/an}bracketle{tσjσj+m;t/an}bracketri}ht. (27)\nNotice that a different, and incomplete, decoupling was adopted in Re f. 10, thus leading to different results with\nrespect to the present work.\n•For sufficiently long times, two-spin correlationfunctions between n-th neighborsare assumed to be independent\nof the initial conditions and to take their static equilibrium values /an}bracketle{tσjσj+n;t/an}bracketri}ht →ηnfort→ ∞. Static two-spin\ncorrelation functions ηn=/an}bracketle{tσjσj+n/an}bracketri}htcan be exactly calculated in 1D via the transfer matrix method.21,22,23,24\nForgA=gBandH0= 0, analytic results21,22can be obtained for ηn: see Appendix A.2 for details.\nUnder these approximations, the master equation for the spin exp ectation value on a generic site jbecomes\n∂\n∂(αt)/an}bracketle{tσj;t/an}bracketri}ht=−/bracketleftig\n1+1\n2γ1γ2(η1+η3)/bracketrightig\n/an}bracketle{tσj;t/an}bracketri}ht\n+1\n2γ1/parenleftig\n1−γ2η2/parenrightig/bracketleftig\n/an}bracketle{tσj−1;t/an}bracketri}ht+/an}bracketle{tσj+1;t/an}bracketri}ht/bracketrightig\n+1\n2γ2/parenleftig\n1−γ1η1/parenrightig/bracketleftig\n/an}bracketle{tσj−2;t/an}bracketri}ht+/an}bracketle{tσj+2;t/an}bracketri}ht/bracketrightig\n+/bracketleftiggA,Bµ0H(t)\nkT/bracketrightig/bracketleftig\n1−γ1η1−γ2η2+1\n2γ1γ2(η1+η3)/bracketrightig\n. (28)\nIn the range of the competition ratio rcorresponding to weak nnn antiferromagnetism 0 < r <1\n2, the ground state\nof the model is ferromagnetic (since we have assumed J1>0), while for strong nnn antiferromagnetism1\n2< r <∞,\nthe ground state is the so-called (2 ,2) antiphase state, consisting of two spins up followed by two spins d own. The\ntwo different regimes shall be investigated separately since they re quire different order parameters.8\nA. Weak nnn antiferromagnetism (competition ratio 0< r <1\n2)\nIn the range of the competition ratio rcorresponding to the ferromagnetic ground state (0 < r <1\n2), owing to the\ndifferent gyromagnetic factors on odd ( gA) and even ( gB) lattice sites, it is necessary to consider the magnetizations\novertwosublattices, like in Eq. (8), as the order parameter. From the mast er equation, Eq. (28), one is thus led to\nconsider a system of two coupled equations of motion, which can be w ritten just like Eq. (9), with the elements of the\n2×2 non-symmetric matrix now being\na11= 1−γ2(1−γ1η1)+1\n2γ1γ2(η1+η3) =a22\na12=−gA\ngBγ1(1−γ2η2)a21=−gB\ngAγ1(1−γ2η2) (29)\nand the temperature dependent coefficient\nN(T) =N\n2µ2\n0H0\nkT/bracketleftig\n1−γ1η1−γ2η2+1\n2γ2γ1(η1+η3)/bracketrightig\n. (30)\nAfter diagonalization, the eigenvalues now turn out to be\nλ1= 1−γ1(1−γ2η2)−γ2(1−γ1η1)+1\n2γ1γ2(η1+η3)\nλ2= 1+γ1(1−γ2η2)−γ2(1−γ1η1)+1\n2γ1γ2(η1+η3), (31)\nindependent of the gyromagnetic factors gAandgB. The relationships between the normal modes Mn(t) and the\nsublatticemagnetizations Mn(t) (n= 1,2)arethesameasin Eqs.(13), (14). Alsotheexpressionsforthe f-coefficients\nare the same, i.e.f1=gB\n2(gB+gA),f2=gB\n2(gB−gA). As before, the general solution for the normal modes takes\nthe form in Eq. (15), where the relaxation times are τn=1\nαλn, with the eigenvalues now given by Eq. (31). Finally,\nin the case of weak nnn antiferromagnetic coupling, the complex sus ceptibility of the ANNNI chain turns out to be\nχ(ω,T) =Nµ2\n0\nkT/bracketleftig\n1−γ1η1−γ2η2+1\n2γ1γ2(η1+η3)/bracketrightig\n×/bracketleftigg/parenleftiggB+gA\n2/parenrightig21\nλ11\n1−iωτ1+/parenleftiggB−gA\n2/parenrightig21\nλ21\n1−iωτ2/bracketrightigg\n. (32)\nIn the limiting case r= 0, the well-known result for the nn Ising chain5,6is correctly recovered. In the case\n0< r <1\n2, we show in Fig. 3 that the approximate static susceptibility, calculat ed from Eq. (32) for zero frequency,\nturns out to be in good agreement with the exact transfer matrix r esult,21,22Eq. (A14), only at high temperatures\n(kT>∼J1). Incontrast, anunphysical(negative)staticsusceptibility isob tainedatlowtemperatures, asaconsequence\nof the negative values assumed by the eigenvalue λ1forkT<∼J1.\nThe low-temperature failure of Eq. (32) can be attributed to the d ecoupling (27) of three-spin correlation functions,\nwhich was made in order to close the set of master equations, Eq. (2 6): in fact, decoupling approximations have the\ndrawback to be uncontrollable, but in principle they are expected to be more accurate the higher the temperature.\nMoreover, at low temperatures one can guess another source of error to lie in the assumption that, for sufficiently long\ntimes, the spin-spin correlation functions take their staticequilibrium values: /an}bracketle{tσjσj+n;t/an}bracketri}ht →ηnfort→ ∞. In fact,\nfor competition ratio in the range 0 < r <1, the 1D ANNNI model with Glauber dynamics is known to be lacking\nin ergodicity at T= 0: the ground state can notbe reached by single spin-flip Glauber dynamics, after a sudden\ncooling of the system down to T= 0 starting from high temperature. The difference between the st atic (r= 1/2)18\nand the dynamic ( r= 1)25ground state phase boundary of the 1D ANNNI model was pointed o ut by Redner and\nKrapivsky25, who showed that for 0 < r <1/2 the ferromagnetic ground state can not be reached because of the\nrepulsion between domain walls which forces them to be at least two lat tice constants apart, while for 1 /2< r <1\nthe (2,2) antiphase ground state can not be reached owing to the persist ence of isolated domains of length ≥3.25In\ncontrast, both for r= 0 (1D nn Ising model)5,26andr >1 (1D ANNNI model with strong nnn AF coupling)25the\nground state can asymptotically ( t→ ∞) be reached at T= 0.\nThe low temperature failure of our approximate theory in the case 0 < r <1\n2prevented us from calculating the\ntemperature dependence of the amplitude of the complex suscept ibility. However, it is worth observing that, since for\nT→0 the zero-field static susceptibility diverges,21a resonant behavior might be expected for low frequency provided\nthat the system admits also a diverging relaxation time for low temper ature.9\nFIG. 3: (color online) Temperature dependence of the static susceptibility χ(ω= 0,T) of an ANNNI chain with J1= 1,\nJ2=−0.35 andgA=gB= 2, corresponding to a value r= 0.35 of the competition ratio (weak nnn antiferromagnetism). The\nthick line is the exact transfer matrix result, while open ci rcles denote the approximate calculation, Eq. (32). The tem perature\ndependence of the eigenvalues λ1andλ2(31) is also shown by the dashed lines. The approximations ma de to close the set of\nmaster equations (26) are found to fail for low temperatures .\nB. Strong nnn antiferromagnetism (competition ratio1\n2< r <∞)\nIn the range of the competition ratio rcorresponding to the (2 ,2)-antiphase state (1\n2< r <∞), it is necessary to\nconsider the magnetizations over foursublattices27\nM1(t) =gAµ0N\n4−1/summationdisplay\nj=0/an}bracketle{tσ1+4j;t/an}bracketri}ht, M2(t) =gBµ0N\n4−1/summationdisplay\nj=0/an}bracketle{tσ2+4j;t/an}bracketri}ht\nM3(t) =gAµ0N\n4−1/summationdisplay\nj=0/an}bracketle{tσ3+4j;t/an}bracketri}ht, M4(t) =gBµ0N\n4−1/summationdisplay\nj=0/an}bracketle{tσ4+4j;t/an}bracketri}ht (33)\nas the order parameter. One is thus led to consider a system of fou r coupled equations of motion, which can be\nwritten in matrix form as\n\n∂\n∂(αt)M1(t)\n∂\n∂(αt)M2(t)\n∂\n∂(αt)M3(t)\n∂\n∂(αt)M4(t)\n+\nA B C B\nD A D C\nC B A B\nD C D A\n\nM1(t)\nM2(t)\nM3(t)\nM4(t)\n=N(T)\ng2\nA\ng2\nB\ng2\nA\ng2\nB\ne−iωt\nwhere\nA=a11=a22=a33=a44= 1+1\n2γ2γ1(η1+η3)\nB=a12=a14=a32=a34=−1\n2gA\ngBγ1(1−γ2η2)\nC=a13=a31=a24=a42=−γ2(1−γ1η1)\nD=a21=a23=a41=a43=−1\n2gB\ngAγ1(1−γ2η2)\nand\nN(T) =N\n4µ2\n0H0\nkT/bracketleftig\n1−γ1η1−γ2η2+1\n2γ2γ1(η1+η3)/bracketrightig\n(34)10\nFIG. 4: (color online) Temperature dependence of the static susceptibility χ(ω= 0,T) of an ANNNI chain with J1= 1,\ngA=gB= 2, for two different values of the nnn exchange constant: (a) J2=−0.75 and (b) J2=−1.25, corresponding to\ncompetition ratio1\n2< r <1 andr >1 respectively (strong nnn antiferromagnetism). The thick line is the exact transfer\nmatrix result, while open circles denote the approximate ca lculation, Eq. (41). The temperature dependence of the eige nvalues\nλ1=λ2,λ3andλ4(36) is also shown by the dashed lines.\nDiagonalizing the matrix of coefficients, the time dependence of the e igenmodes is found to be ( n= 1,2,3,4)\nMn(t) =Mn(t0)e−t−t0\nτi+N(T)fn/integraldisplayt\nt0dt′et′−t\nτie−iωt′(35)\nwhereτn=1\nαλnare the relaxation times and f1=f2= 0,f3=gB\n2(gB−gA),f4=gB\n2(gB+gA). The eigenvalues\n(λn) of the 4 ×4 nonsymmetric matrix of the coefficients turn out to be independen t ofgAandgB\nλ1=λ2= 1+γ2(1−γ1η1)+1\n2γ2γ1(η1+η3)11\nλ3= 1+γ1(1−γ2η2)−γ2(1−γ1η1)+1\n2γ2γ1(η1+η3)\nλ4= 1−γ1(1−γ2η2)−γ2(1−γ1η1)+1\n2γ2γ1(η1+η3) (36)\nFor non vanishing magnetic field, the time dependence of the eigenmo des is\nMn(t) =N(T)fn\nλn1\n1−iωτne−iωt(n= 1,2,3,4) (37)\nThe relationships between the eigenmodes Mn(t) and the sublattice magnetizations Mn(t) (n= 1,2,3,4) are\nM1(t) =1\n2/bracketleftig\nM4(t)−M2(t)/bracketrightig\nM2(t) =1\n2/bracketleftig\nM3(t)−M1(t)/bracketrightig\nM3(t) =1\n4/bracketleftig/parenleftig\nM4(t)+M2(t)/parenrightig\n−gB\ngA/parenleftig\nM3(t)+M1(t)/parenrightig/bracketrightig\nM4(t) =1\n4/bracketleftig/parenleftig\nM4(t)+M2(t)/parenrightig\n+gB\ngA/parenleftig\nM3(t)+M1(t)/parenrightig/bracketrightig\n(38)\nand conversely\nM1(t) =gA\ngB/bracketleftig\nM4(t)−M3(t)/bracketrightig\n−M2(t)\nM2(t) =M4(t)+M3(t)−M1(t)\nM3(t) =gA\ngB/bracketleftig\nM4(t)−M3(t)/bracketrightig\n+M2(t)\nM4(t) =M4(t)+M3(t)+M1(t) (39)\nThe total magnetization is\nMtot(t) =4/summationdisplay\ni=1Mi(t) = 2gB+gA\ngBM4(t)+2gB−gA\ngBM3(t) =χ(ω)H0e−iωt, (40)\nwhere the complex susceptibility χ(ω,T) is given by\nχ(ω,T) =Nµ2\n0\nkT/bracketleftig\n1−γ1η1−γ2η2+1\n2γ2γ1(η1+η3)/bracketrightig\n×/bracketleftigg/parenleftiggB+gA\n2/parenrightig21\nλ41\n1−iωτ4+/parenleftiggB−gA\n2/parenrightig21\nλ31\n1−iωτ3/bracketrightigg\n. (41)\nThe approximate static susceptibility, calculated from Eq. (41) for zero frequency, is shown in Fig. 4a for1\n2< r <1.\nOne immediately notices that, in striking contrast with the case 0 < r <1\n2displayed in Fig. 3, the low temperature\nbehavior of the static susceptibility is correctly reproduced.\nThe latter feature appears at odds with the expectation that a de coupling approximation should work better the\nhigher the temperature. However it is worth noticing that, for the 1D ANNNI model, the T→0 asymptotic behavior\nof thestatictwo-spin correlation functions is very different depending on the va lue ofr. For 0< r <1\n2both the\ninter- and the intra-sublattice spin-spin correlations are strong ( η1≈η2≈η3≈1, see Note 32 later). In contrast, for\nr >1\n2the intersublattice correlations are strong ( η2≈ −1, see Eq. (45) later), whereas the intrasublattice correlations\nare exponentially vanishing ( η1≈η3≈0, see Eq. (45)). At intermediate temperatures intrasublattice c orrelations\nbecome significant, too, and the decoupling approximation becomes less satisfactory; at high temperatures, it works\nwell again, since all correlations (both intra- and inter-sublattice) decrease.\nIt should be remarked that the above considerations about the be havior of staticcorrelation functions can not, on\ntheir own, account for the good agreement found, at low T, in the case1\n2< r <1. In fact, the use of equilibrium\nvalues for the spin correlations might be questionable, since the T= 0 Glauber dynamics does not lead to the ground\nstate of the 1D ANNNI model in the entire region 0 < r <1.25To this regard, first we observe that the physical\nmechanism which at T= 0 prevents the system from reaching the ground state is differen t, for1\n2< r <1, with\nrespect to the case 0 < r <1\n2.25,27,28Next, considering that at T= 0 a 1D model is simultaneously in the ordered\nphaseandat its critical point, while our theory applies at T >0, we believe that some insight into the problem might12\nFIG. 5: (color online) Temperature dependence of the amplit ude of the complex susceptibility |χ(ω,T)|for an ANNNI chain\nwithJ1= 1,J2=−1.25 (r= 1.25), subject to a weak external magnetic field oscillating at frequency ω. Figures (a), (b) refer\nto the compensated case ( gA=gB= 2), while Figures (c) and (d) to the uncompensated case ( gA= 2, gB= 3), for selected\nvalues of the frequency ( ω/α= 0.001 and 10). In Figs. (c,d) the thin (color) lines represent t he contributions to the amplitude\nof the two terms on the r.h.s. of Eq. (41), while the thick (bla ck) line is their sum. No resonant behavior is observed.\nbe provided by a careful study of the role of a small but non-zero t emperature on the coarsening of the 1D ANNNI\nmodel.26,29\nIn Fig. 4b the approximate static susceptibility, calculated from Eq. (41) for zero frequency in the case r >1, is\nreported. A nice overall agreement with the exact transfer matr ix result21,22is obtained. In this case our approximate\nresults are expected to be quite reliable since the long-time approxim ation is well founded (for r >1, the static\nequilibrium state can asymptotically be reached even at T= 0,25and thus the use of staticspin-spin correlation\nfunctions is justified); moreover, the decoupling approximation is e xpected to be satisfactory both at high and low\ntemperatures. Finally it is worth mentioning that, in the limiting case 1 /r= 0 (i.e.,J1= 0), the transfer matrix\nresult for the static susceptibility is exactly reproduced by Eq. (41 ) forω= 0 (not shown).\nIn Fig. 5 the temperature dependence of the amplitude of the comp lex susceptibility |χ(ω,T)|, obtained from\nEq. (41), of an ANNNI chain with nnn antiferromagnetic coupling dom inating over the nn ferromagnetic one (com-\npetition ratio r= 1.25) is reported - for selected values of the oscillation frequency ωof the external magnetic field\n- both in the compensated ( gA=gB= 2) and uncompensated ( gA/ne}ationslash=gB) case. No resonant behavior was observed\neven in the uncompensated case since, in the T→0 limit, both the zero-field static susceptibility and the relaxation\ntimes (τ3andτ4in Eq. (41)) fail to diverge. Thus, for low frequency, a resonance condition - similar to the one in\nEq. (20) - can not be fulfilled. In the case1\n2< r <1 a qualitatively similar behavior for |χ(ω,T)|was found (not\nshown).\nC. Critical dynamics of the 1D ANNNI model for r >1\nThe identification of r= 0,r= 1 and 1 /r= 0 as dynamic critical transition points for the 1D ANNNI model\nwith single spin-flip Glauber dynamics was recently proposed in theore tical studies of T= 0 coarsening25(i.e., the\nrelaxation of the system into the ground state after a quench fro m high temperature) and T= 0 persistence27(i.e.,\nthe probability for a spin to remain in its original state after a quench from high temperature). In such T= 0 studies,13\nthe dynamic critical exponent zis customarily defined as the inverse of the growth exponent nof the domain size\nL(t)≃tn≃t1/z′. (42)\nFor the 1D nn Ising model, analytical calculations26provided z′= 2. For the 1D ANNNI model with r >1 numerical\ncalculations27,30predicted a somewhat higher dynamic exponent, z′≃2.3. Finally, it is worth noting that Sen and\nDasgupta,27in their study of t= 0 persistence in the ANNNI chain, found that the dynamic critical e xponent z′\nundergoes abrupt changes for r= 0 (when a slight amount of nnn interaction is added to the nn one), f or 1/r= 0\n(when a slight amount of nn interaction is added to the nnn one), as w ell as for r= 1.27\nThe fair accuracy of our approximate theoretical approach in des cribing the low temperature static susceptibility\nof the ANNNI chain with r >1, see Fig. 4b, encouraged us to tentatively estimate the dynamic c ritical exponent.\nHowever, since we work at finite temperature, rather than at T= 0, we use a different definition, namely11,31\n1\nατ1=λ1≈/parenleftig1\nξ/parenrightigz\n(43)\nwhereλ1is the smallest eigenvalue of the dynamical matrix, see Eq. (36), and ξis the static correlation length of the\ninfinite system (the lattice constant calong the chain was set to 1). For the compensated case gA=gB, the latter\nquantity can be analytically calculated using the transfer matrix met hod,21see Eq. (A18), and for r >1 its expansion\nin theT→0 limit turns out to be\n/parenleftig1\nξ/parenrightig2\n≈1\n4e2(J1−2|J2|)\nkT, (44)\nwhere we have explicitly taken into account that J1>0 andJ2<0.\nTaking into account the T→0 asymptotic behavior, for r >1/2,32of theγiandηicoefficients\nγ1≈1−2e−4J1\nkTγ2≈ −1+2e−4|J2|\nkT\nη1≈1\n2eJ1−2|J2|\nkTη2≈ −1+eJ1−2|J2|\nkTη3≈ −3\n2eJ1−2|J2|\nkT, (45)\nfor the inverse of the longest relaxation time we obtain, provided th atJ1/ne}ationslash= 0\n1\nατ1=λ1≈eJ1−2|J2|\nkT. (46)\nIn the special case J1= 0 (i.e., 1/r= 0), letting γ1= 0 in Eq. (36) and using the T→0 expansion for γ2in Eq. (45),\nwe obtain\n1\nατ1=λ1≈2e−4|J2|\nkT. (47)\nIn conclusion, within our approximate theoretical scheme, the dyn amic critical exponent of the 1D ANNNI chain\nwith competing nn and nnn exchange interactions was found to be z= 1 for any finite r >1, while in the absence\nof competing interactions ( i.e., forr= 0 and 1 /r= 0) we found z= 2. Notice that, for the 1D Ising model with\nexchange limited to the nn ( r= 0), the value z= 2, obtained using the definition in Eq. (43),31coincides with the\nvaluez′= 2, obtained using the definition in Eq. (42).26This appears notto be the case for the 1D ANNNI model\nwith 1< r <+∞, where the values z= 1 (present work) and z′≃2.3 (References 27,30) were found. In order to\nascertain the origin of this discrepancy, we believe that it would be us eful to study the role of a small but non-zero\ntemperature ( T >0) on the coarsening dynamics of the 1D ANNNI model.29\nIV. CONCLUSIONS\nInconclusion,inthispaperwehavestudiedtheeffectofantiferrom agneticinteractionsonthesinglespin-flipGlauber\ndynamics of two different one-dimensional (1D) Ising models with spin ±1. For the first model, an Ising chain with\nantiferromagnetic exchange interaction limited to nearest neighbo rs and subject to an oscillating magnetic field, the\nsystem of master equations describing the time evolution of sublatt ice magnetizations can easily be solved within a\nlinear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature\n(stochastic resonance) is found, at low frequency, only when spin s on opposite sublattices are uncompensated owing\nto different gyromagnetic factors ( i.e., in the presence of a ferrimagnetic short range order). For the s econd model,14\nthe axial next-nearest neighbor Ising (ANNNI) chain, where the n nn antiferromagnetic exchange coupling is assumed\nto compete with the nn ferromagnetic one, the long time response o f the model to a weak, oscillating magnetic field\nis investigated in the framework of a decoupling approximation for th ree-spin correlation functions, which is required\nto close the system of master equations. Within such approximate t heoretical scheme, the T= 0 dynamics of the\nIsing-Glauber chain with competing interactions is found to be in a diffe rent universality class than that of the Ising\nchain with antiferromagnetic exchange limited to nearest neighbors (r= 0) or limited to next-nearest neighbors\n(1/r= 0). In particular, we find an abrupt change in the T= 0 dynamic behavior of the model in the neighborhood\nof the dynamic critical point 1 /r= 0 since, when a slight amount of ferromagnetic nn exchange is adde d to the\nantiferromagnetic nnn exchange, we find that the critical expone ntz, defined by Eq. (43), changes abruptly from\nz= 2 toz= 1. Considering that z= 2 is also the value of the dynamic critical exponent for the unfrust rated nn Ising\nchain, one might expect similar abrupt changes in zto occur also in the neighborhood of the dynamic critical points\nr= 0 (i.e.when a slight amount of AF nnn exchange is added to the nn F exchang e) andr= 1, as suggested by\nstudies of T= 0 coarsening dynamics25andT= 0 persistence27in the ANNNI chain. Unfortunately, the inaccuracy\nof our approximate theoretical scheme in reproducing the static s usceptibility of the 1D ANNNI model with 0 < r≤1\nfor low temperature prevented us from calculating the dynamic crit ical exponent in this range of the competition\nratio.\nAPPENDIX A: ANALYTIC TRANSFER MATRIX RESULTS FOR THE STATIC PROPERTIES OF 1D\nISING MODELS\n1. The 1D nearest neighbor Ising model with alternating spin s in a static field\nIn this subsection we calculate, within the transfer matrix formalism ,33the static properties of the 1D Ising model,\nEq. (1), with nearest neighbor coupling Jof either sign, subject to a static magnetic field H(i.e.,ω= 0). Two types\nof spins with different gyromagnetic factors ( gA/ne}ationslash=gB) are assumed to alternate along the chain. Taking periodic\nboundary conditions, the partition function of the chain of length N(withNeven without loss of generality) can be\nexpressed as\nZN= Tr/parenleftbig\ne−H\nkT/parenrightbig\n=/summationdisplay\nσ1=±1/summationdisplay\nσ2=±1···/summationdisplay\nσN=±1K(σ1,σ2)L(σ2,σ3)···K(σN−1,σN)L(σN,σ1) (A1)\nwhere, letting J=J/(kT),hA= (gAµ0H0)/(kT),hB= (gBµ0H0)/(kT), the two different kernels KandLare\ndefined as\nK(σ2i−1,σ2i) =eJσ2i−1σ2ie1\n2(hAσ2i−1+hBσ2i)L(σ2i,σ2i+1) =eJσ2iσ2i+1e1\n2(hBσ2i+hAσ2i+1). (A2)\nSumming over the even sites, ZNcan be expressed as\nZN=/parenleftbig\nΛ+/parenrightbigN\n2+/parenleftbig\nΛ−/parenrightbigN\n2(A3)\nin terms of the eigenvalues\nΛ±=e2Jcosh(hA+hB)+e−2Jcosh(hA−hB)±\n±/radicalig\ne4Jcosh2(hA+hB)+e−4Jcosh2(hA−hB)+2cosh( hA+hB)cosh(hA−hB)+2−e4J−e−4J(A4)\nof the real symmetric 2 ×2 matrix\nS=/parenleftbigg\ne2J+hA+hB+e−2J+hA−hB ehB+e−hB\nehB+e−hB e2J−hA−hB+e−2J−hA+hB/parenrightbigg\n. (A5)\nIt is immediate to verify that, in the limit gA=gB, the well-known result for the 1D nn Ising chain in a static external\nfield is recovered.33In the thermodynamic limit N→ ∞, only the larger eigenvalue Λ +matters, ZN→(Λ+)N\n2, and\nthe static susceptibility in zero field can be expressed in terms of its s econd derivative with respect to the field H\nχ(ω= 0,T) =N\n2kT/bracketleftig1\nΛ+∂2Λ+\n∂H2/bracketrightig\nH=0=Nµ2\n0\nkT/bracketleftig/parenleftiggA+gB\n2/parenrightig2\ne2J\nkT+/parenleftiggA−gB\n2/parenrightig2\ne−2J\nkT/bracketrightig\n. (A6)15\n2. The 1D ANNNI model in zero field\nIn this subsection we collect, for the reader’s convenience, some e xact results for the static properties of the 1D\nANNNI model in zero field, Eq. (22), which were obtained by Stephen son21and Harada22in the case of a linear chain\nwithNidentical spins ( gA=gB=gandσ=±1). Using the transfer matrix method, the partition function can b e\nexactly expressed as\nZN= (λ+)N+(λ−)N(A7)\nin terms of the eigenvalues of the symmetric 2 ×2 matrix\nS=/parenleftbigg\na c\nc b/parenrightbigg\n=/parenleftigg\neJ2+J1\nkTeJ2−J1\nkT\neJ2−J1\nkTe−J2\nkT/parenrightigg\n(A8)\nThe eigenvalues take the form21\nλ±=1\n2[a+b±∆] =eJ2\nkT/bracketleftig\ncosh/parenleftigJ1\nkT/parenrightig\n±/radicalbigg\nsinh2/parenleftigJ1\nkT/parenrightig\n+e−4J2\nkT/bracketrightig\n(A9)\nwhere\n∆ =/radicalbig\n(a−b)2+4c2= 2eJ2\nkT/radicalbigg\nsinh2/parenleftigJ1\nkT/parenrightig\n+e−4J2\nkT. (A10)\nBothλ+and ∆ are always real positive quantities.\nIn the thermodynamic limit N→ ∞, the static two spin correlation function ηntake the form21\nηn=/an}bracketle{tσjσj+n/an}bracketri}ht=1\n2/parenleftbig\nλ+/parenrightbign/bracketleftig/parenleftbig\nµ+/parenrightbign/parenleftig\n1+a2−b2\n∆∆′/parenrightig\n+/parenleftbig\nµ−/parenrightbign/parenleftig\n1−a2−b2\n∆∆′/parenrightig/bracketrightig\n(A11)\nwhere the quantities ∆′, defined as\n∆′=/radicalbig\n(a+b)2−4c2= 2eJ2\nkT/radicalbigg\ncosh2/parenleftigJ1\nkT/parenrightig\n−e−4J2\nkT, (A12)\nand\nµ±=eJ2\nkT/bracketleftig\nsinh/parenleftigJ1\nkT/parenrightig\n±/radicalbigg\ncosh2/parenleftigJ1\nkT/parenrightig\n−e−4J2\nkT/bracketrightig\n(A13)\nmay be complex. More precisely, the quantities µ±are real for T < T Dand complex conjugates for T > T D.TDis\nthe so-called disorder point, defined by the equation ∆′(TD) = 0, which has solutions for 0 < r <1/2 at some finite\ntemperature TD. ForT < T Dthe static equilibrium two-spin correlation functions ηn=/an}bracketle{tσjσj+n/an}bracketri}htpresent a monotonic\nexponential decay, while for T > T Dthey have an oscillating exponential decay.21\nSumming over all pair correlations, the exact zero field static susce ptibility can be expressed as21\nχ(ω= 0,T) =/parenleftigg2µ02\nkT/parenrightig/parenleftiga+b\n∆/parenrightig/bracketleftiga(a−b+∆)+2 c2\nb(b−a+∆)+2 c2/bracketrightig\n. (A14)\nThe wave-vector dependent susceptibility, defined as\nχ(q) =Ng2µ2\n0\nkT/summationdisplay\nn/an}bracketle{tσjσj+n/an}bracketri}hteiqn(A15)\npresents a maximum at a wave-vector qm, which is given by22\ncosqm=(µ++µ−)(λ+−λ−)\n4µ+µ−(A16)\nFor 0< r <1/4 one has qm= 0 at all temperatures , while for 1 /4< r <1/2 there is a definite temperature TL\n(/ne}ationslash=TD) above which qm/ne}ationslash= 0, whereas for T < T Lqm= 0. When 1 /2< r, one has qm(T= 0) =π/2. In the limit of16\nT→ ∞,qmtends to the mean field value cos qm= 1/(4r). Expanding χ(q) in the neighborhood of qmup to second\norder in ∆ q=qm−q, one obtains a Lorentzian form, and the correlation length ξcan be defined in terms of its full\nwidth at half maximum as\nχ(q) =χ(qm)\n1+ξ2(∆q)2(A17)\nand turns out to be\n/parenleftig1\nξ/parenrightig2\n=(λ+−λ−−µ+−µ−)2\n(µ++µ−)(λ+−λ−)−4µ+µ−forqm= 0,\n/parenleftig1\nξ/parenrightig2\n=(µ+−µ−)2(λ++λ−)2\n(µ++µ−)2(λ+−λ−)2−16µ2\n+µ2\n−forqm/ne}ationslash= 0. (A18)\nFor 1/4< r <1/2, it turns out that at TLthe correlation length becomes zero, which is a characteristic of th e Lifshitz\npoint.34\n∗Electronic address: mariagloria.pini@isc.cnr.it\n†The authors acknowledge financial support from the Italian M inistry for University and Research.\n1R. Benzi, G. Parisi, A. Sutera and A. Vulpiani, Tellus 34, 10 (1982); SIAM J. Appl. Math. 43, 565 (1983).\n2L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta and S. S antucci, Phys. Rev. Lett. 62, 349 (1989).\n3L. Gammaitoni, P. H¨ anggi, P. Jung and F. Marchesoni, Rev. Mo d. Phys. 70, 223 (1998).\n4U. Siewert and L. Schimansky-Geier, Phys. Rev. E 58, 2843 (1998).\n5R. J. Glauber, J. Math. Phys. 4, 294 (1963).\n6J. J. Brey and A. Prados, Phys. Lett. A 216, 240 (1996).\n7Z. N´ eda, Phys. Rev. E 51, 5315 (1995).\n8Z. N´ eda, Phys. Lett. A 210, 125 (1996).\n9Kwan-tai Leung and Z. N´ eda, Phys. Lett. A 246, 505 (1998).\n10Z. R. Yang, Phys. Rev. B 46, 11578 (1992).\n11J. Kamphorst Leal da Silva, Adriana G. Moreira, M. Silv´ erio Soares, and F. C. S´ a Barreto, Phys. Rev. E 52, 4527 (1995).\n12A. Caneschi, D. Gatteschi, N. Lalioti, C. Sangregorio, R. Se ssoli, G. Venturi, A. Vindigni, A. Rettori, M. G. Pini and M. A .\nNovak, Angew. Chem., Int. Ed. Engl. 40, 1760 (2001).\n13A. Caneschi, D. Gatteschi, N. Lalioti, C. Sangregorio, R. Se ssoli, G. Venturi, A. Vindigni, A. Rettori, M. G. Pini and M. A .\nNovak, Europhys. Lett. 58, 771 (2002).\n14L. Bogani, C. Sangregorio, R. Sessoli and D. Gatteschi, Ange w. Chem. Int. Ed. 44, 5817 (2005).\n15K. Bernot, L. Bogani, A. Caneschi, D. Gatteschi and R. Sessol i, J. Am. Chem. Soc. 128, 7947 (2006).\n16A. Vindigni, N. Regnault and Th. Jolicoeur, Phys. Rev. B 70, 134423 (2004).\n17W. Selke, Physics Reports 170, 213264 (1988).\n18T. Morita and T. Horiguchi, Phys. Lett. A 38, 223 (1972).\n19In contrast with the equilibrium case, the dynamics of the 1D ANNNI model in zero field is notequivalent to that of a 1D\nnearest neighbor Ising model in an effective external field: s ee Ref. 10.\n20J.-J. Kim, S. Mori and I. Harada, Phys. Lett. A 202, 68 (1995).\n21J. Stephenson, Can. J. Phys. 48, 1724 (1970).\n22I. Harada, J. Phys. Soc. Jpn\n23M. G. Pini and A. Rettori, Phys. Rev. B 48, 3240 (1993).\n24J.-J. Kim, S. Mori and I. Harada J. Phys. Soc. Japan 65, 2624 (1996).\n25S. Redner and P. L. Krapivsky, J. Phys. A 31, 9929 (1998).\n26A. J. Bray, J. Phys. A: Math. Gen. 22, L67 (1989).\n27P. Sen and S. Dasgupta, J. Phys. A 37, 11949 (2004).\n28P. Sen and P. K. Das, in Quantum Annealing and Related Optimization Methods , edited by A. Das and B. K. Chakrabarti,\nLecture Notes in Physics 679 (Springer, New York, 2005), (ar Xiv:cond-mat/0505027).\n29While for the 1D nn Ising model such a study was performed by Br ay,26for the 1D ANNNI model it has not yet been\nperformed, to our knowledge, and is deferred to future work.\n30It is worth noting that such a value, n=1\n2.3, for the growth exponent turns out to be in close agreement wi th previous\nanalytical predictions, obtained by Redner and Krapivsky25in the same regime of strong nnn antiferromagnetic exchange .\nIn fact for r >1 the latter authors estimated the density of domains of leng th 3 and 1 to scale as A(t)≃/radicalbiglnt\ntand\nB(t)≃1√\ntlnt, respectively. For t≫1, it turns out that the average of A(t) andB(t) (which is inversely proportional to the\ndomain size L(t)) scales just as1\nt1/2.3.\n31James H. Luscombe, Marshall Luban and Joseph P. Reynolds, Ph ys. Rev. E 53, 5852 (1996).17\n32In the case of ferromagnetic ground state, 0 < r <1/2, theT→0 asymptotic behavior of the correlation functions for the\n1D ANNNI model in zero field is instead: η1= 1−2e−2J1+4|J2|\nkT,η2= 1−4e−2J1+4|J2|\nkT,η3= 1−6e−2J1+4|J2|\nkT.\n33H. E. Stanley, in Introduction to Phase Transitions and Critical Phenomena , (Clarendon Press, Oxford, 1971), p. 115 and\np. 131.\n34S. Redner and H. E. Stanley, Phys. Rev. B 16, 4901 (1977)." }, { "title": "1707.04854v1.Competing_magnetic_and_spin_gap_less_semiconducting_behaviour_in_fully_compensated_ferrimagnet_CrVTiAl__Theory_and_Experiment.pdf", "content": "Competing magnetic and spin gap-less semiconducting behaviour in fully\ncompensated ferrimagnet CrVTiAl: Theory and Experiment\nY. Venkateswara,1,\u0003Sachin Gupta,1, 2,\u0003S. Shanmukharao Samatham,1\nManoj Raama Varma,3Enamullah,4K. G. Suresh,1,yand Aftab Alam4,z\n1Magnetic Materials Laboratory, Department of Physics,\nIndian Institute of Technology Bombay, Mumbai 400076, India\n2WPI-Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan\n3National Institute for Interdisciplinary Sciences and Technology (CSIR), Thiruvananthapuram, India;\n4Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India\n(Dated: July 18, 2017)\nWe report the structural, magnetic and transport properties of polycrystalline CrVTiAl alloy\nalong with \frst principles calculations. It crystallizes in the LiMgPdSn type structure with lattice\nparameter 6.14 \u0017A at room temperature. Absence of (111) peak along with the presence of a weak\n(200) peak indicates the antisite disorder of Al with Cr and V atoms. The magnetization measure-\nments reveal a ferrimagnetic transition near 710 K and a coercive \feld of 100 Oe at 3 K. Very low\nmoment and coercive \feld indicate fully compensated ferrimagnetism in the alloy. Temperature co-\ne\u000ecient of resistivity is found to be negative, indicating a characteristic of semiconducting nature.\nAbsence of exponential dependence of resistivity on temperature indicates a gapless/spin-gapless\nsemiconducting behaviour. Electronic and magnetic properties of CrVTiAl for three possible crys-\ntallograpic con\fgurations are studied theoretically. All the three con\fgurations are found to be\ndi\u000berent forms of semiconductors. Ground state con\fguration is a fully compensated ferrimagnet\nwith band gaps 0.58 eV and 0.30 eV for up and down spin bands respectively. The next higher\nenergy con\fguration is also ferrimagnetic, but has spin-gapless semiconducting nature. The highest\nenergy con\fguration corresponds to a non-magnetic gapless semiconductor. The energy di\u000berences\namong these con\fgurations are quite small ( <1 mRy=atom) which hints that at \fnite temperatures,\nthe alloy exists in a disordered phase, which is a mixture of the three con\fgurations. By taking\ninto account the theoretical and the experimental \fndings, we conclude that CrVTiAl is a fully\ncompensated ferrimagnet with predominantly spin-gapless semiconductor nature.\nPACS numbers: 75.50.Gg, 75.50.Pp, 75.50.Ee, 78.40.Fy, 61.43.-j, 85.75.-d, 31.15.A\nI. INTRODUCTION\nSpintronics is an emerging branch of electronics in\nwhich the spin degree of freedom is added to the charge\ndegree of electron to realize many advantages such as\nnon-volatility, high processing speed, low power con-\nsumption, high storage density etc. over the conventional\nelectronics.1{7The utilization of the spin degree of free-\ndom i.e., in spintronic devices, can be found in spin diodes\nused in magnetic hard disks, read heads, magnetoresis-\ntive random access memory (MRAM), spin transistors,\ntunnel diodes, vortex oscillators etc.8{12For realization\nof spintronic devices, special materials are required, for\nexample, their electrical conduction should be restricted\nto one type of spin carriers. Such a phenomenon is seen\nin half metallic ferromagnets (HMF), spin gap-less semi-\nconductors (SGS), semiconducting spin \flters etc.3,13{16\nAmong the discovered materials, fully compensated fer-\nrimagnetic (FCF) materials have gained a lot of inter-\nest recently.17{19Leuken and Groot showed theoretically\nthat this new class of materials can show 100 % spin po-\nlarization without having a net magnetic moment, and\nhence given the name half metallic antiferromagnetic ma-\nterials or fully compensated ferrimagnets.20However, for\nthe antiferromagnets, symmetry demands the same den-\nsity of states (DOS) for spin up and spin down bands.19,21Due to symmetric bands and DOS, both the spin chan-\nnels equally contribute to electrical conductivity, which\nresults in zero net spin-polarized current. Such a scenario\nis not always true for FCF materials, which usually con-\ntain three or more magnetic ions with moments aligned\nin such a way that the net magnetization is nearly zero.\nSome of the unique properties and advantages of FCF\nmaterials are (i) nearly zero magnetic moment which cre-\nates no external stray \felds, resulting in low energy losses\n(ii) spin sensitivity without stray magnetic \felds, which\nallows them not to disturb the spin character and make\nthem ideal for spin polarized scanning tunnelling micro-\nscope tips and improved density of circuit integration in\na chip22(iii) low shape anisotropy, which helps in appli-\ncations in spin injection etc. These properties make them\nsuperior compared to HMF materials and are very much\nin demand today.\nThough Heusler alloys are known for many decades,\nthey gained renewed interest because of the develop-\nments in the \feld of spintronics.4,23{27Ozdogan et al.\nstudied the electronic and magnetic properties of qua-\nternary Heusler alloys (QHAs) theoretically by using\nthe full-potential non-orthogonal local-orbital minimum-\nbasis band structure scheme (FPLO).28Among the stud-\nied alloys, CrVTiAl (CVTA) has attracted a lot of inter-\nest and the preliminary band structure studies indicated\nit to be an antiferromagnetic semiconductor. Later, itarXiv:1707.04854v1 [cond-mat.mtrl-sci] 16 Jul 20172\nFIG. 1. Schematic density of states (DOS) of various types of\nsemiconductors. a) conventional semiconductor (CS) in which\nboth up ( \") and down ( #) spin bands have \fnite and equal\nband gaps. b) a gapless semiconductor (GS) where both the\nspin bands have vanishing band gap. c) a magnetic semicon-\nductor (MS) in which the band gaps of up and down spin\nbands are \fnite but unequal. d) a spin-gapless semiconduc-\ntor in which any one band (up or down) is gapless while the\nother is with \fnite gap.\nwas found that it is a fully compensated ferrimagnet with\ndistinct magnetic moments at Cr, V and Ti ions.29,30\nSchematic density of states of di\u000berent classes of semi-\nconductors are displayed in Fig. 1. Figure 1 (a) is a\nconventional semiconductor (CS) in which both spin up\nand spin down bands have equal band gap (\u0001 Eg). In\nthermal equilibrium, the intrinsic charge carrier concen-\ntration is given by31\nni= 2\u0012kBT\n2\u0019~2\u00133=2\n(memh)3=4e\u0000(\u0001Eg=2kBT):(1)\nHereme(mh) is the e\u000bective mass of electron(hole).\nThe conductivity in CS is dominated by exponential\nterm. Figure 1 (b) is the special case in which gap closes\n(\u0001Eg\u00190) (for example HgTe) in which nivaries as\nT3=2.32Figure 1 (c) is a typical magnetic semiconduc-\ntor in which band gap for each spin band is \fnite but\nnot equal, resulting in spin polarized intrinsic carriers\nand hence used in spin \flters. Figure 1(d) is the special\nclass of magnetic semiconductors in which one of the spin\nbands encounters zero gap (\u0001 Eg\"\u00190), while the other\nhas a \fnite gap.14,33,34In this case, the concentration of\nintrinsic spin up carriers ( n\"), which varies as T3=2dom-\ninates in comparison to that of spin down carries ( n#),\nwhich varies in an exponential manner and as a result,\nthe temperature dependence of the total concentration of\nintrinsic carriers slightly deviates from pure T3=2.\nAn experimental investigation carried out by\nStephen et al., has shown CVTA to be a magnetic\nsemiconductor,35but they attributed the resistivity be-\nhaviour to a combination of metallic and semiconducting\ncontributions. According to them, the magnetization\ndepends linearly on the \feld, indicating the antifer-\nromagnetic behaviour. However, a close inspection of\ntheir XRD pattern reveals small peaks near (220), which\nis indicative of secondary phase(s). In addition, theirsample shows a (111) peak with considerable intensity,\nunlike ours. The nearly equal electronegativities of Al\nand Cr/V causes the antisite disorder between these\nsites resulting the absence of superlattice (111) peak\nin the XRD.36In view of these di\u000berences and with\nthe aim of shedding more light into the anomalous\nproperties exhibited by this alloy, we have carried out\na combined theoretical and experimental study, which\npredicts entirely di\u000berent set of properties than what is\nreported earlier.\nII. EXPERIMENTAL AND THEORETICAL\nDETAILS\nPolycrystalline CrVTiAl alloy was prepared by arc\nmelting the stoichiometric proportions of constituent el-\nements with purity at least 99.99 %. Room temperature\nX-ray di\u000braction (XRD) patterns were collected using\nX'Pert Pro di\u000bractometer using Cu K \u000bradiation. The\ncrystal structure was analyzed by Rietveld re\fnement us-\ning FullProf suite.37\nThe crystal structure of QHAs of type XX0YZ (where\nX, X0, Y are transition elements and Z is the main group\nelement), can be described by three distinct (symme-\ntry inequivalent) possible arrangements of atoms.4,36The\nstructure consists of 4 wycko\u000b sites 4a, 4b, 4c and 4d. By\n\fxing Z at 4a site, the distinct con\fgurations are\n(I) X at 4b, X0at 4c and Y at 4d sites ,\n(II) X at 4c , X0at 4b and Y at 4d sites,\n(III) X at 4d , X0at 4c and Y at 4b sites\nrespectively. The structure factor for the \frst con\fgura-\ntion is given by\nFhkl= 4(fz+fye\u0019i(h+k+l)+fxe\u0019i\n2(h+k+l)+fx0e\u0000\u0019i\n2(h+k+l))\n(2)\nwith unmixed (hkl) values. Here fz,fy,fxandfx0are the\natomic scattering factors for the atoms Z, Y, X and X0\nrespectively. Therefore, the magnitudes of\nF111= 4[(fz\u0000fy)\u0000i(fx\u0000fx0)] (3)\nF200= 4[(fz+fy)\u0000(fx+fx0)] (4)\nF220= 4[fz+fy+fx+fx0] (5)\nare used to classify the order of the crystal structure.\nMagnetization measurements (from 2-400 K) were per-\nformed implementing zero \feld cooled warming (ZFCW)\nand \feld cooled warming (FCW) protocols in 500 Oe\nusing vibrating sample magnetometer (VSM, Quantum\nDesign). High temperature magnetization measurement\nwas carried out using (MPMS) in \feld warming (FW)\nmode at 1 kOe. Resistivity ( \u001a) measurements were car-\nried out using Physical Property Measurement System\n(PPMS) by four probe method applying 5 mA current.3\nA. Theoretical Details\nSpin-resolved Density Functional Theory (SDFT) as\nimplemented in Quantum Espresso (QE) package38was\nused to calculate the band structure and magnetic prop-\nerties of CVTA. The exchange-correlation functional\nwas taken within the generalized gradient approxima-\ntion (GGA) in the parametrization of Perdew-Burke-\nErnzerhof (PBE).39The pseudo potentials with Pro-\njector Augmented-Wave method40were generated using\nPSlibrary and QE. Self consistent calculations were car-\nried out using 24\u000224\u000224 k-point mesh with Methfessel-\nPaxton smearing of width 0.005 Ry, resulting in 413 k-\npoints in the irreducible wedge of the Brillouin zone. The\nenergy convergence criterion was set to 10\u00009Ry. The ki-\nnetic energy of the plane wave expansion (energy cuto\u000b\nEcut) was restricted to 60 Ry and charge density expan-\nsion to 700 Ry. Non-self consistent \feld calculations were\ncarried out using 48 \u000248\u000248 k-point grid. Projected den-\nsity of states (DOS) were extracted with an energy width\nof 0.0025 Ry.\nThermal charge carrier concentration was calculated\nusing theoretical DOS, D(E). The electron density\nabove the Fermi energy ( EF) at \fnite temperature T is\nD(E)f(E). Hence the total number of thermally created\nelectrons is\nne(T) =Z1\nE=0D(E)f(E)dE: (6)\nHereEFis taken as the reference level, f(E) = 1=(1 +\nexp(\u0000E=k BT)) is the Fermi function. In a similar man-\nner, the total number of thermally created holes can be\nfound by the expression\nnh(T) =Z0\nE=\u00001D(E)[1\u0000f(E)]dE: (7)\nIn an intrinsic semiconductor, at thermal equilibrium,\nthe number of thermal electrons is equal to the number\nof created holes i.e., nei=nhi=pnenh. In addition to\nthe thermally created charge carriers, there exists \fnite\nnumber of charge carriers ne0even atT= 0. So the total\nnumber of intrinsic carriers at a given Tisn= 2pnenh+\nne0. To obtain spin resolved total carriers one has to\nreplaceD(E) with spin resolved density of states. The\nintrinsic spin polarization is obtained by the following\nexpression\nP(T) =n\"(T)\u0000n#(T)\nn\"(T) +n#(T)\u0002100: (8)\nIII. EXPERIMENTAL RESULTS\nA. Crystal Structure\nCVTA is found to crystallize in the LiMgPdSn (space\ngroup F \u001643m, # 216) prototype structure (or Y struc-\nture) with a lattice parameter of aexp= 6.14 \u0017A. Figure\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s50/s52 /s50/s53 /s50/s54 /s50/s55 /s50/s56 /s50/s57 /s51/s48 /s51/s49\n/s40 /s41\n/s32/s32\n/s32/s73\n/s111/s98/s115\n/s32/s73\n/s99/s97/s108\n/s32/s73\n/s111/s98/s115/s45/s73\n/s99/s97/s108\n/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110/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/s49/s49/s41\n/s40/s50/s48/s48/s41\n/s40/s50/s50/s48/s41\n/s40/s51/s49/s49/s41\n/s40/s50/s50/s50/s41\n/s40/s52/s48/s48/s41\n/s40/s51/s51/s49/s41\n/s40/s52/s50/s48/s41\n/s40/s52/s50/s50/s41\n/s40/s53/s49/s49/s41/s40/s51/s51/s51/s41\n/s40/s52/s52/s48/s41\n/s40/s53/s51/s49/s41\n/s40/s52/s52/s50/s41 /s40/s54/s48/s48/s41\n/s40/s54/s50/s48/s41\n/s40/s53/s51/s51/s41/s67/s114/s86/s84/s105/s65/s108\n/s40 /s41\n/s40/s50/s32/s48/s32/s48/s41\n/s32/s32FIG. 2. Rietveld re\fned room temperature XRD of CVTA.\nThe super-lattice (111) re\rection is absent whereas a weak\n(200) peak is present (see inset).\n2 shows the XRD pattern for CVTA. Inset shows the\nzoomed region near (200) peaks. The absence of (111)\nand the presence of (200) superlattice peaks usually in-\ndicate the existence of B2 type disorder. For this type of\nantisite disorder to occur, there should be simultaneous\ndisorder between two pairs of atoms occupying octahe-\ndral sites and tetrahedral sites i.e., disorder between one\npair of X and X0and another pair of Y and Z. Because\nof this, the resulting structure resembles the CsCl type\nstructure. Rietveld re\fnement with B2 disorder did not\n\ft well for (200) peak. Observed peak intensity was much\nless than the calculated value for this peak. Subsequently\nit was \ftted to DO 3type anti-site disorder which yielded\ngood agreement between the experimental data and the\ntheoretical pattern. As Cr, V and Al have nearly same\nelectronegativity values, it is more probable to have anti-\nsite disorder among these atoms. Due to the large ionic\nradius and least electronegativity, Ti ions are less prone\nto have antisite disorder with other atoms. However, it is\nto be noted that XRD analysis alone cannot completely\nresolve the structural disorder in this alloy.\nB. Magnetic and Transport Properties\nFor QHAs composed of atleast two elements having less\nthan half-\flled d electrons, the saturation magnetization\nobeys the Slater Pauling (SP) rule,28,41,42\nM=N\u000018\u0016B=f:u: (9)\nwhereNis the total number of valence electrons in the\nalloy.\nFigure 3 (a) and (b) show the M-T and M-H data\nwhich clearly indicate a very small moment of CVTA\n(\u001810\u00003\u0016B=f:u:) and the magnetic ordering tempera-\nture is high (\u0018710 K, see the inset of Fig. 3(a)). M-H\ncurve, as shown in Fig. 3(b), has a low, nonzero hystere-\nsis (see also the inset). The behaviour remains almost4\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s50/s54/s48/s50/s55/s48/s50/s56/s48/s50/s57/s48/s51/s48/s48/s32/s40 /s45/s99/s109/s41/s67/s114/s86/s84/s105/s65/s108/s32/s32/s32/s48/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101\n/s32/s50/s48/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s72/s32/s61/s32/s53/s48/s48/s32/s79/s101/s90/s70/s67/s87/s32 /s32/s77/s32/s40/s49/s48/s45/s52\n/s32\n/s66/s47/s102/s46/s117/s46/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s70/s67/s87\n/s67/s114/s86/s84/s105/s65/s108\n/s72/s32/s61/s32/s49/s32/s107/s79/s101\n/s32/s32\n/s32/s32/s70/s87/s77/s32/s40/s49/s48/s45/s52\n/s32\n/s66/s47/s102/s46/s117/s46/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s77/s32/s40/s49/s48/s45/s51\n/s32\n/s66/s47/s102/s46/s117/s41\n/s72/s32/s40/s107/s79/s101/s41/s64/s51/s32/s75\n/s67/s114/s86/s84/s105/s65/s108/s72\n/s99/s111/s101/s114/s126/s32/s49/s48/s48/s32/s79/s101/s32/s32\n/s40/s98/s41/s40/s99/s41\n/s40/s97/s41\nFIG. 3. (a) Temperature dependence of magnetization in\nZFCW and FCW modes. The inset shows the temperature\ndependence of magnetization in high temperature regime. (b)\nThe \feld dependence of magnetization. The inset shows the\nmagnetization in zoomed scale. (c) The temperature depen-\ndence of electrical resistivity in di\u000berent \felds.\nthe same at high T. Observation of extremely low mo-\nment with \fnite hysteresis indicate the strong possibility\nof fully compensated ferrimagnetic nature, as also found\nin our simulation. The outcome of nearly zero moment\nis consistent with the SP rule, which is a prerequisite\nfor spintronics materials. Stephen et. al. ,35on the other\nhand, reported a linear M-H curve, which may be due to\nthe presence of small impurities present in their sample.\nFigure 3(c) shows temperature dependence of resis-\ntivity in di\u000berent \felds. The resistivity shows negative\ntemperature coe\u000ecient, suggesting semiconducting be-\nhaviour. In intrinsic semiconductors, the variation of \u001a\nwith T is dominated by exponential term, as shown in Eq.\n1. Hence the absence of such a term in CVTA indicates\neither gapless or spin-gapless semiconducting nature.\nIV. THEORETICAL RESULTS\nWe have fully relaxed the experimentally formed crys-\ntal structure (space group # 216) in three distinct con\fg-\nurations, I, II and III, as described in Sec. II. The whole\nidea of performing these simulations was to get a better\nunderstanding of the XRD results (e.g. absence of (111)\npeak) which is not enough to clarify a few of the struc-\ntural aspects. Table I shows the relaxed lattice param-\neter (a0), total and atom projected magnetic moments\nand the total energy ( E0) for the three con\fgurations.TABLE I. Relaxed lattice parameter ( a0), atom-projected\nmagnetic moments, total moments ( \u0016B) and total energy ( E0)\nfor the three con\fgurations I, II and III of CVTA.\nTypea0(\u0017A)mCrmVmTimTotalE0(Ry/atom)\nI 6.08 0.00 0.00 0.00 0.00 -171.370934\nII 6.15 2.25 -1.26 -0.98 0.00 -171.370997\nIII 6.19 2.80 -2.29 -0.49 0.00 -171.371658\nAmong these, the con\fguration III was found to be\nenergetically the most stable one with lattice parameter\na0= 6:19\u0017A. The total energy di\u000berence among the three\ncon\fgurations is less than 1 mRy =atom which hints that\nat \fnite temperature CVTA could be a mixture of these\nthree con\fgurations, responsible for the observed disor-\nder. In order to understand the e\u000bect of this disorder,\nwe studied all the three con\fgurations in detail.\n/s45/s50/s45/s49/s48/s49\n/s87 /s32/s32/s76/s32 /s32/s32/s32/s88/s32/s85/s44/s75/s32/s32 /s32/s32/s32/s87 \n/s32/s45/s50/s45/s49/s48/s49\n/s87 /s32/s32/s76/s32 /s32/s32/s32/s88/s32/s85/s44/s75/s32/s32 /s32/s32/s32/s87 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s45/s50/s45/s49/s48/s49\n/s56 /s54 /s52 /s50 /s48 /s50 /s52 /s54 /s56/s45/s50/s45/s49/s48/s49/s48 /s50 /s52/s45/s50/s45/s49/s48/s49\n/s48 /s50 /s52/s45/s50/s45/s49/s48/s49\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s45/s50/s45/s49/s48/s49\n/s48 /s50 /s52/s45/s50/s45/s49/s48/s49\n/s48 /s50 /s52/s45/s50/s45/s49/s48/s49/s69/s110/s101/s114/s103/s121/s32/s69/s40/s107/s41/s45/s69\n/s70/s32/s40/s101/s86/s41/s73\n/s73/s73\n/s73/s73/s73\n/s77/s111/s109/s101/s110/s116/s117/s109/s32/s40/s107/s41/s32/s32/s32/s32/s32/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s45/s102/s46/s117/s46/s41/s32/s32/s32/s32/s32/s32/s77/s111/s109/s101/s110/s116/s117/s109 /s32/s40/s107/s41\nFIG. 4. Spin polarized band structure and DOS for the con\fg-\nurations, I, II and III of CVTA at relaxed lattice parameters\n(a0). Left-side bands correspond to spin up while right-hand\nside is for spin down.\nCalculated spin polarized band structure and DOS for\nall the three con\fgurations are shown in Fig. 4. Cal-\nculations for all the three con\fgurations were initiated\nwith a ferrimagnetic arrangement of spins with moments\nat Cr-atoms aligned antiparallel to those of V and Ti.5\nFIG. 5. (a) Fermi surface (FS) is same for both up and\ndown spin bands of con\fguration I. It has spherical shape\nand crosses at X point. (b) Fermi surface for the spin down\nband of con\fguration II. It has oblate shape and crosses at X\npoint. Con\fguration III does not have any FS.\nThis was done keeping in mind the vanishingly small ex-\nperimental net moment (see previous section) and other\ntheoretical reports where ferrimagnetic arrangement was\nproposed to be the stable phase. In our case, con\fgura-\ntion I converges to a non-magnetic phase with identical\nspin up and down bands, and consequently has the lowest\nmagnetic ordering temperature. Both spin up and down\nbands are gapless with nearly zero DOS at EF, indicating\nthe gapless nature. It acquires an indirect band gap with\nconduction band minima touching at X-point and valence\nband maxima at other k-point. Figure 5(a) shows the\nFermi surface plot for con\fguration I. As expected, both\nspin up and down Fermi surfaces are identical, with tiny\nspherical shape and are shared by neighbouring Brillouin\nzone. The essential features of DOS and band structure\nremain unchanged at aexpand hence its physical proper-\nties (transport and magnetic) are robust.\nIn the case of con\fguration II, irrespective of the ini-\ntial magnetic moments at each site, the calculations con-\nverged in a ferrimagnetic arrangement with Cr moments\naligned antiparallel to V and Ti. For this con\fguration,\nDOS and band structure for spin down channel mostly\nresemble that of con\fguration I, except for the shape\nand size of Fermi surface at X-point, indicating gapless\nnature for spin down channel. The shape of the Fermi\nsurface, as shown in Fig. 5(b) is oblate, centred at X-\npoint and equally shared by neighbouring Brillouin zone.\nThe size of the surface is more than double that of con-\n\fguration I. There is almost no change in the DOS and\nband structure of spin down channel with a0andaexp,\nindicating that its spin down gapless nature is robust\nagainst small changes in a. On the other hand, DOS and\nband structure for spin up channel shows a clear indirect\nband gap of \u0001 E\"\ng= 0:36 eV, revealing semiconducting\nnature. There is no observable change in \u0001 E\"\ngwith lat-\ntice parameter indicating that its semiconducting nature\nis also robust against small changes in a. Due to the ab-\nsence of symmetric DOS and band structure along with\nrelatively high absolute magnetization (4.64 \u0016B=f:u:), itsmagnetic ordering temperature is expected to be very\nhigh. Such a phase with zero gap in one spin band and\n\fnite gap in the other gives rise to a fully compensated\nferrimagnetic, spin-gapless semiconductor.\nSimilar to con\fguration II, con\fguration III is also\nfound to be ferrimagnetic. However the later has \u0001 E\"\ng=\n0:58 eV and \u0001 E#\ng= 0:30 eV ata0, indicating that it is\na magnetic semiconductor. Spin up and down gaps are\nreduced to 0.55 eV and 0.25 eV respectively at aexp. Ab-\nsence of Fermi surfaces (no states at EF) for both spin\nchannels also con\frms the magnetic semiconducting na-\nture. Presence of large exchange splitting gaps and a\nlarge absolute magnetization (5.92 \u0016B=f:u:) indicates a\nlarge magnetic ordering temperature.\nAll the properties of con\fguration III (such as mag-\nnetic state, \u0001 E\";#\ngetc.), which corresponds to ground\nstate one, are in good agreement with the earlier reports\nby Ozdogan et. al with exception of sign of moments on\nall ions. As a result, DOS and band structure are in-\nterchanged for spin up and down electrons. In addition,\nthey reported a small positive moment on Al ions which\nalso has an opposite sign in the current study. Notably,\nCr and V moments are aligned in opposite directions and\nthe resulting moment is compensated by Ti ions (see Ta-\nble I).\nIt is important to note that all the con\fgurations show\ndi\u000berent forms of semiconductors, with two of them con-\nverged into the ferrimagnetic state. In order to under-\nstand the true nature of the semiconductor, it is quite\nrelevant to study the temperature dependence of the\ntransport properties i.e., intrinsic carrier concentration\n(n) and spin polarization ( P). Figure 6 shows the tem-\nperature dependence of these quantities for the three con-\n\fgurations. Con\fguration I, being almost non magnetic,\nhas negligible spin polarization ( P), although they indeed\nhave a \fnite carrier concentration due to small states at\nthe Fermi level ( EF), For con\fguration II and III, the spin\nup carrier concentration is negligibly small due to vanish-\ning states at EF. The spin down carrier concentration,\non the other hand, is large but has very di\u000berent nature\nof T-dependence for the two con\fgurations (II and III).\nInterestingly, in the case of con\fguration III, nshows an\nalmost straight line behaviour, which is neither exponen-\ntial nor T3=2. The magnitude, however, is much smaller\n(n\u00182) compared to the other two con\fgurations. Spin\npolarization values for these two con\fgurations are high.\nV. DISCUSSION AND CONCLUSION\nExperimental results reveal that CrVTiAl is a fully\ncompensated ferrimagnet with DO 3disorder among Al,\nCr and V atoms. First principle calculations within GGA\napproximation predict con\fguration III as the ground\nstate, which is a fully compensated ferrimagnet with un-\nequal band gaps for spin up and down channels. How-\never, the energy di\u000berences among all the three con\fg-\nurations (I, II and III) are small, which indicates the6\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s53/s49/s48/s49/s53\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48\n/s32/s110/s110/s32/s32/s32/s110/s32 /s40/s49/s48/s49/s57\n/s47/s99/s99/s41/s67/s114/s86/s84/s105/s65/s108/s45/s73/s73\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s57/s57/s46/s55/s57/s57/s46/s56/s57/s57/s46/s57/s49/s48/s48/s46/s48\n/s80\n/s32/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s110 /s32/s40/s49/s48/s49/s55\n/s47/s99/s99/s41\n/s32/s32\n/s110\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s32/s32\n/s32/s32/s110 /s32/s40/s49/s48/s49/s57\n/s47/s99/s99/s41/s32/s83/s112/s105/s110/s45/s117/s112/s32/s67/s97/s114/s114/s105/s101/s114/s115/s32/s40/s110 /s41\n/s32/s83/s112/s105/s110/s45/s100/s110/s32/s67/s97/s114/s114/s105/s101/s114/s115/s32/s40/s110 /s41\n/s32/s84/s111/s116/s97/s108/s32/s67/s97/s114/s114/s105/s101/s114/s115/s32/s40/s110/s41/s67/s114/s86/s84/s105/s65/s108/s45/s73\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s50/s52/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s110/s110/s32\n/s32/s110 /s32/s40/s49/s48/s49/s57\n/s47/s99/s99/s41/s67/s114/s86/s84/s105/s65/s108/s45/s73/s73/s73\n/s57/s57/s46/s48/s57/s57/s46/s50/s57/s57/s46/s52/s57/s57/s46/s54/s57/s57/s46/s56/s49/s48/s48/s46/s48\n/s80\n/s32/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s40/s37/s41/s32/s32\n/s110/s110 /s32/s49/s48/s49/s54\n/s47/s99/s99\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s40/s97/s41/s40/s98/s41/s40/s99/s41\nFIG. 6. Spin resolved intrinsic carrier concentrations (left axis) and spin polarization (right axis) vs. temperature (between\n3-400 K) for con\fguration I (left), II (middle) and III (right panel) for CVTA.\npossibility of a mixed phase or disordered phase at \f-\nnite T. One can consider the disordered state as a lin-\near combination of all the three independent con\fgura-\ntions (with probabilities proportional to their Boltzmann\nfactors) yielding either spin-gapless, gapless semiconduc-\ntor or magnetic semiconductor with reduced band gaps.\nThis concept originates from the fact that the disordered\nKohn Sham (KS) orbitals (\b dis) can be written as linear\ncombination of KS orbitals (\b) of each con\fguration i.e.,\n\bdis=cI\bI+cII\bII+cIII\bIIIwhereci/exp(\u0000Ei\n0=kBT).\nSuch a linear combination is possible due to the fact that\nall the three pure con\fgurations have vanishing states\nat the Fermi level, unlike most of the reported cases\nin which certain con\fgurations alone have \fnite states\natEF. At the observation level, the above linear com-\nbination presents itself as having a predominantly SGS\nproperty, because the \frst term is energetically least at-\ntainable and non magnetic while the third term is not\ne\u000bective as the DOS ( D\";#(E)) is zero at the Fermi level\n(E\";#\ng\u001dkBT). The last scenario can change if there are\nimpurities, which will alter the SGS nature seen in our\nsample.35In conclusion, we identify the true crystallographic and\nmagnetic ground states of CrVTiAl with the help of a\njoint theoretical and experimental investigation. While\nthe magnetic ground state is uniquely identi\fed as a fully\ncompensated ferrimagnet, the balance between spin gap-\nless nature and the magnetic semiconducting nature ap-\npears to be quite delicate according to the theoretical\ncalculations. 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Jin, Journal of Magnetism and Magnetic\nMaterials 324, 3099 (2012).\n42Q. Gao, H.-H. Xie, L. Li, G. Lei, J.-B. Deng, and X.-R.\nHu, Superlattices and Microstructures 85, 536 (2015)." }, { "title": "2007.02983v1.Ferrimagnetism_in_EuFe4As12_revealed_by_153Eu_NMR_and_75As_NQR_measurements.pdf", "content": "arXiv:2007.02983v1 [cond-mat.str-el] 6 Jul 2020Ferrimagnetism in EuFe 4As12revealed by153Eu NMR and75As NQR measurements\nQ.-P. Ding,1,2H.-C. Lee,1,2K. Nishine,3Y. Kawamura,3C. Sekine,3and Y. Furukawa1,2\n1Ames Laboratory, U.S. Department of Energy, Ames, Iowa 5001 1, USA\n2Department of Physics and Astronomy, Iowa State University , Ames, Iowa 50011, USA\n3Muroran Institute of Technology, Muroran, Hokkaido 050-85 85, Japan\n(Dated: July 8, 2020)\nFilled skutterudite compound EuFe 4As12shows the highest magnetic ordering temperature of TC\n= 154 K among Eu-based skutterudite compounds, but its magne tic ground state has not been\ndetermined yet. Here, we performed153Eu nuclear magnetic resonance (NMR) and75As nuclear\nquadrupole resonance (NQR) measurements on EuFe 4As12to reveal its magnetic ground state as well\nas the physical properties from a microscopic point of view. From the temperature and magnetic\nfield dependence of153Eu NMR spectrum in the magnetically ordered state, we found t hat the\nEu ions are in Eu2+state with a nearly 7 µBcorresponding to S= 7/2 spins. Combined with\nthe magnetization measurements which show the reduced satu ration moments of 4.5 µB/f.u., we\ndetermined the ground magnetic structure in EuFe 4As12to be ferrimagnetic where the Eu2+4f\nand the Fe 3 dordered moments are ferromagnetically aligned in each subl attice but the moments\nbetween the sublattices are antiferromagnetically aligne d. We also found the local distortion at the\nEu site from the cubic symmetry in the magnetically ordered s tate. The relationship between the\nrattling motion of Eu atoms and the local symmetry of the Eu io ns is discussed. From the75As\nNQR nuclear spin-lattice relaxation time measurements as w ell as153Eu NMR measurements, we\nfound that the 4 felectrons of the Eu ions are well described by the local momen t picture in both\nthe magnetic and paramagnetic metallic states.\nI. INTRODUCTION\nThe interplay of 4 fand itinerant electrons in rare-\nearth containing metallic systems provides fascinating\nphysicalphenomenasuchassuperconductivity,ferromag-\nnetism, antiferromagnetism, non-Fermi-liquid behavior,\nheavy-fermion states, and so on [1–6]. Among many ma-\nterials bearing rare-earth elements, the europium con-\ntaining Fe-based superconducting material EuFe 2As2\nstands as one of the interesting materials since, with car-\nrier doping or pressure application, the system shows the\ncoexistence of superconductivity and magnetism origi-\nnating from Eu 4 fmoments, where the magnetism of\nitinerant Fe moments also plays an important role [7–\n10]. EuFe 2As2at ambient pressure exhibits two distinct\nmagnetic phase transitions at T= 189 K and 19 K. The\nfirst magnetic order is a spin density wave [stripe-type\nantiferromagnetic (AFM) state] associated with the itin-\nerant Fe moments while the second one is due to Eu2+\n4fmoments, making an A-type AFM structure where\nthe Eu ordered moments are ferromagnetically aligned\nin theabplane but the moments in adjacent layers along\nthecaxis are antiferrmagnetically aligned [7]. Although\nthe Eu and Fe moments order at different temperatures,\nthe strong coupling between the 4 fand itinerant elec-\ntrons has been pointed out, which will be responsible for\nthe interesting and complicated magnetic properties ob-\nserved in Eu(Fe 1−xCox)2As2where the A-type magnetic\nstructure changes to the A-type canted AFM structure\nat intermediate Co doping levels around x∼0.1, and\nthen to the ferromagnetic order along the caxis atx∼\n0.18 [9].\nThe importance of the magnetic interaction between\nEu 4felectrons and itinerant delectrons has also beenpointed out in an Eu-based filled skutterudite com-\npound EuFe 4As12[11]. However, the magnetism of\nEuFe4As12can be quite different from the aforemen-\ntioned Eu(Fe 1−xCox)2As2. EuFe 4A12exhibits a mag-\nnetic phase transition at TC���152 K where both Eu 4 f\nanditinerant Fe momentsorderatthe sametime [11, 12].\nIt is pointed out that the transition temperature of TC∼\n152 K is relatively high in comparison with other related\ncompounds [11]. When Fe is replaced by Ru or Os, the\nmagnetic ordering temperature is suppressed to 25 K for\nEuOs4As12and no magnetic order is observed down to 2\nK in EuRu 4As12[11]. On the other hand, when Eu2+is\nreplaced by other divalent Sr or Ba ions, no magnetic or-\nder is observed although ferromagnetic spin fluctuations\nwere reported [13–15]. In the case of La for the replace-\nment, an itinerant ferromagnetism associated with Fe 3 d\nmoments is observed below a Curie temperature of ∼5.2\nK in LaFe 4As12[16, 17] . The difference of the magnetic\nordering temperature for those compounds indicates the\nstrong exchange coupling between Eu 4 felectrons and\nitinerant Fe 3 delectrons [11]. Such strong exchange cou-\nplings have also been reported in similar Eu-containing\niron skutterudite compounds EuFe 4Sb12(TC= 85±4\nK) [18–22] and EuFe 4P12(TC= 100±3 K) [23, 24].\nAs for the spin structure of the magnetic state, mag-\nnetic susceptibility χ(T) measurements on EuFe 4As12\nsuggest either a canted ferromagnetic or ferrimagnetic\nstructurebelow TC[11]. TheeffectivemomentandCurie-\nWeiss temperature in the paramagnetic state estimated\nfromχ(T) measurement are reported to be 6.93 µB/f.u.\nand 46 K, respectively. The value of the effective mo-\nments is slightly smaller than that expected for divalent\nEu2+(S=7/2)of µeff=7.94µB[11]. The magneticfield\ndependence ofmagnetizationat2K forEuFe 4As12shows2\na saturationmoment of4.5 µB/f.u. at 1 T, which is much\nsmaller than 7 µBexpected for Eu2+[11]. Similar mag-\nnetic properties have been reported in the isostructural\ncompound EuFe 4Sb12which was initially considered as a\nferromagnet below TC[18–20]. Later on, the x-ray mag-\nnetic circular dichroism spectroscopy (XMCD) and x-ray\nabsorption spectroscopy (XAS) measurements suggest a\nferrimagneticstateinEuFe 4Sb12whereferromagnetically\naligned Eu spins are ordered antiferromagnetically with\nordered Fe moments [25], however, the possibility of a\ncanted ferromagnetic state was not fully ruled out. In\nthe case of EuFe 4A12, such measurements have not been\nperformed yet. Therefore, the detailed studies of the lo-\ncal magnetic and electronic properties of the magnetic\nions are important to reveal the magnetic structure as\nwell as the magnetic properties of the system.\nNuclear magnetic resonance(NMR) is a powerful tech-\nnique to investigate magnetic properties of materials\nfrom a microscopic point of view. In particular, one can\nobtain direct and local information of magnetic state at\nnuclear sites, helping to determine the magnetic struc-\nture of magnetic systems. Although the magnetic state\nof the Eu ions in EuFe 4As12is a key to understand the\nmagnetic propertiesofthe system, therehavebeen no Eu\nNMR studies of this compound up to now to our knowl-\nedge.\nInthispaper,wehavecarriedout153EuNMRand75As\nnuclear quadrupole resonance (NQR) measurements to\ninvestigate the magnetic properties of EuFe 4As12, espe-\ncially focusing on the magnetic state of the Eu ions, from\na microscopic point of view. From the external magnetic\nfielddependenceof153EuNMRspectrum, theEuionsare\nshown to be in divalent state with nearly 7 µBwhich or-\ndered ferromagnetically without any canting component,\nrevealingaferrimagneticorderedstateinEuFe 4As12. We\nalso report the local distortion at the Eu site from the\ncubic symmetry in the magnetically ordered state, which\nwill be important to understand the physics of motion\nfor Eu ions called rattling in the cage formed by Fe 4As12\nunits.\nII. EXPERIMENTAL\nPolycrystalline EuFe 4As12samples were prepared at\nhigh temperature and high pressures using a wedge-\ntype cubic anvil high-pressure apparatus [11]. The\nchemical composition of the samples was determined by\nenergy-dispersive x-ray analysis (EDX) using a JEOL\nscanning electron microscope and was found to be\nEu1.06Fe4.00As12.2showing no obvious deficiency of Eu\nions. Good chemical homogeneity was found in the\npowder samples. The crystal structure of the sample\nwas characterized by powder x-ray diffraction (XRD) us-\ning MoKαradiation and silicon as a standard. The\npowder samples are loosely packed into a sample case,\nwhich allows the crystallites to be oriented along the ap-\nplied magnetic field direction. NQR measurements of75As (I=3\n2,γN\n2π= 7.2919 MHz/T, Q= 0.29 barns)\nand NMR measurements of153Eu (I=5\n2,γN\n2π= 4.632\nMHz/T, Q= 2.49 barns) nuclei were conducted using\na homemade phase-coherent spin-echo pulse spectrome-\nter.153Eu NMR spectra in zero and nonzero magnetic\nfieldsHin the magneticallyordered state and75As-NQR\nspectra were measured in steps of frequency fby mea-\nsuring the intensity of the Hahn spin echo. The75As\nand153Eu nuclear spin-lattice relaxation rate 1/ T1was\nmeasured with a saturation recovery method. 1 /T1at\neach temperature was determined by fitting the nuclear\nmagnetization Mversus time tusing the exponential\nfunction 1 −M(t)/M(∞) =e−(3t/T1)for75As NQR,\nand 1−M(t)/M(∞) = 0.029e−(t/T1)+ 0.18e−(6t/T1)+\n0.79e−(15t/T1)for153Eu NMR, where M(t) andM(∞)\nare the nuclear magnetization at time tafter the satura-\ntion and the equilibrium nuclear magnetization at t→\n∞, respectively.\nIII. RESULTS AND DISCUSSION\nA.153Eu zero field NMR in the Ferrimagnetic state\nFigure 1 shows the frequency-swept153Eu NMR spec-\ntrum in zero magnetic field at 4.3 K in the magnetically\nordered state. The observation of the153Eu NMR sig-\nnals clearly evidences that the magnetic moments of Eu\n4felectrons order in the magnetic state. The relatively\nsharp peaks in the observed spectrum indicate a high\nquality of the powder samples. The peak positions of\nthe spectrum are well explained by the combination of a\nlarge Zeeman interaction due to magnetic field [for the\npresent case, an internal magnetic induction ( Bint) at the\nEusite]andasmallquadrupoleinteractionwhosenuclear\nspin Hamiltonian is given as follows;\nH=−γn¯hI·Bint+hνQ\n6[3I2\nZ−I2+1\n2η(I2\n++I2\n−)],(1)\nwhere\nIz=1\n2(I+e−iφ+I−eiφ)sinθ+IZcosθ.(2)\nHerehisPlanck’sconstant, and νQis nuclearquadrupole\nfrequency defined by νQ= 3e2QVZZ/2I(2I−1)h(=\n3e2QVZZ/20hforI= 5/2) where Qis the electric\nquadrupole moment of the Eu nucleus, VZZis the elec-\ntric field gradient (EFG) at the Eu site in the coordinate\nof the principal X,Y, andZaxes of EFG, and ηis the\nasymmetry parameter of the EFG [26]. θandφare the\npolar and azimuthal angles between the Zaxis of EFG\nand the direction of Bint, respectively, where the quanti-\nzationaxis ( zaxis) forthe Zeemaninteractionis pointing\nalong the Bintdirection.\nIn the case of I= 5/2, when η= 0, the NMR spectrum\nis composedof acentral transitionline ( Iz= 1/2↔-1/2)\nand two pairs of satellite lines shifted from the central\ntransitionlineby ±1\n2νQ(3cos2θ−1)(forthetransitionsof3\nTABLE I: BintandνQfrom153Eu NMR measurements at 4.2 K for EuFe 4As12, the helical antiferromagnets (AFM) EuCo 2P2\nand EuCo 2As2, and the A-type antiferromagnet EuGa 4and the magnetic moments of Eu ions MEufrom neutron diffraction\nmeasurements on EuCo 2P2[31] and EuCo 2As2[32].\nBint(T) νQ(MHz) MEu(µB) Ground state (ordered temperature) Ref.\nEuFe4As12 -28.14(5) T 2.90(5) Ferrimagnet ( TC= 154 K) This work\nEuCo 2P2 -27.5(1) T 30.6(1) 7.26 Helical AFM ( TN= 45 K) [28]\nEuCo 2As2 -25.75(2)T 30.2(2) 6.9(1) Helical AFM ( TN= 66.5 K) [29]\nEuGa 4 -27.08 T 30.5 A-type AFM ( TN= 16 K) [30]\nIz= 3/2↔1/2 and -3/2 ↔-1/2), and ±νQ(3cos2θ−1)\n(forIz= 5/2↔3/2 and -5/2 ↔-3/2). The five solid\nlines shown in Fig. 1 are the calculated positions using\n|Bint|= 28.14(5) T, νQ= 2.90(5) MHz, η= 0 andθ= 0◦\n(and, thus, φ= 0◦) which reproduce the observed posi-\ntions very well. Since Bintmainly originates from core\npolarization from 4 felectrons and is oriented in a di-\nrection opposite to that of the Eu spin moments, the\nsign ofBintis considered to be negative [27]. The ob-\nservedBint= -28.14(5) T is close to the values of Bint\nreported from153Eu zero-field NMR in the helical anti-\nferromagnets EuCo 2P2[28] and EuCo 2As2[29], and the\nA-type antiferromagnet EuGa 4[30] as shown in Table\n1. The values of Eu magnetic moments in EuCo 2P2and\nEuCo2As2determined by neutron diffraction measure-\nments are close to 7 µBexpected for Eu2+ion (S= 7/2)\n[31, 32]. Since the Eu ordered moment MEuis propor-\ntional to |Bint/Ahf|whereAhfis the hyperfine coupling\nconstant mainly originating from the core polarization,\nthe similar value of Bint= -28.14(5) T in EuFe 4As12in\ncomparison with those in other Eu compounds leads to\nthe conclusion that the Eu ions are in Eu2+state with S\n= 7/2.\nIt is noted that, from the XRD measurements [11],\nEuFe4As12crystallizes in a body-centered cubic struc-\nture (the space group Im3, see the inset of Fig. 1 [ ?\n]) with a lattice constant of 8.3374 ˚A at room tempera-\nture. Since the local symmetry of the Eu site is cubic in\nthe structure( Th), oneexpectsnoquadrupoleinteraction\nbecause the EFG due to the charges on the neighboring\nions is zero. In general, there is another contribution to\nthe EFG at the Eu site originating from the on-site f\nelectrons. However, this EFG contribution is also zero\nbecause of the spherical charge distribution of 4 f7elec-\ntrons for Eu2+ions with zero angular momentum L= 0,\nwhich is independent of crystal structure. Therefore, the\nfinite quadrupole interaction at the Eu site observed in\nthe magnetically ordered state must be attributed to the\ncontribution from the neighboring ions. Thus, the exper-\nimental data clearly evidence the lowering symmetry at\nthe Eu site from cubic. We will discuss this issue later.\nThe direction of Bintfrom the NMR spectrum is not\ndetermined in the present case. Usually, one can deter-\nmine the direction from the value of θif we know the\nprincipal axis of the EFG. However, as described above,\nour NMR spectrum indicates that the local symmetryat the Eu site in the magnetically ordered state is dif-\nferent from what expected from the high temperature\ncubic one. Therefore, the EFG direction cannot be de-\ntermined, making the determination of the direction for\nBintimpossible at present.\nIt is also important to point out that the the satellite\n/s49/s50/s48 /s49/s50/s50 /s49/s50/s52 /s49/s50/s54 /s49/s50/s56 /s49/s51/s48 /s49/s51/s50 /s49/s51/s52 /s49/s51/s54 /s49/s51/s56 /s49/s52/s48\n/s72/s32 /s32/s61/s32/s48/s32/s32/s84/s84/s32 /s61/s32/s52/s46/s51/s32/s75/s32/s49/s53/s51\n/s69/s117/s32/s78/s77/s82\n/s32/s83/s112/s105/s110/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s72/s32 /s32/s61/s32/s48/s46/s53/s32/s84\n/s32/s32/s83/s112/s105/s110/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s102/s32/s32/s40/s77/s72/s122/s41\nFIG. 1: Top:153Eu-NMR spectrum in zero magnetic field\natT= 4.3 K in the magnetic ordering state. The lines are\nthe calculated positions of153Eu-NMR spectrum using Bint\n= -28.14 T and νQ= 2.90 MHz. The inset shows the crystal\nstructure. Bottom:153Eu-NMR spectrum under H= 0.5 T.\nThe curves are calculated spectrum using the same param-\neters where a line broadening of 0.35 MHz is introduced for\neach line.4\nlines of the observed153Eu zero-field NMR are asymmet-\nric and are tailing toward to the central transition line.\nThis asymmetric shape is reminiscent of a so-called pow-\nder pattern of NMR spectrum. Although, in general, we\ndo not expect the powder-pattern like shape in zero-field\nNMR spectrum, we have the following three possibili-\nties: the first one is due to the distribution of νQ, the\nsecond one comes from the slight distribution of θ, and\nthe last one originates from domain walls. The first sce-\nnario has been used to explain the similar asymmetric\n153Eu zero-field NMR lines observed in EuGa 4where a\nlog-normal distribution of νQwas used [30]. However,\nwe do not find such large distribution of νQin our case,\nas will be shown below. As for the second scenario, as\nthe small deviation in θfrom 0 degree makes the satellite\nline shift toward the central transition line as described\nin the above equations for the satellite line positions, one\nmay explain the characteristic asymmetric shape of the\nsatellite lines by taking the small deviation from θ= 0◦\ninto consideration. In fact, such scenario has been ap-\nplied to explain the asymmetric lines observed in EuAl 4\nwhere the distribution of θis estimated to be at most 6◦\nor less [34]. However, for our case, this seems to be not\nthe case. Since the signal intensity does not go down to\nzero between the satellite lines, to reproduce the spec-\ntrum, one needs to have a relatively large distribution\nofθup to at least ∼20◦, which seems to be too large.\nIn contrast, the third scenario can be the main reason\nfor the observed asymmetric shape since the asymmetric\nshape of the lines becomes more symmetric when mag-\nnetic field is applied (a typical spectrum measured at H\n= 0.5 T is shown at the bottom of Fig. 1). The red\ncurves in the figure are the calculated spectrum at H\n= 0.5 T with the same values of BintandνQwhere we\nassume a line broadening of 0.35 MHz for each line. It\nis noted that, in a magnetic field of 0.5 T, the values\nof line width (full width at half maximum, FWHM ∼\n0.4-0.5 MHz) for each line are nearly the same, as the\ncalculated spectrum reproduces the observed spectrum\nvery well. This indicates that the broadening of each\nline originates from the slight distribution of Bint(less\nthan∼0.4 %) and the effect of the distribution in νQ\ninto the line width is almost negligible. It is known that\nthe deviation of rare-earth filling factor from unity in\nfilled skutteruride compounds produces the distribution\nofνQand thus broadening of spectrum at the rare-earth\nsite [35–37]. Therefore, the negligible small distribution\nofνQin EuFe 4As12suggests that the Eu filling factor is\nclose to unity which is consistent with the results of the\nEDX measurements, confirming again the high quality of\nour samples.\nB. Magnetic field dependence of153Eu NMR\nspectrum\nIn orderto gain more insight into the magnetic proper-\nties of EuFe 4As12, especially, the Eu ordered moments in/s49/s48/s48 /s49/s49/s48 /s49/s50/s48 /s49/s51/s48 /s49/s52/s48\n/s48 /s50 /s52 /s54 /s56/s57/s48/s49/s48/s48/s49/s49/s48/s49/s50/s48/s49/s51/s48/s49/s52/s48/s49/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54\n/s40/s98/s41/s84 /s32/s61/s32/s52/s46/s51/s32/s75/s49/s53/s51\n/s69/s117/s45/s78/s77/s82\n/s54/s32/s84\n/s52/s32/s84\n/s50/s32/s84\n/s49/s32/s84\n/s48/s46/s53/s32/s84\n/s48/s46/s50/s53/s32/s84\n/s32/s32/s83/s112/s105/s110/s45/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32\n/s102/s32/s32/s40/s77/s72/s122/s41/s48/s32/s84/s40/s97/s41\n/s32/s32/s102\n/s99/s32/s40/s77/s72/s122/s41\n/s72 /s32/s40/s84/s41\n/s32/s32/s77 /s32/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72/s32 /s32/s40/s84/s41\nFIG. 2: (a) Field dependence of153Eu-NMR spectrum at 4.3\nK. (b) Field dependence of153Eu-NMR central line frequency\n(fc) at 4.3 K. The solid line is a linear fit whose slope is exactly\nsame as - γ/2πof153Eu nucleus. The inset shows the field\ndependence of magnetization measured at 2 K.\nthe magnetically ordered state, we have measured153Eu\nNMR spectrum usingthe looselypackedsampleunderan\nexternal magnetic field H. Figure 2(a) shows the mag-\nnetic field dependence ofthe153Eu NMR spectrum at 4.3\nK. The spectrum shifts to lower frequency as magnetic\nfield increases. At the same time, as discussed above,\nthe asymmetric shape of the satellite lines become more\nsymmetric under magnetic field, indicating the main rea-\nson for the asymmetry can be due to domain walls where\nthe Eu ordered moments change the direction making\nthe distribution of θ. It is also noted that the spacings\nbetween the lines keep constant, indicating θdoes not\nchange with the application of magnetic field. Given the\ntotal magnetization saturates around 1 T at T= 2 K\nas shown in the inset of Fig. 2(b), these results indi-\ncate that most of the small particles are aligned along\nthe magnetic field direction. This means there must be a\nfinite magnetic anisotropy in the magnetic ordered state,\nalthough we cannot estimate the magnitude of it.\nThe magnetic field dependence of the resonance fre-5\nquency for the central line ( fc) is shown in Fig. 2(b)\nand the slope of the Hdependence of fcis found to be\n-4.63MHz/T which is exactlythe same as - γ/2πof153Eu\nnucleus. Since the effective field at the Eu site is given\nby the vector sum of BintandH, i.e.,|Beff|=|Bint+H|,\nthe resonance frequency is expressed as f=γ/2π|Beff|.\nTherefore,thevalueofthe slopeclearlyindicatesthatthe\ndirection of Bintis antiparallel to that of Hand also that\nall Eu ordered moments align along Hwithout any ap-\npreciable deviation. Thus, one can conclude that the fer-\nromagnetically aligned Eu ordered moments do not have\nany canting components up to 8 T, excluding clearly a\ncanted ferromagntic state as a possible ground state. To\nexplain the saturated magnetic moment of 4.5 µB/f.u. in\nthe magnetic ordering state [see the inset of Fig. 2(b)],\nthe Fe 3delectrons must be magnetically ordered in an-\ntiparallel direction with respect to the Eu2+ordered mo-\nments, producing the ferrimagnetic ground state. Since\nthe Eu ordered moments are estimated to be ∼7µB\nfrom the magnitude of Bintas discussed above, the Fe\n3dordered moments are estimated to be ∼0.6µBfor\neach Fe ion. A similar ferrimagnetic state has also been\nreported in the isostructural compound EuFe 4Sb12by\nXMCD and XAS measurements [25]. The ferrimagnetic\nstate is consistent with the results from density func-\ntion theory based calculations for EuFe 4As12[38] as well\nas other similar compounds such as EuFe 4P12[39] and\nEuFe4Sb12[40].\nC. Temperature dependence of153Eu zero-field\nNMR\nFigure 3(a) shows the temperature dependence of\n153Eu zero-field NMR spectra, where the spectra shift\nto lower frequency with increasing temperature. This\nis due to the reduction of |Bint|which decreases from\n28.14 T at 4.3 K to 21.19 T at 90 K. The red squares\nshown in Fig. 3(b) exhibit the temperature dependence\nof|Bint|. In the figure, we also plotted the M/Hdata\nmeasured at 0.1 T by black circles [11]. As shown by\nthe red curve, the temperature dependence of M/His\nwell reproduced by the Brillouin function which was cal-\nculated based on the Weiss molecular field model with\nJ=S= 7/2 and TC= 154 K. This suggests that, al-\nthough both the Eu and Fe ordered moments contribute\nto the total magnetization, the temperature dependence\nofM/Hcan be mainly characterized by the properties\nof Eu ordered moments. In addition, the results indi-\ncate that the Eu ordered moments are well described by\na local moment picture even in a metallic state as re-\nvealed by the resistivity measurements [11, 12] as well\nas nuclear relaxation measurements described below. As\nshown in Fig. 3(b), on the other hand, the temperature\ndependence of |Bint|slightly deviates from that of M/H.\nAs|Bint|is proportional to AhfMEu,|Bint|should scale\nwithM/Hsince its temperature dependence is well ex-\nplained by the S=7/2 local moment picture. Although/s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s49/s50/s51/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53 /s49/s53/s48 /s49/s55/s53 /s50/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s48 /s53/s48 /s49/s48/s48/s48/s49/s48/s48/s48/s50/s48/s48/s48/s72 /s32/s61/s32/s48\n/s40/s100/s41/s40/s98/s41\n/s40/s99/s41/s84/s32 /s61/s32/s57/s48/s32/s75/s32\n/s84 /s32/s61/s32/s56/s48/s32/s75/s32\n/s84 /s32/s61/s32/s55/s48/s32/s75/s32\n/s84 /s32/s61/s32/s54/s48/s32/s75/s32\n/s84 /s32/s61/s32/s53/s48/s32/s75/s32\n/s84 /s32/s61/s32/s52/s48/s32/s75/s32\n/s84 /s32/s61/s32/s51/s48/s32/s75/s32\n/s84 /s32/s61/s32/s50/s48/s32/s75/s32\n/s84 /s32/s61/s32/s49/s48/s32/s75/s32/s32\n/s32/s83/s112/s105/s110/s45/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s102/s32/s40/s77/s72/s122/s41/s49/s53/s51\n/s69/s117/s32/s78/s77/s82\n/s84 /s32/s61/s32/s52/s46/s51/s32/s75/s32/s40/s97/s41\n/s32/s32/s81/s32/s40/s77/s72/s122/s41\n/s84 /s32/s40/s75/s41/s48/s53/s49/s48/s49/s53/s50/s48/s32\n/s32\n/s84 /s32/s40/s75/s41\n/s77 /s47/s72 /s32/s40/s101/s109/s117/s47/s109/s111/s108/s41/s124/s66\n/s105/s110 /s116/s124/s32/s32/s40/s84/s41\n/s84 /s32/s40/s75/s41/s84\n/s99/s61/s32/s49/s53/s52/s32/s75\n/s32/s32/s49/s47 /s84\n/s49/s32/s40/s115/s45/s49\n/s41\n/s84\nFIG. 3: (a) Temperature dependence of153Eu-NMR spectrum\nunder zero magnetic field in the magnetic ordering state. The\ncurves are the calculated153Eu-NMR spectra. (b) Tempera-\nture dependence of |Bint|. The black circles show the temper-\nature dependence of M/H measured at H= 0.1 T reported\npreviously [11]. The red curve is the Brillouin function wit h\nJ=S= 7/2 and TC= 154 K. (c) Temperature dependence\nofνQat the153Eu site. (d) Temperature dependence of153Eu\n1/T1under zero magnetic field. The straight line shows the\nKorringa relation 1 /T1T= 28 (sK)−1.\nthe reason for the deviation is not clear at present, it\nmay be due to the local distortion at the Eu site in the\nferrimagnetic state. As described above, the |Bint|is\nmainly determined by the hyperfine coupling originated\nfrom the 4 felectron core-polarization mechanism which\nis atomic in nature. Therefore, the hyperfine coupling\nconstant will not be affected by the local environment\nat the Eu site. On the other hand, if one considers the\neffects of the transferred hyperfine field Btrans\nintat the Eu\nsite from the nearest neighbor Fe ordered moments, the\nBtrans\nintcan be affected by the local distortion at the Eu\nsite since the transferred hyperfine field largely depends\non the strength of Fe-As-Eu covalent bond. Although we\ncannot conclude the origin of the deviation, it would be\ninteresting if it originates from the effects of Btrans\nintsince\nthis may be experimental evidence showing the coupling\nbetween the 3 delectrons of Fe and the 4 felectrons of Eu\nin EuFe 4As12.\nThe temperature dependence of νQat the153Eu site\nis shown in Fig. 3 (c) where νQdecreases from 2.90\nMHz at 4.3 K to 1.2 MHz at 90 K. Such a huge reduc-\ntion ofνQby∼59 % cannot be explained by thermal\nlattice expansion [41] which is normally described by an\nempirical relation νQ(T) =νQ(0)(1−αQT3/2) with a fit-\nting parameter αQ. As described above, one does not6\nexpect the finite value of νQat the Eu site in the high-\ntemperature body-centered cubic structure. In fact, νQ\nseems to disappear around T∼125 K estimated from\nthe smooth extrapolation using the experimental data at\nT= 30 - 90 K, and νQwill be zero in the paramagnetic\nstate above TC= 154 K where the cubic structure ( Im3)\nwas determined by the XRD measurements at room tem-\nperature [11].\nSince the finite values of νQreflecting the degree of\nthe local distortion of the cubic symmetry at the Eu site\nsuggest a structural phase transition, we havecarriedout\nXRD measurements in the temperature range of T= 113\n- 300 K. We did not observe clear evidence for structural\nphase transition in the XRD patterns down to 113 K\nwhich is the lowest temperature we can achieve by us-\ning our XRD spectrometer with a nitrogen gas flow type\ncryostat. Although the results may suggest no structural\nphase transition at least down to 113 K, this does not\nfully rule out the possibility in the compound. One of\nthe possibilities is that the lowest temperature of 113 K\nmay not be low enough to detect the structural phase\ntransition which could occur at TCestimated from the\ntemperature dependence of νQ. The situation here could\nbe similarto the electrondiffraction measurements ofthe\nskutterudite compound PrRu 4P12where the superlattice\nspots due to the structural phase transition were clearly\nobserved at 12 K, however, the intensity of the super-\nlattice spots becomes very weak at 40 K even though\nthe structural transition temperature is around 60 K\n[43]. Another possibility is that ordinary XRD measure-\nments using powder samples may not be able to detect\nthe structural phase transition. This has also been re-\nported in PrRu 4P12where only synchrotron radiation\nXRDmeasurementsusingsinglecrystalsrevealedastruc-\nturalphasetransition[44,45]. Thestructuralphasetran-\nsition was reported to change only in the space group\n(fromIm3 for the hight temperature phase to Pm3 for\nthe low temperature phase) while the cubic crystal sym-\nmetry of structure is unchanged [43–45]. It is interesting\nto point out that, even in the cubic symmetry, the slight\ndisplacements of P and Fe atoms have been reported in\nthe low temperature phase of PrRu 4P12[43–45]. Such\ndisplacement may produce the local distortion of the cu-\nbic symmetry at the rare-earth sites, giving rise to the\nfinite EFG. This could explain our results in EuFe 4As12,\nof course, we cannot conclude it though. Thus, it is im-\nportant to carry out the low temperature XRD measure-\nments, especiallyusing singlecrystalsifavailable, to clar-\nify whether or not the crystal structure in the magneti-\ncally ordered stat is different from the high temperature\none. Such experiments are highly called for.\nFrom the behavior of the temperature dependence\nofνQ, we consider that the structural phase transition\nwould a second-order type of phase transition if the local\ndistortion at the Eu sites were due to a structural phase\ntransition. Since the specific heat measurements do not\nshow any anomaly other than at TC[12], one may spec-\nulate that the magnetic ordering takes place with a pos-sible concomitant structural phase transition, although\nthe estimated T∼125 K is a little bit far from TC= 154\nK.\nIt is also interesting to point out that the specific heat\nmeasurements [12] suggest that the rattling motion of\nthe Eu atoms is not prominent in EuFe 4As12. This has\nbeen generally explained in terms of the ionic radius of\nEu2+ions greater than those of trivalent rare-earth ions,\nwhich gives rise to a less space for the rattling motion\nof the Eu2+ions in the Fe 4As12cage. However, as the\ncubic electric field at the rare-earthsite is alsoconsidered\nto be one of the keys for the rattling motion, the lower-\ning of the cubic symmetry at the Eu site in EuFe 4As12\nmay result in restricting the rattling motion of the Eu\natoms. Even from this point of view, as described above,\nfurther detailed studies on the crystal structure at low\ntemperatures are requested, which may provide further\ninsights into the ”rattling” physics in filled skutterudite\ncompounds.\nAt the end of this sub-subsection, we show the temper-\nature dependence of 1 /T1measured at the central line of\nthe153Eu zero-field NMR spectrum. As shown in Fig.\n3(d), 1/T1is proportional to temperature, obeying a Ko-\nrringa law 1 /T1T= 28 (sK)−1expected for a metallic\nstate. This result confirms the metallic ground state in\nthe ferrimagnetic state from a microscopic point of view,\nconsistent with the resistivity measurements [11, 12].\nD.75As NQR in the paramagnetic state\nFigure 4 (a) shows the temperature dependence of the\n75As NQR spectrum from 160 K to 300 K in the para-\nmagnetic state. In NQR spectrum under zero magnetic\nfield for75As nuclei with I= 3/2, one expects a single\ntransition line at a frequency of νNQR=νQ/radicalbig\n1+η2/3.\nThe observed lines are sharp with the nearly tempera-\nture independent line width ( FWHM ∼65 kHz), which\nis smaller than FWHM ∼140 kHz and is greater than\n∼33 kHz for high quality samples of SrFe 4As12[15]\nand LaFe 4As12[42], respectively, but is much smaller\nthan∼400 kHz in SrOs 4As12having a lesser degree of\nhomogeneity of crystals [46]. This indicates again the\nhigh quality of the samples and also the Eu filling factor\nclose to unity. As shown in the figure, the peak position\nslightly shifts to lower frequency with increasing temper-\nature without showing any sudden changes, indicating\nthat there is no any structural anomaly in the paramag-\nnetic state of EuFe 4As12. On the other hand, below TC,\nthe spectrum becomes broader and the signal intensity\ndecreases. Although we could not measure the spectrum\naroundTCbecause of the poor signal intensity due to the\nshortening of nuclear spin-spin relaxation time T2origi-\nnating from the phase transition, we were able to mea-\nsurethe spectrum at T= 90K. The verybroadspectrum\nwith the two-peak structure is observed as shown in the\ninset of Fig. 4(a) where we also plot the spectrum ( T=\n160 K) observed in the paramagnetic state for compari-7\n/s53/s52/s46/s53 /s53/s53/s46/s48 /s53/s53/s46/s53 /s53/s54/s46/s48 /s53/s54/s46/s53\n/s50/s48/s48/s48 /s51/s48/s48/s48 /s52/s48/s48/s48 /s53/s48/s48/s48/s53/s53/s46/s48/s53/s53/s46/s50/s53/s53/s46/s52/s53/s53/s46/s54/s52/s53 /s53/s48 /s53/s53 /s54/s48 /s54/s53 /s55/s48\n/s40/s98/s41\n/s32/s32/s51/s48/s48/s32/s75\n/s32/s50/s57/s48/s32/s75\n/s32/s50/s56/s48/s32/s75\n/s32/s50/s55/s48/s32/s75\n/s32/s50/s54/s48/s32/s75\n/s32/s50/s53/s48/s32/s75\n/s32/s50/s52/s48/s32/s75\n/s32/s50/s51/s48/s32/s75\n/s32/s50/s50/s48/s32/s75\n/s32/s50/s49/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s49/s57/s48/s32/s75\n/s32/s49/s56/s48/s32/s75\n/s32/s49/s55/s48/s32/s75\n/s32/s49/s54/s48/s32/s75/s83/s112/s105/s110/s45/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s102/s32/s40/s77/s72/s122/s41/s40/s97/s41\n/s32/s32/s78/s81/s82/s32/s40/s77/s72/s122/s41/s32\n/s32 /s84/s51/s47/s50\n/s32/s40/s75/s51/s47/s50\n/s41/s32/s49/s54/s48/s32/s75\n/s32/s57/s48/s32/s75\n/s32/s32/s32/s83/s112/s105/s110/s45/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s102/s32/s32/s40/s77/s72/s122/s41\nFIG. 4: (a) Temperature dependence of the75As-NQR spec-\ntrum for EuFe 4As12in the paramagnetic state. The inset\nshows a typical75As-NQR spectrum measured at T= 90 K\nin the magnetically ordered state. The red curve is a calcu-\nlated result with νNQR = 58. 7 MHz and |Bint|= 5.1 kOe at\nthe As site. (b) Temperature dependence of75As-NQR fre-\nquencyνNQR in the paramagnetic state as a function of T3/2.\nThe solid line is the fitting result (see text).\nson. One possible explanation of the two-peak structure\nis to introduce an internal field at the As site from the\nEu and Fe ordered moments in the ferrimagnetic state.\nIt is also important to point out that the center of mass\nof the spectrum shifts to higher frequency, indicative of\nan increase of νNQR. Based on this consideration, we\nhave calculated75As NQR spectrum assuming |Bint|=\n5.1 kOe and νNQR= 58.7 MHz with a line broadening\nof 4 MHz. The red curve is the calculated spectrum,\nwhich seems to capture the characteristic shape of the\nspectrum. Although we cannot determine the direction\nof|Bint|as well as its sign, the large change of νNQR\nfrom 55.5 - 55.1 MHz in the paramagnetic state to 58.7\nMHz at 90 K would be consistent with the observation of\nthe local distortion related to a possible structural phase\ntransition suggested by the153Eu NMR measurements.\nThetemperaturedependenceof νNQRdeterminedfrom\nthe peak positions of the NQR spectra is shown in Fig. 4\n(b). Similartemperature dependences of νNQRhavebeen\nobserved in SrFe 4As12[15], SrOs 4As12[46] and also in/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s49/s49/s49/s48/s49/s48/s48/s49/s48/s48/s48\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s48/s48/s48/s52/s48/s48/s48/s54/s48/s48/s48/s56/s48/s48/s48/s49/s48/s48/s48/s48\n/s84\n/s99/s32/s126/s32/s49/s53/s52/s32/s75\n/s32/s32\n/s32/s69/s117/s70/s101\n/s52/s65/s115\n/s49/s50\n/s32/s76/s97/s70/s101\n/s52/s65/s115\n/s49/s50\n/s32/s83/s114/s70/s101\n/s52/s65/s115\n/s49/s50/s49/s47 /s84\n/s49/s84 /s32/s32/s40/s115/s75/s41/s45/s49\n/s84 /s32/s32/s40/s75/s41/s84\n/s99/s32/s61/s32/s53/s46/s50/s32/s75/s55/s53\n/s65/s115/s32/s78/s81/s82\n/s32/s32/s49/s47 /s84\n/s49/s32/s32/s40/s49/s47/s115/s41\n/s84/s32 /s32/s40/s75/s41\nFIG. 5: Temperature dependence of75As 1/T1Tin EuFe 4As12\n(black circles), together with those in LaFe 4As12(magenta\ncircles) from Ref. [52] and SrFe 4As12(green triangles) from\nRef. [15]. The inset shows the temperature dependence of\n1/T1.\notherfilled skutteruditecompounds[42,47–51]wherethe\ntemperature dependence is found to obey an empirical\nrelationνNQR(T) =νNQR(0)(1−αQT3/2) with a fitting\nparameter αQ[41]. As shown by the curve in Fig. 4\n(b), the temperature dependence of νNQRat As sites in\nEuFe4As12alsofollowstherelationwith αQ= 2.24×10−6\nK−3/2. The value of αQ= 2.24×10−6K−3/2is similar\nto the values for SrFe 4As12(3.21×10−6K−3/2) [15] and\nSrOs4As12(2.09×10−6K−3/2) [46].\nIt is noted that one cannot determine the values of\nνQandηfor the As nuclei separately from only the NQR\nspectrum measurements. The values of η= 0.4-0.45have\nbeen reported in the isostructural compounds Sr M4As12\n(M= Fe, Os) [15, 46]. Since the crystal structure of\nEuFe4As12in the paramagnetic state is the same with\nthose compounds, we expect the similar value of ηin the\npresent compound.\nTo investigate the magnetic fluctuations in the param-\nagnetic state in EuFe 4As12, we have measured the tem-\nperature dependence of75As spin-lattice relaxation rate8\n1/T1in zero magnetic field. Figure 5 shows the temper-\nature dependence of 1/ T1T. For comparison, we also\nplotted the 1/ T1Tdata for the itinerant ferromagnet\nLaFe4As12reported previously [52] where a clear peak in\n1/T1Tcan be observed at a Curie temperature of 5.2 K.\nWith decreasing temperature, 1/ T1Tfor EuFe 4As12in-\ncreases gradually and shows a divergent behavior around\n153 K, corresponding to the ferrimagnetic phase transi-\ntion atTC∼154 K. The values of 1/ T1Tfor EuFe 4As12\naboveTCare almost one order of magnitude greaterthan\nthose in LaFe 4As12at the temperature region. Since\nthere is no 4 felectrons in LaFe 4As12, the largely en-\nhanced75As relaxation in EuFe 4As12could mainly orig-\ninate from the fluctuations of the Eu 4 felectron mo-\nments. According to Moriya [53], when the magnetic\nfluctuations are dominated by the paramagnetic fluctua-\ntions of local moments, 1/ T1is expected to be a constant\nwell above magnetic ordering temperature. As shown\nin the inset of Fig. 5, 1/ T1is nearly independent of\ntemperature at high temperatures. This indicates that\nthe magnetic fluctuations in the paramagnetic state of\nEuFe4As12are characterized by the fluctuations of the\nlocal moment nature of the Eu 4 felectron spins. This\nis consistent with the local moment picture of the Eu\nspins indicated by the temperature dependence of M/H\ndiscussed above.\nIt is also interesting to compare the 1 /T1Tdata\nfor LaFe 4As12with those for the paramagnetic metal\nSrFe4As12(plotted by the green triangles in Fig. 5)\nwhere the existence of ferromagnetic spin fluctuations\nwasreported [46]. The values of1/ T1TofLaFe 4As12well\naboveTC= 5.2 K are much less than those in SrFe 4As12.\nAlthough we do not have the detailed information about\nthe local density of states at the As sites for both com-\npounds, it is interesting if such reduction of 1/ T1Tin\nLaFe4As12could be attributed to the suppression of fer-\nromagnetic spin fluctuations, resulting in the ferromag-\nnetic order at TC= 5.2 K. Further studies, especially\nelectronic structure calculations, are required to address\nthis issue.\nIV. SUMMARY\nIn summary, we performed75As NQR and153Eu\nNMR measurements on the filled skutterudite compoundEuFe4As12withTC= 154 K. We observed the153Eu\nNMR spectrum in the magnetically ordered state, which\nreveals that the Eu2+ordered moments are close to 7\nµB. From the external magnetic field dependence of\n153Eu NMR spectrum observed in the magnetically or-\ndered state, we found that the Eu ordered moments fer-\nromagnetically align the magnetic field direction without\nany canting component. Taking the magnetization data\ninto consideration, we determined the magnetic ground\nstate ofEuFe 4As12to be ferrimagneticin which the Fe 3 d\nmoment and the Eu 4 fmoment aremagnetically ordered\nwith antiferromagnetic coupling. The observed153Eu\nNMR spectrum shows quadrupole split lines which are\nnot expected at the Eu site in the cubic structure ( Im3)\ndetermined by the XRD measurements at room temper-\nature, suggesting the lowering of the local symmetry at\nthe Eu site at low temperatures. The temperature de-\npendence of75As 1/T1Tsuggests that the magnetic fluc-\ntuations in the paramagnetic state are dominated by the\nEu4felectronspins which arewelldescribed by the local\nmoment picture. It is shown that153Eu NMR can be a\nunique tool in determining the magnetic structure in the\nEu compound. It is interesting to study other Eu based\nmagnetic compounds, such as EuFe 4Sb12, EuOs 4Sb12,\nEuFe4P12, and EuOs 4P12to gain deeper understanding\nabout the interaction between the delectrons and the Eu\n4felectrons.\nV. ACKNOWLEDGMENTS\nThe authors would like to acknowledge S. Pakhira and\nD. C. 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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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In a mag-\nnetic medium, according to Korteweg-Helmholtz\nformula [2], magnetostrictive forces should arise\nand lead to phonon-magnon interaction. Here\nwe report such a coupled phonon-magnon system\nbased on ferrimagnetic spheres, which we term\nas cavity magnomechanics, by analogy to cav-\nity optomechanics. Coherent phonon-magnon in-\nteractions, including electromagnetically induced\ntransparency and absorption, are demonstrated.\nExcitingly, due to strong hybridization of magnon\nand microwave photon modes and their high tun-\nability, our platform exhibits new features in-\ncluding parametric ampli\fcation of magnons and\nphonons, triply resonant photon-magnon-phonon\ncoupling and phonon lasing. Our work demon-\nstrates the fundamental principle of cavity mag-\nnomechanics and its application as a new in-\nformation transduction platform based on co-\nherent coupling between photons, phonons and\nmagnons.\nMechanical oscillators have been recently widely stud-\nied as a transducer mediating the coherent signal conver-\nsion between di\u000berent systems [1]. Particularly, radiation\nforce [3{7], electrostatic force [8{10] and piezoelectric\nforce [11, 12] have been utilized for coupling phonon with\noptical or microwave photons. Such interaction mecha-\nnisms lead to the fast development of a variety of cavity\nelectro- and opto-mechanical systems [1], but they all\nintrinsically lack good tunability. The magnetostrictive\nforce [2] provides an alternative mechanism to allow a\ndi\u000berent information carrier { magnon { to couple with\nphonon. Magnon is a collective excitation of magnetiza-\ntion, whose frequency can be tuned at will by adjusting\nbias magnetic \feld [13{15]. The magnetostrictive inter-\naction has been long overlooked for information process-\ning as it is negligibly weak in commonly used dielectric\nor metallic materials. However, in magnetic materials\nsuch as yttrium iron garnet (YIG, Y 3Fe5O12), the mag-\nnetostrictive force becomes dominant, which provides a\ngreat opportunity to establish an highly tunable hybrid\nsystem for coherent information processing. Thanks to\n\u0003To whom correspondence should be addressed. E-mail:\nhong.tang@yale.eduthe excellent material property of YIG, the magnome-\nchanical system can be further integrated with opto- or\nelectro-mechanical elements, providing an excellent plat-\nform for quantum state transfer among di\u000berent physical\nsystems.\nHere, we demonstrate an intriguing cavity magnome-\nchanical system in which magnon couples with phonon\nthrough magnetostrictive interaction, resulting in hall-\nmark coherent phenomena such as magnomechanically\ninduced transparency/absorption (MMIT/MMIA) and\nmagnomechanical parametric ampli\fcation (MMPA).\nDuring such processes, magnons are in the hybridized\nstate with cavity microwave photons as they are strongly\ncoupled to each other [16{20]. Therefore coherent sig-\nnal conversions among these three di\u000berent information\ncarriers are realized in a single device. The magnetic\n\feld dependence of magnon provides our system with un-\nprecedented tunability compared with opto- or electro-\nmechanical systems. Moreover, the great \rexibility of\nthis system allows us to achieve triple-resonance among\nmagnon, phonon and photon, which drastically enhances\nthe magnomechanical interaction. The principles demon-\nstrated in our room temperature experiments can be\nreadily applied to the quantum regime at millikelvin tem-\nperature, opening up great opportunities in various ap-\nplications, such as tunable microwave \flter and ampli\fer\n[21], long-lifetime quantum memories [22], microwave-to-\noptics conversion [9].\nMagnetostrictive interaction\nThe device used in our experiments is schematically\nshown in Fig. 1A. The key component is a highly polished\nsingle crystal YIG sphere glued to the end of a silica \fber\nfor supporting purpose (Fig. 1B). With an external mag-\nnetic \feldHbiased along zdirection, a uniform magnon\nmode resonates in the YIG sphere at frequency !m=\rH,\nwhere\ris the gyromagnetic ratio. The YIG sphere is\nalso an excellent mechanical resonator (Fig. 1C) thanks\nto its superior material and geometrical properties. The\nvarying magnetization induced by the magnon excitation\ninside the YIG sphere causes deformation of its spherical\ngeometry (and vise versa), introducing the coupling be-\ntween magnon and phonon modes (Fig. 1D). Considering\nthe large frequency mismatch between the magnon and\nthe phonon modes (gigahertz v.s. megahertz) with our\nexperiment parameters, a strong parametric drive is used\nto compensate their frequency di\u000berence. In this case,\nthe system is described by an radiation pressure-like, dis-\npersive interaction Hamiltonian H=~gmb^my^m(^b+^by),\nwhere ~is the reduced Planck's constant, ^b( ^m) is thearXiv:1511.03680v2 [quant-ph] 15 Nov 20152\nboson operator of the phonon (magnon) mode, and gmb\nis the single magnon-phonon coupling strength.\nSpheroidal phonon modes\nThe magnetostrictive coupling strength is determined by\nthe mode overlap between the uniform magnon mode and\nthe phonon modes. In a YIG sphere, there exist various\nphonon modes, each with a di\u000berent displacement pro-\n\fle and consequently a di\u000berent coupling strength with\nthe magnon mode. Figure 2A plots the typical pro\fles\nof the lowest order spheroidal phonon modes S 1;l;ma(l\nandmaare the angular and azimuthal mode numbers,\nrespectively), among which the S 1;2;2mode shows the\nhighest coupling strength when the bias \feld is along the\ndirection of maximum displacement (Fig. 2B). Therefore\nin our experiments we focus only on the S 1;2;2mode.\nAlthough a YIG sphere with a smaller diameter is favor-\nable for achieving larger coupling strengths (Fig. 2B), it\nalso results in a higher frequency for the phonon mode\n(Fig. 2C), which in turn leads to lower responsivity to\nthe parametric drive, so a trade-o\u000b has to be made when\nchoosing the sphere size. In our experiments, a 250- \u0016m-\ndiameter YIG sphere is used, corresponding to a phonon\nfrequency!b=2\u0019= 11:42 MHz and a coupling strength\ngmb=2\u0019\u00149:9 mHz. With an external drive of 0 dBm,\nthe linear magnon-phonon coupling can be enhanced to\naround 30 kHz, which is two orders of magnitude larger\nthan the phonon dissipation rate \u0014b.\nMagnetostriction mediates the coupling between\nmagnons and photons. However, in order to achieve co-\nherent magnon-phonon coupling, it is further required\nthat phonon mode should have relatively long lifetime.\nSingle crystal YIG has a garnet structure that is known to\nexhibit very low mechanical damping and therefore sup-\nports a material-limited phonon lifetime over a millisec-\nond [23]. The supporting \fber that is glued to the YIG\nsphere reduces the phonon lifetime (Fig. 2D). In our ex-\nperiments, the measured linewidth of S 1;2;2phonon mode\nwith a 125-\u0016m-diameter supporting \fber is 2 \u0014b=2\u0019= 300\nHz, which is su\u000eciently small for observing coherent\nmagnon-phonon coupling phenomena.\nCoherent magnomechanical interaction\nFigure 1E plots the schematics of our measurement setup\nat room temperature ambient condition. The YIG sphere\nis placed inside a three-dimensional microwave cavity\n(Fig. 1A). A weak probe signal is sent into the cavity\nthrough a coaxial probe, and by sweeping its frequency !s\nand measuring the re\rection, we can infer the interaction\namong photon, magnon and phonon inside the cavity.\nThe YIG sphere is positioned at the maximum microwave\nmagnetic \feld of the cavity TE 011mode, which resonates\nat!a=2\u0019= 7:86 GHz. By controlling the bias magnetic\n\feld, we tune the magnon close to resonance with the\ncavity photon mode. This leads to the hybridization be-\ntween magnon and photon [16{19], which shows up in\nthe re\rection spectrum as a pair of split normal modes\n(Fig. 1F). Because each of the two hybrid modes con-tains magnon components, it coherently couples with the\nphonon modes when the cavity is parametrically driven\nby a strong microwave signal at !d.\nWe \frst study the coherent magnomechanical cou-\npling for each individual hybrid mode by applying an\no\u000b-resonance microwave drive. In this case, the cavity\nmagnomechanical system can be described by\nHmb=~gmb(^b+^by)(cos2\u0012^Ay\n+^A++ sin2\u0012^Ay\n\u0000^A\u0000);(1)\nwhere the two hybrid modes interact with the phonon\nmode independently. Here, ^A+= cos\u0012^a+ sin\u0012^mand\n^A\u0000=\u0000sin\u0012^a+ cos\u0012^mare quantized boson operators\nfor hybridized excitations constituted by magnon and\nmicrowave photon (^ a), with\u0012=1\n2arctan2gma\n\u0001mavaries\nwith photon-magnon coupling strength gmaand photon-\nmagnon detuning \u0001 ma=!m\u0000!a. In our system,\nboth the magnon and the cavity photon modes have a\nrelatively narrow linewidth (2 \u0014m=2\u0019= 1:12 MHz and\n2\u0014a=2\u0019= 3:35 MHz). As a result, the hybrid mode\nlinewidth is well below the phonon frequency, leading\nour system deep inside the resolved sideband regime, by\nanalogy with optomechanical systems [1]. In this case,\nthe nonlinear interaction can be converted either into\nthe linear beam splitter model ~(G\u0006^Ay\n\u0006^b+G\u0003\n\u0006^A\u0006^by) or\nthe parametric oscillator model ~(G\u0006^Ay\n\u0006^by+G\u0003\n\u0006^A\u0006^b)\nwith the presence of an external drive, where G\u0006=\nA\u0006;ssgmb(1\u0007cos2\u0012)=2 is the enhanced coupling strength.\nHere,A\u0006;ssis the steady state amplitude of the hybrid\nmode, corresponding to the e\u000bective pumping of the mi-\ncrowave drive on magnon due to the magnon-photon hy-\nbridization.\nFigures 3A and B plot the measured re\rection spec-\ntra for a series of bias magnetic \felds with a microwave\ndrive at a \fxed frequency !d. To avoid the in\ruence of\nthe other hybrid mode, the driving signal is red (blue)\ndetuned for the lower (upper) hybrid mode, as illus-\ntrated by the top insets. For the red-detuned drive,\nthe power is held constant at 26 dBm. In the spec-\ntra, the broad Lorentzian-shaped resonance dip corre-\nsponds to the hybrid mode ^A\u0000, while the very sharp\nmodi\fcation of the spectra at the two-photon detuning\n\u0001sd=!bis evidence of coherent magnomechanical in-\nteraction. The zoomed-in spectra in Fig. 3A show that\nthese phonon-induced resonances have a Fano-like shape\nthat varies with bias magnetic \feld. When the drive-\nresonance detuning \u0001 d\u0000=!d\u0000!\u0000=\u0000!b, the Fano-like\nresonance changes into a symmetric Lorentzian-shaped\ntransparency peak (MMIT). In contrast, the Fano-like\nresonances in the spectra for the blue-detuned drive\n(with a constant power of 22 dBm) show an opposite\nsymmetry (Fig. 3B). When the drive is blue detuned to\n\u0001d+=!d\u0000!+=!b, such a resonance becomes a\nLorentzian-shaped absorption dip (MMIA).\nOne distinct advantage of magnon is that its frequency\nis determined by the external bias magnetic \feld and\ntherefore can be conveniently tuned in a broad range.\nBy varying \u0001 ma, the percentage of magnon component\nin the hybrid mode changes. Therefore, the hybrid mode3\nexperiences di\u000berent e\u000bective dissipation rate, external\ncoupling rate, as well as e\u000bective coupling strength with\nthe phonon mode. As a result, the coherent magnome-\nchanical interaction is magnetically controllable, which\ncan be quanti\fed by the dependence of the cooperativity\nC=G2\n\u0006=\u0014\u0006\u0014bon the bias magnetic \feld. The measured\nC-Hrelation is plotted in Fig. 3D. For each measurement\nunder a speci\fc bias condition, the drive frequency is de-\ntuned from the hybrid mode by \u0001 d\u0006=\u0006!b, as indicated\nby the crosses in Fig. 3C, while the driving power is \fxed\nconstant at 30 dBm. We can see there exists an optimal\ncondition for a maximum C, as a the result of the compe-\ntition between the magnon and photon components in the\nhybrid mode: more magnon component yields stronger\nmagnetostrictive coupling, while more photon component\nleads to a higher driving e\u000eciency. From these measure-\nment results we can extract the magnon-phonon coupling\nstrengthgmb=2\u0019= 4:1 mHz, in accordance with our the-\noretical prediction (Fig. 2B).\nTriply resonant cavity magnomechanics\nThe great \rexibility of our system leads to tremendous\nadvantages. For instance, it allows us to work in the\ninteresting triple-resonance condition, where both max-\nimum hybridized modes simultaneously couple with the\nphonon mode, as described by\nHmb=1\n2~gmb(^b+^by)(^Ay\n+^A\u0000+^Ay\n\u0000^A+): (2)\nBy adjusting the direction of bias \feld or the position\nof the YIG sphere inside the cavity, we can tune the hy-\nbrid mode splitting to match the phonon frequency !b.\nIn this case, both the drive and probe photons can be\napplied on-resonance with the hybrid modes (top inset\nof Figs. 4A and B), resulting in a drastically enhanced\nmagnomechanical coupling. For the red-detuned drive,\nthe transparency windows at various driving powers are\nplotted in Fig. 4A. In addition to the red shift of the cen-\nter frequency, the linewidth of the transparency windows\nexhibits a clear linear dependence on the driving power\n(Fig. 4C, red squares). With a driving power of only\n8.0 dBm, the linewidth increases from its intrinsic value\n0:62 kHz to 2 :12 kHz, corresponding to a cooperativity\nC= 2:4. As a comparison, a driving power of 34 dBm\nis used to achieve the same cooperativity when the drive\nis applied o\u000b-resonance, indicating the drastic enhance-\nment of the magnomechanical interaction induced by the\ntriple-resonance condition. The re\rection signal for the\nblue detuning situation is plotted in Fig. 4B at various\ndriving powers. As the driving power increases, the cen-\nter frequency of the small phonon-induced resonance in-\nside the hybrid mode is blue shifted, and its linewidth\nlinearly decreases (Fig. 4C, blue circles).\nA direct comparison of Figs. 4A and B reveals dis-\ntinctly di\u000berent spectral lineshapes of the phonon-\ninduced resonances. The same as in the case of o\u000b-\nresonance drive, we observed MMIT for the red-detuned\ndrive in the triply resonant system, with the peak heightand linewidth of the transparency window increasing\nwith the driving power. While for the blue-detuned drive,\nwe observed the transition from MMIA to MMIT, and\nthen to MMPA and eventually self-sustained oscillation\nas we increase the driving power. These observations\nlead to a uni\fed explanation about the modi\fed spec-\ntral lineshape (which is not limited to the triple reso-\nnance situation): the coupling with phonon introduces\nadditional dissipation and phase shift to the hybridized\nmodes and therefore changes their lineshapes. With the\npresence of a parametric drive, the e\u000bective dissipation\nrate of the hybrid mode is modi\fed from \u0014to\u0014(1\u0006C),\nwhich increases for the red-detuned drive while decreases\nfor blue-detuned drive. Given a \fxed external coupling\nrate\u0014e, the on-resonance re\rectivity of the cavity is\nr=1\u0006C\u00002\u0014e\n\u0014\n1\u0006C: (3)\nTherefore, depending on the external coupling condition\nand the driving power, the re\rection spectra lineshape\nvaries among MMIT, MMIA or MMPA.\nThe measured on-resonance re\rectivity for an under-\ncoupled hybrid mode agrees well with our theoreti-\ncal model (Fig. 4D). For the red-detuned drive, the in-\ncreasing linewidth with elevated driving power causes\nthe mode further under-coupled and therefore the on-\nresonance re\rectivity increases. On the contrary, for\nthe blue-detuned situation, the decreasing linewidth \frst\nleads to critical coupling and then over coupling condi-\ntion, yielding a dip in the re\rectivity followed by a rapid\nincrease which diverges as C\u00191 at a driving power of\n6.2 dBm. The deviation of the measured re\rectivity from\nthe theoretical prediction can be attributed to thermal\ninstability or gain-bandwidth-product limitation, which\nalso limit the highest measurable parametric gain to 3\ndB. When the hybrid mode is tuned to over coupled, the\nincrease of the parametric gain with the driving power\nis more gradual, and therefore a much higher paramet-\nric gain up to 25 dB is achieved before reaching insta-\nbility (Fig. 4E). The observed MMPA is similar to the\nelectromechanical parametric ampli\fers [24] but with un-\nprecedented tunability. Further increasing the driving\npower leads the system into the instable regime where the\nphonon mode experiences self-sustained oscillation. The\nthreshold behavior of the measured emission power from\nthe Stokes sideband, as shown by the inset of Fig. 4D,\nindicates the onset of the phonon lasing [25].\nConclusion\nThe demonstration of the coherent magnon-phonon inter-\naction, including the MMIT (MMIA) and MMPA, pro-\nvides a versatile platform for the coherent information\nprocessing. Besides, as YIG also possesses great opti-\ncal properties such as low optical loss and optomagnetic\nnonreciprocity, our study shows great potential for inte-\ngrating di\u000berent systems, including microwave, optical,\nmechanical and magnonic systems, in a single device and\nrealizing information inter-conversion among these dif-4\nferent information carriers. Distinguished from opto- or\nelectro-mechanical systems, our cavity magnomechanical\nsystem shows high level of tunability which allows the\nresonance be externally controlled in a wide frequency\nrange. 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Tabuchi et al. , Coherent coupling between a ferromag-\nnetic magnon and a superconducting qubit. Science 349,\n405{408 (2015).\n[27] S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, D. Vi-\ntali, Reversible optical-to-microwave quantum interface.\nPhys. Rev. Lett. 109, 130503 (2012).\n[28] Y.-D. Wang, A. A. Clerk, Using interference for high \f-\ndelity quantum state transfer in optomechanics. Phys.\nRev. Lett. 108, 153603 (2012).\nACKNOWLEDGEMENTS\nWe thank N. Zhu for gluing the YIG sphere to the\nsilica \fber. This work was supported by DARPA\nMTO/MESO program (N66001-11-1-4114). C.Z., L.J.\nand H.X.T. acknowledge support from LPS through an\nARO grant (W911NF-14-1-0563) and Packard Founda-\ntion. L.J. also acknowledges support from the Alfred P.\nSloan Foundation.5\ny xz\nHA\nC B\nu\n01\nFDE\n7.82 7.84 7.86 7.88 7.90-50-40-30-20\n \nFrequency (GHz)Reflection (dB)Δd-\nΔsd\nωd ωs ω-ω+\nSync AMP BPF\nBSFSRCCIR\nPWRIVNA\nDUTBSF\ny xz\nxyMagnon mode\nDeformation (top view)\nFigure 1. Device schematic and measurement setup. (A) Schematic of the device that consists of a three-dimensional\ncopper cavity (only bottom half is shown) and a YIG sphere. The YIG sphere is placed near the maximum microwave magnetic\n\feld (along ydirection) of the cavity TE 011mode. A uniform external magnetic \feld ( H) is applied along zdirection to bias\nthe YIG sphere for magnon-photon coupling. (B) Optical image of the highly polished 250- \u0016m-diameter YIG sphere that\nis glued to a 125- \u0016m-diameter supporting silica \fber. The gluing area is minimized to reduce the contact damping to the\nphonon mode. Scale bar: 100 \u0016m.(C) Simulated mechanical displacement ( u) of the S 1;2;2phonon mode in the YIG sphere\nwhich has the strongest magnomechanical interaction with the uniform magnon mode. (D) An intuitive illustration of the\nmagnomechanical coupling. Top panel shows the uniform magnon excitation in the YIG sphere. Bottom panel shows that\nthe magnon-induced magnetization (vertical yellow arrows) causes the deformation (compression along ydirection) of the YIG\nsphere (and vise versa). (E)Schematic illustration of the measurement setup. VNA: vector network analyzer; SRC: microwave\nsource for driving; AMP: microwave ampli\fer; BPF: bandpass \flter; PWR: microwave power meter; CIR: circulator; BSF:\nbandstop \flter; DUT: device-under-test. (F)Black curve: cavity re\rection spectrum when magnon is on-resonance with the\ncavity photon mode. The two dips represent the maximum hybridized modes ^A\u0006=1p\n2(^a\u0006^m). Red and blue vertical lines\nindicate the applied drive and probe, respectively. The probe is swept across the hybrid mode resonance. \u0001 sd: two photon\n(probe-drive) detuning; \u0001 d\u0000: drive-resonance detuning.6\nAB\n100 200 300 400 500020406080\n Frequency (MHz)\nSphere diameter ( µm) S1,0,0\n S1,2,x\n S1,3,x\n S1,4,x\n Measured S1,2,2 D C100 200 300 400 5000102030405060\n Coupling strength (mHz)\nSphere diameter ( µm)S1,2,0Top view:\nHS1,2,2HS1,2,2\n100 150 20010-2100102104106\n Phonon linewidth (Hz)\nSupporting fiber diameter ( µm) S1,2,0\n S1,2,1\n S1,2,2\nS1,0,0S1,2,2\nS1,3,3 S1,4,4S1,2,0 S1,2,1\nFigure 2. Analysis of the phonon modes and magnetostrictive coupling strengths. (A) Simulated displacement\npro\fles of the low order phonon modes in the YIG sphere (with a small supporting \fber). S 1;l;marepresents the spheroidal\nmode with a radial mode number of 1, an angular mode number of l, and an azimuthal mode number of ma. Only one of\nthe 2l+ 1 degenerate modes is plotted for each l.(B) Theoretical prediction of the magnomechanical coupling strength as\na function of YIG sphere diameter for the S 1;2;0(black) and S 1;2;2modes (red and blue, corresponding to di\u000berent bias \feld\ndirections). Solid lines are numerical calculations while symbols are analytical \fttings. (C) Phonon mode frequency as a\nfunction of the YIG sphere diameter. Solid lines are the theoretical calculations, showing an inverse proportional dependence,\nwhile red circles are the measurement results. (D) Simulated phonon linewidth due to clamping loss as a function of the\nsupporting \fber diameter for the S 1;2;mamodes. Black dot indicates the experiment parameter, showing an anchor-loss-limited\nlinewidth of 20 Hz.7\n2720 2730 27407.827.847.867.887.90 \nMagnetic field (Oe)Frequency (GHz)C D\n2720 2730 2740012 Cooperativity\nMagnetic field (Oe)\nReflec/g415on (dB)-50-20\n-30\n-40AB\n11.41 11.42-50-40-50-40-50-40-50-40-50-40 \nΔsd (MHz) \n-18 -16 -14 -12 -10 -8 -6-40-20-40-20-40-20-40-20-40-20 2715.34 Oe\n \nΔsd (MHz) 2716.49 Oe\n 2717.64 Oe\n Reflection (dB) 2718.84 Oe\n 2720.00 Oe\n \n6 8 10 12 14-60-40-60-40-60-40-60-40-60-40 \n \nΔsd (MHz) Reflection (dB) 2719.17 Oe\n2717.03 Oe\n \n2716.40 Oe2717.79 Oe2718.47 Oe\n-25-24-26-25-80-60-40-25-24-24-23\n-11.417 \nΔsd (MHz)-11.415 Zoom-in Zoom-in\nA- +Aωdωs\nA- +A\nωd ωs\nFigure 3. Tunable magnomechanically induced transparency/absorption .(A) Measured re\rection spectra near the\nlower hybrid mode ^A\u0000as a function of the two-photon detuning \u0001 sdfor a series of di\u000berent bias magnetic \felds. The broad\ndip corresponds to the lower hybrid mode resonance, whose lineshape changes with bias magnetic \feld because of the change\nin the ratio between magnon and photon components. A strong (26 dBm) microwave drive is red-detuned with a \fxed driving\nfrequency!d. A Fano-like narrow resonance line shows up inside the hybrid mode, which turns into a Lorentzian transparency\npeak when \u0001 d\u0000=\u0000!b. Zoom-in shows detailed spectra of the magnomechanically induced resonances (shaded area in (A)).\n(B)Measured re\rection spectra near the upper hybrid mode ^A+with a blue-detuned strong drive (22 dBm) for various bias\nmagnetic \felds. When \u0001 d+=!b, the magnomechanically induced narrow resonance shows up as a Lorentzian absorption\ndip. Zoom-in shows detailed spectra of the shaded area in (B). (C) Re\rection spectra of the hybrid magnon-photon modes\nat various bias magnetic \felds. The crosses indicate the drive frequency and bias magnetic \feld used for each data point in\n(D).(D)The magnomechanical cooperativity as a function of bias magnetic \feld. For each measurement, the microwave drive\nis detuned from the hybrid mode by \u0001 d\u0006=\u0006!b, while the probe is swept across the hybrid mode resonance. Red squares\n(blue circles) are for the red (blue) detuning situation. Solid lines in (D) and in the zoom-in of (A) and (B) are theoretical\ncalculations using only a single \ftting parameter gmb.8\n02 0 4 0 6 0 8 0 1 0 0 1 2 0-10010203040506070\n Gain (dB)\nDrive power (mW)11.410 11.415 11.420 11.425-40-20020406080100120140 \n Reflection (dB)\nΔsd (MHz)0.28 mW0.38 mW0.56 mW0.83 mW1.29 mW2.00 mW3.16 mW5.01 mW6.17 mW\n0.08 mW0.13 mW0.21 mW0.25 mW0.29 mW0.34 mW0.41 mW0.49 mW0.89 mW1.70 mW3.31 mW\n0.09 mW0.15 mWAB\nCEωdωs ωsωd\n7.72 7.74 7.76-30-20-100\nFrequency (GHz)Drive51 0 1 5024Signal (μW)\nPd (mW)\n-11.425 -11.420 -11.415 -11.410-40-20020406080100120140Reflection (dB)\nΔsd (MHz)0123456-60-40-200204060 \n Reflection (dB)\nDrive power (mW)\n01234560.00.51.01.52.0\nDrive power (mW)\n Linewidth (kHz)D\nA- +A A- +A\nFigure 4. Enhanced magnomechanical coupling in the triply resonant system. (A) Magnomechanically induced\ntransparency (MMIT) signal for a red-detuned drive at various driving powers. (B) Magnomechanically induced absorption\n(MMIA) and magnomechanical parametric gain (MMPA) signal for a blue-detuned drive at various driving powers. (C)The\nlinewidth of the magnomechanically induced resonance as a function of the drive power. (D) Magnomechanical-interaction-\nmodi\fed on-resonance re\rectivity of the hybrid mode as a function of the drive power. Shaded area indicates the instable\nregime. Inset: Measured power of the Stokes sideband of the driving signal. The threshold behavior indicates the onset of\nphonon lasing. (E)Magnomechanical parametric gain as a function of the drive power in an over-coupled hybrid system. Inset:\nMeasured re\rection spectrum that shows a 25-dB gain. In the main panels of (C){(E), blue circles (red squares) are for the\nblue (red) detuning, and solid lines are theoretical calculations." }, { "title": "2107.07939v2.Influence_of_inter_sublattice_coupling_on_the_terahertz_nutation_spin_dynamics_in_antiferromagnets.pdf", "content": "Influence of inter-sublattice coupling on the terahertz nutation spin dynamics in\nantiferromagnets\nRitwik Mondal1;2\u0003and Peter M. 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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M.\nOppeneer, A.Kirilyuk, andV.Kapaklis,Phys.Rev.Mater.\n4, 104418 (2020)." }, { "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 iзацiюферим��гн 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": "1110.4905v1.Exchange_spring_behavior_in_bimagnetic_CoFe2O4_CoFe2_nanocomposite.pdf", "content": "1 Exchange-spring behavior in bimagnetic \nCoFe 2O4/CoFe 2 nanocomposite \nLeite, G. C. P.1, Chagas, E. F. 1, Pereira, R. 1, Prado, R. J. 1, Terezo, A. J. 2 , \nAlzamora, M. 3, and Baggio-Saitovitch, E. 3 \n1Instituto de Física , Universidade Federal de Mato Grosso, 78060-900, Cuiabá-\nMT, Brazil 2Departamento de Química, Universidade Federal do Ma to Grosso, 78060-900, \nCuiabá-MT, Brazil \n3Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150 Urca. Rio de \nJaneiro, Brazil. \nPhone number: 55 65 3615 8747 \nFax: 55 65 3615 8730 \nEmail address: efchagas@fisica.ufmt.br \n \nAbstract \nIn this work we report a study of the magnetic beha vior of ferrimagnetic oxide CoFe 2O4 and \nferrimagnetic oxide/ferromagnetic metal CoFe 2O4/CoFe 2 nanocomposites. The latter compound is \na good system to study hard ferrimagnet/soft ferrom agnet exchange coupling. Two steps were used \nto synthesize the bimagnetic CoFe 2O4/CoFe 2 nanocomposites: (i) first preparation of CoFe 2O4 \nnanoparticles using the a simple hydrothermal metho d and (ii) second reduction reaction of cobalt \nferrite nanoparticles using activated charcoal in i nert atmosphere and high temperature. The phase \nstructures, particle sizes, morphology, and magneti c properties of CoFe 2O4 nanoparticles have \nbeen investigated by X-Ray diffraction (XRD), Mossb auer spectroscopy (MS), transmission \nelectron microscopy (TEM), and vibrating sample mag netometer (VSM) with applied field up to \n3.0 kOe at room temperature and 50K. The mean diame ter of CoFe 2O4 particles is about 16 nm. \nMossbauer spectra reveal two sites for Fe3+. One si te is related to Fe in an octahedral coordination \nand the other one to the Fe3+ in a tetrahedral coor dination, as expected for a spinel crystal \nstructure of CoFe 2O4. TEM measurements of nanocomposite show the format ion of a thin shell of \nCoFe 2 on the cobalt ferrite and indicate that the nanopa rticles increase to about 100 nm. The \nmagnetization of nanocomposite showed hysteresis lo op that is characteristic of the exchange \nspring systems. A maximum energy product (BH) max of 1.22 MGOe was achieved at room \ntemperature for CoFe 2O4/CoFe 2 nanocomposites, which is about 115% higher than th e value \nobtained for CoFe 2O4 precursor. The exchange-spring interaction and th e enhancement of product \n(BH) max in nanocomposite CoFe 2O4/CoFe 2 have been discussed. \nKeywords: Exchange-Spring, Ferrite, Nanocomposite, (BH) max product, \nCoercivity 2 Introduction \n \nThe figure of merit for a permanent magnet material , the quantity ( BH )max to the ideal hard \nmaterial (rectangular hysteresis loop) is given by /g4666/g1828/g1834 /g4667/g3040/g3028/g3051 /g3404/g4666/uni0032/g2024/g1839/g3020/g4667/g2870. For materials with high \ncoercivity ( HC) the magnetic energy product is limited by the sat uration magnetization ( MS). \nAiming to overpass this limitation, and in order to obtain a material with high ( BH)max product, \nKneller and Hawig (1991) [1] proposed a nanocomposi te formed by both hard (high HC) and soft \n(high MS) magnetic materials exchange coupled. These materi als, called exchange spring or \nexchange-hardened magnets, combine the high coercit ivity of the hard material with the high \nsaturation magnetization of the soft material, maki ng possible the increase of the ( BH )max product \nof the nanocomposite when compared with any individ ual phase that form the nanocomposite. [1-\n6]. \nThe increase of the MS is caused by the exchange coupling between grains of nanometer size. \nKneller and Hawig [1] derived a relationship that p redicts how to reach a significant remanence \nenhancement using the microstructural and magnetic properties of this new kind of material, as the \ndistribution of soft and hard magnetic phases and t he fraction of soft magnetic phase, indicating \nthe possibility of developing nanostructured perman ent magnetic materials. \nAccording to the exchange spring model of Kneller a nd Hawig, the critical dimension ( bcm ) for the \nm-phase (soft material) depends on the magnetic cou pling strength of the soft phase Am and the \nmagnetic anisotropy of the hard phase Kh, according to the following equation: \n/g1854/g3030/g3040 /g3404/g2024/g4672/g3002/g3288\n/g2870/g3012/g3283/g4673/g2869/g2870/g3415\n equation (1). \nTo obtain a sufficiently strong exchange coupling, the grain size of the soft phase must be smaller \nthan 2 bcm . In a general way, a good magnetic coupling of the hard and soft components is achieved \nin materials with grain sizes of about 10–20 nm [7] , the approximate value of the domain wall \nwidth in the hard magnetic materials. \nCobalt ferrite, CoFe 2O4, is a hard ferrimagnetic material that has interes ting properties like high HC \n[8, 9] moderate MS [10, 11], high chemical stability, wear resistance , electrical insulation and \nthermal chemical reduction [12, 13]. The latter pro perty allows the transformation of CoFe 2O4 in \nCoFe 2 (a soft ferromagnetic material with high MS value of about 230 emu/g [14]) in \nmoderate/high temperature. This property was used b y Cabral et. al. [13] to obtain the \nnanocomposite CoFe 2O4/CoFe 2 and by Scheffe et al. to hydrogen production [12]. Also, the \nCoFe 2O4/CoFe 2 nanostrutured bimagnetic material was formerly stu died as layered thin films by \nJurca et. al. [15] and Viart et. al. [16]. \nIn this work we describe an original process of che mical reduction used for the synthesis of the \nCoFe 2O4/CoFe 2 nanocomposite materials, as well as the magnetic an d structural characterization \nof both precursor and nanocomposite materials. Fina lly, the enhancement obtained for the (BH) max \nproduct of the CoFe 2O4/CoFe 2 nanocomposite compared with that of the CoFe 2O4 precursor is \nreported. 3 Experimental procedure \nSynthesis of CoFe 2O4 \nThe hydrothermal method was used to synthesize coba lt ferrite. This method provides different \nclasses of nanostructurated inorganic materials fro m aqueous solutions, by means of small Teflon \nautoclaves and has a lot of benefits such as: clean product with high degree of crystallinity at a \nrelative low reaction temperature (up to 200ºC). Al l the reagents used in this synthesis are \ncommercially available and were used as received wi thout further purification. An appropriate \namount of analytical-grade ammonium ferrous sulfate ((NH 4)2(Fe)(SO 4)2·6H 2O (0.5 g, 1.28 mmol) \nand sodium citrate Na 3C6H5O7 (0.86 g, 4.72 mmol) was dissolved in 20 ml of ultra pure water and \nstirred together for 30 min at room temperature, th en stoichiometric CoCl 2.6H 2O (0.15 g, 0.64 \nmmol) was added and dissolved, followed by the addi tion of an aqueous solution of 5M NaOH. \nThe molar ratio of Co (II) to Fe (II) in the above system was 1:2. The mixtures were transferred \ninto an autoclave, maintained at 120 °C for 24 h an d then cooled to room temperature naturally. A \nblackish precipitate was separated and several time s washed with ultra pure water and ethanol. \n \nSynthesis CoFe 2O4/CoFe 2 Nanocomposite \n \nTo obtain the nanocomposite we mixed the nanopartic les of cobalt ferrite with activated charcoal \n(carbon) and subjected the mixture to heat treatmen t at 900 °C for 3 hours in inert atmosphere \n(Ar), promoting the following chemical reduction: \n/g1829/g1867/g1832/g1857 /g2870/g1841/g2872/g3397/uni0032/g1829\n/uni2206/g1372/g1829/g1867/g1832/g1857/g2870/g3397/uni0032/g1829/g1841 /g2870 \nThe symbol ∆ indicates that thermal energy is necessary in the process. \nThe similar process was used by Cabral et. al . [13] to obtain the same nanocomposite and by \nScheffe et. al . to produce hydrogen[12]. \nTheoretically, varying the molar ratio between acti vated carbon and cobalt ferrite we can control \nthe formation of CoFe 2 phase in the nanocomposite. However, the process i s difficult to control \ndue the residual oxygen in the inert atmosphere. \nTwo samples were prepared using the process describ ed here: a full and another partially reduced. \nThe molar ratio between activated charcoal and coba lt ferrite was 2:1 and 10:1, to the partially and \nfully reduced samples respectively. \n \nStructural and magnetic measurements \n \nThe crystalline phases of the calcined particles we re identified by the powder X-ray diffraction \n(XRD) patterns of the magnetic nanoparticles were o btained on a Siemens D5005 X-ray \ndiffractometer using Cu-K radiation (0.154178 nm). \nMagnetic measurements were carried out using a VSM (VersaLab Quantum Design) at room \ntemperature and 50K. 57 Fe Mossbauer spectroscopy experiments were performe d in two \ntemperatures, 4.2 and 300 K to CoFe 2O4 samples. 4 The morphology and particle size distribution of th e samples were examined by direct observation \nvia transmission electron microscopy (TEM) (model J EOL-2100, Japan). \nResults and Discussion \nThe XRD analysis of the synthesized powder after ca lcination (figure 1) shows that the final \nproduct is CoFe 2O4 with the expected inverse spinel structure (JCPDS No. 00-022-1086), \npresenting the Fd3m spatial group with a lattice pa rameter a = 8.403Å ± 0.0082 Å. Value close to \nthat is expected for the bulk CoFe 2O4 (a = 8.39570) [17]. The XRD pattern also reveals trace s of \nCo and Co 7Fe 3 crystalline phases (indicated in figure 1). \nFigure 2 shows the diffraction profile obtained for the sample completely reduced. The XRD \nprofile is similar to that to the CoFe 2 (JCPDS No. 03-065-4131), indicating the expected c hemical \nreduction occurred. Due the small quantity of the s ample partially reduced obtained we could not \nperform XRD measurements. \nTo analyze the cation distribution of the precursor compound (CoFe 2O4), Mossbauer spectroscopy \nexperiments at room temperature and 4.2 K were perf ormed, as shown in the figure 3. The \nMossbauer measurements at 4.2 K reveals two sites f or Fe 3+ related to both octahedral and \ntetrahedral coordination, respectively, as expected for the spinel crystal structure of CoFe 2O4 [18]. \nThe morphology and dimension of nanoparticles were analyzed by TEM measurements. The \nmeasurement of the cobalt ferrite sample (precursor material) shows formation of aggregates. This \nresult is expected to samples prepared by hydrother mal method [19, 20]. Figure 4 shows a TEM \nimage of cobalt ferrite particles. The TEM measurem ent reveals that the CoFe 2O4 nanoparticles \nform a polidisperse system with approximately spher ical nanoparticles. The of particle size \ndistribution indicates that ferrite cobalt particle s have mean diameter of 16 nm and the standard \ndeviation of about 4.9 nm. The particle size histog ram obtained by TEM measurements of the \ncobalt ferrite sample is shown in figure 5. \nThe TEM measurements of the nanocomposite (CoFe 2O4/CoFe 2) are shown in figures 6 and 7. In \nfigure 6a one can see there is roughness at the sur face of the nanoparticle. Note that similar \nroughness was not observed at the surface of the pr ecursor material (figure 4). In addition, figure \n6b shows that the superficial material connects the nanoparticles and the most part of this material \nis in the interface of the nanoparticles. In figure 7a one can see that the nanoparticle is composed \nof two parts a big core and a thin shell (thickness about 1.5 nm). Similar pictures are observed to \nother nanoparticles (not shown). As previously ment ioned, the shell does not cover each \nnanoparticle but the aggregates of nanoparticles. W e attribute the core to the CoFe 2O4 (hard \nmaterial) and the shell to the CoFe 2 (soft material). Thus the nanocomposite obtained i s constituted \nof spheres of magnetically hard material in a soft matrix. \nThe inserts in figures 7a and 7b show details of th e interplanar distance of both core and shell, \nrespectively. The interplanar distance observed to the core is about 0.49 nm (insert of figure 7a). \nThis value is the same obtained by Chen et. al [21] to the (111) plane of CoFe 2O4. The insert in \nfigure 7b shows an interplanar distance of about 0. 3 nm, obtained to the shell. But due the small \nthickness of the shell we consider necessary measur ements of high-resolution TEM (HRTEM) to \nmore precise results. 5 TEM analysis indicates that the nanoparticles of na nocomposite are larger than the originals \nnanoparticles, indicating the reduction process inc reases the mean size (diameter) of the \nnanoparticles to about 100 nm. Also, TEM measuremen ts showed that the dimension of the soft \nphase (CoFe 2) is larger than the critical size obtained by equa tion 1 (see figure 6b). Using the \nmagnetic parameters available for CoFe 2O4 and CoFe 2 (Am ~ 1.7 × 10 −11 J/m [22, 23], Kh ~ 2.23 × \n10 5 J/m 3)[24], the calculated critical grain size bcm for the soft CoFe 2 phase is about 20 nm. \nThe cobalt ferrite sample studied in this work has shown coercivity about 1.69 kOe, at room \ntemperature. This result is higher than the coerciv ity obtained by Cabral et. al. (1.32 kOe) [13] but \nlower than those reported by Ding et. al. [8] and Liu et. al. [9] to samples treated by thermal \nmagnetic annealing and mechanical milling, respecti vely. \nThe hysteresis loop at 50K shows a strong increase of coercivity (8.8 kOe) compared with the \nvalue obtained at room temperature (see the figure 8). Similar behavior of coercivity was observed \nby Maaz et. al. [25] and Gopalan et. al. [26]. Another effect observed by theses authors an d also \nobserved in this work is the increase of the remane nce ratio (M r/M S). The saturation magnetization \n(MS) and remanent magnetization ( Mr) obtained here were, respectively, 445 emu/cm 3 (82 emu/g) \nand 181 emu/cm 3 (33 emu/g) at room temperature, while at 50 K were 477 and 323 emu/cm 3 (88 \nand 60 emu/g). These values indicate an increase fo r remanence ratio (M r/M S), from 0.42 to 0.68 , \nwhen the temperature is decreased from 300 K to 50 K. In these results there are two important \nfacts: first the increase of M r/M S value; and second, the M r/M S value obtained at room temperature \nis close to the theoretical value expected (0.5) to non interacting single domain particles with \nuniaxial anisotropy [27] even the cobalt ferrite ha s a cubic structure. Kodama [28] attribute the \nexistence of an effective uniaxial anisotropy in ma gnetic nanoparticles to the surface effect. The \nstrong anisotropy that produces a high coercivity c an also caused by surface effect [28]. Golapan \net. al. [26] suggest that the increase in the value of the Mr/M S ratio is associated with an enhanced \nof cubic anisotropy contribution at lower temperatu re. \nFigure 9 shows the hysteresis loop of the sample pa rtially reduced (CoFe 2O4/CoFe 2) at room \ntemperature and 50K. The hysteresis curves of the n anocomposite can be described by a single-\nshaped loop (no steps in the loop) similar to that of a single phase indicating that magnetization of \nboth phases reverses cooperatively. \nThe same behavior observed to coercivity for the co balt ferrite was also seen for the \nnanocomposite. The coercivity increased from 1.34 k Oe (at 300K) to 6.0 kOe (at 50K). This \nenormous increase of coercivity deserves more inves tigation. \nThe MS obtained at room temperature was about 146 emu/g , a value is higher than the MS obtained \nfor precursor material and lower than the expected for pure CoFe 2 (230 emu/g ) [14]. \nThe CoFe 2O4/CoFe 2 nanocomposites demonstrate inter-phase exchange co upling between the \nmagnetic hard phase and the magnetic soft phase, wh ich lead to magnets with improved energy \nproducts. We obtained an energy product (BH) max of 1.22 MGOe to the nanocomposite. This value \nis about 115% higher than the value obtained for Co Fe 2O4. To room temperature we obtained \n0.568 MGOe to the product (BH) max , assuming the theoretical density for CoFe 2O4 [29]. The value \nof (BH) max to the nanocomposite is higher than the best value obtained by Cabral et. al. (0.63 \nMGOe) to same nanocomposite (but with different mol ar ratio). Also, the precursor sample 6 prepared in this work showed higher coercivity and saturation magnetization than the precursor \nsample of Cabral et. al .. This observation suggest that the (BH) max product depends of the \nmagnetic properties of precursor material. \nConsidering the nanocomposite formed only by the mi xture of CoFe 2O4 and CoFe 2, we expect that \nthe value of M S is the sum of individual saturation magnetization of these two compounds. \nUsing the values of M S = 230 emu/g to CoFe 2 [14] and 82 emu/g to cobalt ferrite (result of this \nwork), the saturation magnetization of the nanocomp osite (146 emu/g ) suggests that content of \nCoFe 2 in the nanocomposite is about 40% and that of CoFe 2O4 60 %. \nTo better visualization the improvement obtained in magnetic properties, figure 10 show both of \nhysteresis curve of the nanocomposite CoFe 2O4/CoFe 2 and cobalt ferrite at room temperature. The \nsmall decrease of coercivity and the increase of Mr and MS are expected. These behaviors can be \nqualitatively explained by the simple one-dimension al model proposed by Kneller and Hawig. \nConclusion \nWe synthesize nanocomposite of hard ferrimagnetic C oFe 2O4 and soft ferromagnetic CoFe 2 with \nexchange spring behavior at room temperature. This assertion is confirmed by the hysteresis curve \nof the nanocomposite, which do not show steps in th e loop. The thermal treatment at 900 °C used \nin the synthesis method increases the mean size of nanoparticles to about 100 nm (indicated by \nTEM measurements). However the thermogravimetric an alysis (not shown) indicates that the \nsimilar treatment can be used at temperature about 600°C, increasing the time. \nThe chemical reduction process described in this wo rk is a good pathway to obtain CoFe 2O4/CoFe 2 \nwith exchange spring behavior, but the residual oxy gen in the argon commercial gas makes \ndifficult the control the CoFe 2 molar ratio in the nanocomposite. \nThe magnetic energy product was greatly improved in the nanocomposite when compared with the \nferrite precursor. However, other studies show that the coercivity of this precursor material can be \nincreased by thermal annealing, thermal and magneti c annealing or mechanical milling. This \nincrease of coercivity may also improve the magneti c energy product of the nanocomposite, but \nthis assumption deserves more investigation. \nAcknowledgments \nThis work has been supported by Brazilian funding a gency CAPES (PROCAD-NF 2233/2008). \nThe authors would like to thank the LME/LNLS for te chnical support during electron microscopy \nwork. 7 \nFigure 1 – XRD diffraction patterns of the CoFe 2O4. Rietveld fits (solid line) are displayed. \n \nFigure 2 - XRD diffraction patterns of the CoFe 2 produced by reduction reaction of cobalt ferrite \nnanoparticles blended with activated charcoal in th e molar ratio 1:10. \n8 \nFigure 3 – Mossbauer spectra of CoFe 2O4 at room temperature (RT) and 4,2 K (He). \n \nFigure 4 - Transmission electron microscopy of as-p repared CoFe 2O4 by hydrothermal method. \n9 \nFigure 5– Histogram of the particle size distributi on, fitted by a log normal distribution (solid line) . \nParticles have mean diameters of 16 nm. \n \nFigure 6 – Transmission electron microscopy of nano composite CoFe 2O4/CoFe 2. View of a) one \nnanoparticle and b) two nanoparticles. \na b 10 \nFigure 7– TEM measurements of nanocomposite CoFe 2O4/CoFe 2. (A) Show the an interplanar \ndistance of CoFe 2O4 the insert show details of marked area. (B) Show t he an interplanar distance \nof CoFe 2 the insert show details of marked area. \n \n0.3 nm 11 Figure 8 – Hysteresis loops CoFe 2O4 nanoparticles at room temperature (300 K) and 50 K at \nmaximum applied field of 30kOe. \n \nFigure 9 - Hysteresis loops for CoFe 2O4/CoFe 2 nanocomposites at room temperature (300 K) and \n50 K at maximum applied field of 20 kOe \n \nFigure 10 – Hysteresis loops to CoFe 2O4/CoFe 2 and CoFe 2O4 nanocomposites both at room \ntemperature (300 K). \n \nReference \n \n[1] E.F. Kneller, R. Hawig, The Exchange-Spring Mag net: A New Material Principle for \nPermanent Magnets, Journal of Magnetism and Magneti c Materials, 27 (1991). \n[2] L. Withanawasam, A.S. Murphy, G.C. Hadjipanayis , R.F. Krause, Nanocomposite \nR2Fe14B/Fe exchange coupled magnets, J. Appl. Phys, 76 (1994) 7065-7067. \n12 [3] T. Schrefl, J. Fidler, H. Kronmüller, Remanence and coercivity in isotropic nanocrystalline \npermanent magnets, Phys. Rev. B, 49 (1994) 6100-611 0. \n[4] R. 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Zhang, Chara cterizing the magnetic anisotropy constant of \nspinel cobalt ferrite nanoparticles, Appl. Phys. Le tt., 76 (2000). \n[25] K. Maaz, A. Mumtaz, S.K. Hasanain, A. Ceylan, Synthesis and magnetic properties of cobalt \nferrite (CoFe2O4) nanoparticles prepared by wet che mical route, Journal of Magnetism and \nMagnetic Materials, 308 (2007) 289-295. \n[26] E.V. Gopalan, I.A. Al-Omari, D.S. Kumar, Y. Yo shida, P.A. Joy, M.R. Anantharaman, \nInverse magnetocaloric effect in sol–gel derived na nosized cobalt ferrite, Applied Phisics A, 99 \n(2010) 497-503. \n[27] T. Ibusuki, S. Kojima, O. Kitakami, Y. Shimada , Magnetic Anisotropy and Behaviors of Fe \nNanoparticles, IEEE Transactions on Magnetics, 37 ( 2001) 2223-2225. \n[28] R.H. Kodama, Magnetic nanoparticles, Journal o f Magnetism and Magnetic Materials, 200 \n(1999) 359-372. \n[29] A. Rafferty, T. Prescott, D. Brabazon, Sinteri ng behavior of cobalt ferrite ceramic, Ceram Int, \n34 (2008) 15-21. \n \n " }, { "title": "1607.07142v2.Observation_of_an_anisotropic_Dirac_cone_reshaping_and_ferrimagnetic_spin_polarization_in_an_organic_conductor.pdf", "content": "ARTICLE\nReceived 4 Dec 2015 |Accepted 21 Jul 2016 |Published 31 Aug 2016\nObservation of an anisotropic Dirac cone reshaping\nand ferrimagnetic spin polarization in an organic\nconductor\nMichihiro Hirata1,2,w, Kyohei Ishikawa1, Kazuya Miyagawa1, Masafumi Tamura3, Claude Berthier2, Denis Basko4,\nAkito Kobayashi5, Genki Matsuno5& Kazushi Kanoda1\nThe Coulomb interaction among massless Dirac fermions in graphene is unscreened around\nthe isotropic Dirac points, causing a logarithmic velocity renormalization and a conereshaping. In less symmetric Dirac materials possessing anisotropic cones with tilted axes,the Coulomb interaction can provide still more exotic phenomena, which have not beenexperimentally unveiled yet. Here, using site-selective nuclear magnetic resonance, we find anon-uniform cone reshaping accompanied by a bandwidth reduction and an emergentferrimagnetism in tilted Dirac cones that appear on the verge of charge ordering in an organiccompound. Our theoretical analyses based on the renormalization-group approach and theHubbard model show that these observations are the direct consequences of the long-rangeand short-range parts of the Coulomb interaction, respectively. The cone reshaping andthe bandwidth renormalization, as well as the magnetic behaviour revealed here, can beubiquitous and vital for many Dirac materials.DOI: 10.1038/ncomms12666 OPEN\n1Department of Applied Physics, University of T okyo, Bunkyo-ku, Tokyo 113-8656, Japan.2Laboratoire National des Champs Magne ´tiques Intenses, UPR\n3228 CNRS, EMFL, UGA, UPS and INSA, Boite Postale 166, Grenoble, Cedex 9 38042, France.3Department of Physics, Faculty of Science and T echnology,\nT okyo University of Science, Noda, Chiba 278-8510, Japan.4Universite ´Grenoble Alpes and CNRS, Laboratoire de Physique et Mode ´lisation des Milieux\nCondense ´s UMR 5493, 25 rue des Martyrs, Grenoble 38042, France.5Department of Physics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan.\nwPresent address: Institute for Materials Research, T ohoku University, Aoba-ku, Sendai 980-8577, Japan. Correspondence and requests for material s should\nbe addressed to M.H. (email: michihiro_hirata@imr.tohoku.ac.jp) or to K.K. (email: kanoda@ap.t.u-tokyo.ac.jp).\nNATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunications 1Dirac materials1are a novel class of solid-state systems, in\nwhich the low-energy electronic excitations are described\nby pseudo-relativistic massless Dirac fermions (DFs) with\nlinear energy dispersion around the Fermi energy EF. Triggered\nby the studies in two-dimensional (2D) graphene2and the surface\nof three-dimensional topological insulators3, extended now to\nthree-dimensional Weyl and Dirac semimetals with strong\nspin–orbit coupling4–6, many intriguing properties of DFs have\nbeen revealed and have constituted active topics in modern\ncondensed matter physics. The role of Coulomb interaction is one\nof such issues of particular interest7–17. For instance, in\ncharge-neutral 2D massless DF systems, composed of two\ngapless points at EF, the long-range (LR) part of the Coulomb\npotential V(q)(q: wave vector) is unscreened owing to the\nvanishing density of states (DOS) at EF. Consequently, the LR\ncharacter of the interaction ( V(q)p1/|q|) is preserved at low\nenergy, which couples to the fermionic excitations and induces a\nlogarithmic correction to the Fermi velocity vFand associated\nphysical quantities7–9. The logarithmic velocity renormalization\ninduces a nonlinear reshaping of the cones around each of Dirac\npoints (DPs), as observed in graphene near the charge neutrality\npoint11–14.\nHowever, graphene is a special case of 2D massless DF systems,\nin which isotropic Dirac cones with vertical axes have the DPs at\nparticularly symmetric points on the Brillouin zone boundary2.\nIndeed, theoretical studies have revealed that massless DFs\npossessing anisotropic cones and the DPs at arbitrary k-points\nemerge more generally in a broad class of materials1,18–20.\nA typical example in 2D is the organic-layered compound\na-(BEDT-TTF) 2I3(a-I3) (BEDT-TTF ¼bis(ethylenedithio)-\ntetrathiafulvalene) (Fig. 1a), which has a pair of Dirac cones\noccurring at two distinct points ( ±kD) in the 2D first Brillouin\nzone (Fig. 1c)20–30. The electronic structure of a-I3, described on\nthe base of molecular orbitals as usual in this type of\ncompounds31, is rather involved compared with graphene due\nto the presence of four sites per unit cell (Fig. 1b) with anisotropic\nhopping amplitudes32,33. The system has only the inversion\nsymmetry32–34, which, in conjunction with the anisotropic\nhopping, brings about a tilt of Dirac cones and drives the\n2D DPs away from high crystallographic symmetry\npositions20,21,23,25,35,36(Fig. 1c). A remarkable feature is that,\nbecause of the 3/4-filled nature of the electronic bands23–26,33, the\ntwo gapless points are anchored at EFby this band filling in a-I3.\nAnother issue of great physical interest in a-I3in terms of the\nCoulomb interaction problems is that, within the pressure–\ntemperature ( P–T) phase diagram (Fig. 1g)32,37–41, the 2D\nmassless DF phase appears in the vicinity of an insulating\nphase with charge order, as first pointed out by transport\nmeasurements22. This contrasts with the case of graphene, in\nwhich no phase transitions have been observed at least in the\nabsence of a quantizing magnetic field10. The charge-ordered\nphase in a-I3, which is induced by the strong short-range (SR)\nelectron correlations in this 3/4-filled system40,41, is suppressed\nwhen applying a Pabove a critical value of PCE1.2 GPa (Fig. 1g)\nand turns into the 2D massless DF phase22,39. Once the high- P\nphase is reached, the Dirac cones become stable against further\npressurization; in fact, the gapless point is fixed at EFon varying\nthe hopping integrals in a finite range by virtue of the 3/4-filled\nnature of the electronic bands, as revealed by band-structure\ncalculations23,25,26,33,42–45. The presence of such a phase\ntransition in this system potentially offers the possibility to test\nthe impact of the SR electron correlations on the behaviours of\n2D massless DFs. Moreover, the tilt of anisotropic Dirac cones36\ncoupled with the SR and LR parts of the Coulomb interactionopens new possibilities in the physics of 2D massless DFs. For\ninstance, it is predicted to bring about a non-uniform reshapingof titled cones46, novel non-Fermi liquid behaviours near the\nquantum critical point16,17, where two DPs merge47,48, enhanced\nshot noise for quantum transport49and anomalous charge/spin\ntextures inside the unit cell48,50. Studying the electronic structures\nand the role of the Coulomb interaction in pressurized a-I3,\nwhich remains unclear up to date, is thus of primary importance\nto understand the various effects of the Coulomb interaction in\n2D massless DFs.\nIn this article, we focus on the 2D massless DF phase in a-I3\nemerging under a hydrostatic pressure ( P4PC) and present\nexperimental evidence for interaction effects of massless DFs.\nEmploying site-selective nuclear magnetic resonance (NMR),\nwe uncover three distinct interaction phenomena induced by\nthe electron–electron Coulomb interaction. First, NMR-shift\nmeasurements in conjunction with renormalization-group (RG)\nanalyses reveal a T-driven cone reshaping around each of the DPs\ndue to the LR part of the Coulomb interaction. Because of this\nreshaping, tilted cones become effectively isotropic at low\nenergies. Second, quantitative RG analyses establish that the best\nfit to the data inevitably requires a strong bandwidth reduction\ninherent to the SR electron correlation, as often discussed in\nstrongly correlated materials. Finally, an anomalous ferrimagnetic\nspin polarization is observed, which is accounted for by the onsite\nCoulomb repulsion, as revealed by a simulation based on the\nHubbard model presented here. These experimental and\ntheoretical investigations demonstrate that a-I3under Pis an\nintrinsically interacting 2D massless DF system, in which both the\nLR and SR parts of the Coulomb interaction strongly influence\nthe electronic behaviours.\nResults\nBasic principles to probe tilted Dirac cones . Our strategy to\ninvestigate tilted Dirac cones in a-I3is as follows. The crystal\nstructure of a-I3has a 2D unit cell with four molecular sites\n(dubbed sites A,A’(¼A),BandC), each of which constitutes a\nsublattice in the crystalline ab-plane (Fig. 1a,b). The four\nmolecular orbitals on these sites form a pair of tilted Dirac cones\nnear EF(Fig. 1c). Around the gapless point at EF, a very unique\nsituation is realized where the Bloch state has different weights in\nthe amplitudes of the four molecular orbitals. The band-structure\ncalculation25revealed that these weights, dubbed site-spectral\nweights hereafter, show anisotropic kdependence around each of\n2D DPs with a clear contrast between non-equivalent sites. The\ncorresponding site-spectral weight for the sublattice j¼A(¼A’),\nBandCaround the DP at kD,nz\njqðÞ(equation (11)), is shown in\nFig. 1d–f, where q¼(qx,qy) is defined as q¼k/C0kDandz¼±is\nthe band index (Fig. 1c). (For details, see Methods.) Notably, the\nanisotropy of the site-spectral weights makes a particular\ndistinction between the site Band C. Namely, the Bloch\nelectrons with large vF(in the steep slope of the tilted cones)\nhave a large weight on the B-site wavefunction, whereas the Bloch\nstates with small vF(on the opposite side of the cones in the\ngentle slope) have a large weight on the C-site wavefunction\n(Fig. 1e,f and Supplementary Fig. 1a,c). Thus, if one probes the\nlocal electronic states on sites BandCseparately by means of a\nsite-selective measurement, it is possible to reveal the electronic\nnature of the Bloch states in the steep part and the gentle part of\nthe tilted Dirac cones individually. Taking advantage of this\nfeature, we performed a site-selective NMR in this compound to\nseparately elucidate the electronic states in the two slopes of the\nDirac cones. Specifically, the Knight shift, derived from the NMR\nline shift measured at a temperature T, is converted into the local\nelectron-spin susceptibility on the site j,wj\nsTðÞ, which is given by a\nthermal average of the site-spectral weight nz\njqðÞaround EF\nsummed over all qfor both electrons ( z¼þ ) and holes ( z¼/C0 ).\nHence, the site-selective NMR in a-I3works as an effectivelyARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666\n2 NATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunicationsq-resolved probe of 2D DFs thermally excited around each of\nDPs. Indeed, the electronic excitations in the steep and gentle\nparts of the Dirac cones can be almost independently probed by\nwB\nsandwC\ns, respectively, as we will demonstrate below.\nNMR observation of tilted Dirac cones . To address the electron\ninteraction issues of 2D massless DFs, we have carried out13C-NMR measurements in a-I3atP¼2.3 GPa ( 4PC; see\nFig. 1g) on the four molecular sites in the unit cell, j¼A,A’\n(¼A),BandC. The two13C nuclei (spin I¼1/2) introduced at\nthe centre of BEDT-TTF (ET) molecules (inset of Fig. 1a) areused for13C NMR, which are known as a sensitive probe of\nelectronic states at EFin this class of compounds51. Figure 2a\nshows the typical NMR spectra observed at a magnetic field of\nP (GPa)a\nBEDT-TTF (ET)\nmolecule\nbc\na\nA′ (= A)B\nC\nbaj = A\nNon-magnetic\ninsulating layerbConducting layers\nc\n–0.50.5 000.5\n–0.5(EF =) 00.2\n–0.2\n–0.4\n–0.6E (eV)\nkx (r.l.u.)ky (r.l.u.)\nkDΓ\nΓ –kD\n X\nY M\nTCO/afii9825-(BEDT-TTF)2I3\nAnomalous metal\n050100150200T (K)\n0 0.5 1.0 1.5 2.0 2.5Charge ordered\ninsulatorMasslessDirac fermionj = A (A′)\nq\nx (r.l.u.)0.012 0–0.012qy (r.l.u.)0.012\n06\n04\n–2\n0.006 –0.006 –0.0122\n–4\n–6E (meV)d\nj = B\nqx (r.l.u.)0.012 0–0.0120.012\n0\n6\n04\n–2\n0.006 –0.006 –0.0122\n–4\n–6e\nj= C\nqx (r.l.u.)0.012 0–0.0120.012\n0\n6\n04\n–2\n0.006 –0.006 –0.0122\n–4\n–6kD1\n0E (meV) E (meV)\nfζ = +\nζ = +\nζ = –\nζ = +\nζ = –ζ = –\ngqy (r.l.u.)\nqy (r.l.u.)kD\nkDζ = –ζ = +NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666 ARTICLE\nNATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunications 3H¼6 T, applied in the crystalline ab-plane. Eight lines are\nobserved in the spectra, associated with the four molecular sites in\nthe unit cell. By changing the field orientation in the crystalline\nab-plane and examining the angular dependence of the line\npositions, the eight lines are to be assigned to two doublets from\nsites BandCand one quartet from the site A(¼A’)37. (The site\nAand A’are not distinguished hereafter). Figure 2b shows the\ntypical angular dependence of the NMR total shift for eachmolecular site, defined as the centre-of-mass position of the\ndoublet or the quartet. The j-site total shift for a given Tand a\nfield-angle c(measured from the crystalline aaxis; Fig. 2c) is\nexpressed as SjT;cðÞ ¼ /C22AjcðÞwj\nsTðÞ þ sjcðÞ, where the first term is\nthe conduction-electron term (Knight shift) and the second term\nstands for the core-electron contribution (chemical shift). Note\nthat the so-called hyperfine coupling constant, /C22AjcðÞ, and sj(c)\nare strongly c-dependent (with little TandPdependence), while\n02468/afii9851s j (10–5 /afii9839B/kOe) d/afii9851s j /dT (10–4 /afii9839B/kOe)\n0 50 100 150 200 250 30000.20.40.60.81.0\nT (K)c\nab\nH\n0 200 400 600 800\n0 30 60 90 120 150 180Intensity (arb. units)\nShift from TMS (p.p.m.)Shift from TMS (102 p.p.m.)\nAngle /afii9820 (degree)02468\nE0DOSkBTEDOSkBTa\nbd\ne\n0A (= A′)\nBC\nBC\nA (= A′)c\n300 K\n200 K\n110 K\n70 K\n50 K\n40 K\n20 K\n10 K\n3K/afii9825-(BEDT-TTF)2I313C NMR\n13C NMR T = 260 K\nP = 2.3 GPa/afii9825-(BEDT-TTF)2I3\nP = 2.3 GPa ( /afii9820= 60°)\nP = 2.3 GPa13C NMR\nI\nIITC\nflex\nTBflexT A(A′)\nflex/afii9820\nFigure 2 | Site-selective NMR probes anisotropic tilted Dirac cones. (a) T emperature, T, dependence of the13C-NMR spectra measured under\nP¼2.3 GPa at the magnetic field of H¼6 T applied in the crystalline ab-plane (for the direction c¼60o). Symbols denote the13C lines associated to the\nthree non-equivalent sites in the unit cell: j¼A(¼A’) (circles), B(triangles) and C(squares). ( b) Field angular dependence of the centre-of-mass positions\nof the corresponding13C lines for each sublattice. Lines indicate the sinusoidal fitting curves. ( c) Definition of the field-angle cin the ab-plane, measured\nfrom the aaxis. ( d) Local electron-spin susceptibility wj\nsTðÞplotted against T, which are determined from the spectra measured at c¼60ofor the site\nA(¼A’) and c¼120ofor the site BandC. Lines are the guide to the eyes. ( e) The Tdependence of the first derivative d wj\ns/dT. The inflection point, Tj\nflex,\n(TA\nflexE50 K, TB\nflexE120 K and TC\nflexE60 K) is indicated by arrows and vertical dashed lines. Lines stand for the guides to the eyes. Inset: Schematic\nillustrations of thermal excitations of electron–hole pairs around the gapless point at EF(¼0) for high T(T4Tj\nflex) (I) and low T(ToTj\nflex) (II).\nFigure 1 | Tilted Dirac cones in a-(BEDT-TTF) 2I3.(a) Layered structure of a-(BEDT-TTF) 2I3(a-I3), with alternatingly stacked BEDT-TTF (ET) and triiodide\n(I3) layers. Balls-and-stick diagram represents the molecular structures, where sticks indicate bonds; red, blue and green balls stand for carbon atom s; grey\nballs are sulphur atoms; and big purple balls indicate iodine atoms. One electron per two ETs is donated to a I 3molecule due to charge transfer, constituting\na quasi-2D (hole) conducting system in (ET) 2þlayers and non-magnetic insulting I 3/C0layers. Inset of ( a): The structure of a ET molecule.13C isotopes,\nsubstituted for13C-NMR, are indicated by arrows. ( b) 2D unit cell with four ET sites in the ab-plane, distinguished as the site j¼A(¼A’) (blue), B(green)\nand C(red). Same colour correspondence as in a. The dashed rectangle indicates the unit cell, and crosses stand for the inversion centre. ( c) Electronic\nband structure of a-I3at high pressures ( P4PC; see g)25. As the band is 3/4-filled owing to the charge transfer, the Fermi energy EF(¼0) is present\nbetween the first ( z¼þ ) and second ( z¼/C0 ) bands from the top. Gapless points appear at EFand locate at ±kDin the first Brillouin zone, around which a\npair of tilted Dirac cones are visible (indicated by circles). Wave vectors ( kxandky) are given in the r.l.u. ( d–f) The calculated titled Dirac cone around EF\n(¼0) plotted as a function of the wave vector q¼k/C0kD¼(qx,qy). The label bar stands for the size of the site-spectral weight, nz\njqðÞ, for the\nnon-equivalent site j¼A(¼A’)(d),B(e) and C(f)25(equation (11) in Methods). The magnitude is normalized to the maximum value of nz\njqðÞforj¼B.\n(g) The pressure–temperature ( P–T) phase diagram. The charge ordering temperature TCO(squares) is determined from the electrical resistance ( R)\nmeasurements39, defined as the maximum of /C0d(lnR)/T. Error bars follow those provided in ref. 39. At T¼0, the phase boundary locates at PCE1.2 GPa\nas determined from a linear extrapolation (solid line), corresponding to d TCO/dPE–110 KGPa/C01. r.l.u., reciprocal lattice unit.ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666\n4 NATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunicationsthe susceptibility wj\nsTðÞis isotropic in this compound52. Because\nthe electronic excitations around the gapless point (at EF) are\nvanishingly small at low temperatures, the total shift Sjat the\nlowest T(¼3 K in the present experiment) is expected to provide\nthe chemical shift term sj. Thus, subtracting sj(c) from SjT;cðÞ\nand employing the value of the hyperfine coupling constant /C22AjcðÞ\nreported at ambient pressure37, the total shift SjT;cðÞ is\nconverted into the local electron-spin susceptibility wj\nsTðÞ (for\ndetails, see Methods).\nFigure 2d shows the temperature dependence of wj\nsTðÞat the\nsitej¼A,Band C, which arises from the inter-band thermal\n(electron–hole) excitations across the gapless point at EFand is\nproportional to the kBTaverage of the j-site DOS around EF:\nwj\nsTðÞ / DjEðÞ/C10/C11\nkBT. Decreasing from T¼300 K, wj\nsTðÞexhibits\nT-independent features down to TE200 K, followed by a rapid\ndecrease with a clear difference in the size of the susceptibility\nbetween non-equivalent sites, wC\ns4wA\ns4wB\ns(see also\nSupplementary Fig. 2), and finally becomes vanishingly small at\nall sites. The observation, in particular the crossover from\nwj\nsBconst. to wj\nsB0 on cooling, indicates that there is an\nenergy-independent large Dj(E) at high energies (inset I of\nFig. 2e) and a vanishingly small Dj(E) at low energies around EF\n(inset II of Fig. 2e). This is consistent with the band-structure\ncalculations24,25,50, where a flat DOS is predicted at high energies\nabove the van-Hove singularity (locating at about 12 meV off EF)\nand linear energy dependence is suggested around the\nband-crossing point at EF. Note that Dj(E) around the gapless\npoint ( E¼EF¼0) is given by a qsummation of the site-spectral\nweight nz\njqðÞ(Fig. 1d–f) at a given energy Efor the band z\n(equation (14)). Then, the anisotropic q-dependence of nz\njqðÞ\naround the DPs of tilted Dirac cones (Supplementary Fig. 1) is\nexpected to bring about DC(E)4DA(E)4DB(E) at low energy.\nThis is in excellent agreement with the observed relation\nwC\ns4wA\ns4wB\nsbelow TE200 K (Fig. 2d) and is the direct\nconsequence of the fact that the site Band Cselectively probes\nthe steep and gentle slopes of titled cones, respectively, consistent\nwith the prediction of the effective tight-binding (TB) modelgiven in ref. 25. All these observations demonstrate the existence\nof tilted Dirac cones with E\nFlocated around the gapless point.\nFermi velocity renormalization . However, the strong nonlinear\ntemperature dependence in wj\nsTðÞ below TE150 K does not\ncomply with the expectation of TB calculations, which leads to\nwj\nspTat low temperatures25,50. To better visualize this point, we\nplot the first derivative of the susceptibility (d wj\ns/dT), as shown in\nFig. 2e. With decreasing T,dwj\ns/dTexhibits a peak at TB\nflexE120 K\nfor the site B, and at TA;C\nflexE50–60 K for the site AandC, and then\ndrops continuously to zero at all sites towards low temperatures.\nThese features are in striking contrast to the TB calculation25,\nwhere d wj\ns/dTincreases on cooling but saturates at low T\n(Supplementary Fig. 3d–f). Indeed, wj\nsTðÞvaries almost quadratic\ninTat all sites below the inflection point ( Tj\nflex), which suggests a\nnonlinear suppression of Dj(E) around EF(¼0) in an energy\nrange of DEj/C25kBTj\nflex. As the total DOS, D(E), is proportional\nto the inverse square of vFin 2D massless DF systems for the\nnon-interacting case DEðÞ ¼ Ejj=p‘2v2\nF(refs 2,20), a suppression\nof DOS in turn corresponds to an enhancement of vF. Thus, the\nobserved peak structure in d wj\ns/dTstrongly indicates that a\nT-driven renormalization of vFgrows below TETj\nflex. The most\nprobable origin of this effect is the LR part of the Coulomb\ninteraction between electrons, which is unscreened at EFin\ncharge-neutral massless DF systems and is known to cause a\nlogarithmic correction to vFeither driven by tuning carrier\ndensities1,7–13or temperatures9,46. We recall, however, the value\nofTj\nflexis twice higher for the site B(TB\nflexE120 K) than for thesiteAand C(TA;C\nflexE50–60 K). At first glance, this may add an\nextra complication to the data interpretation but in fact turns out\nto be a direct consequence of the anisotropy of the site-spectral\nweight nz\njqðÞ(Fig. 1d–f) and the tilt of Dirac cones, as we shall\nsee below.\nRenormalization-group analyses . To further understand the\nnonlinear temperature dependence of wj\nsTðÞat each site, we have\nexamined the self-energy correction effect due to the LR Coulomb\ninteraction. For this, we employed a RG approach based on an\neffective Hamiltonian near the gapless point20,25, whose\nenergy-momentum dispersion is given by\nE/C6qðÞ ¼ ‘w0/C1q/C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nv2\nxq2xþv2\nyq2yq /C16/C17\n; ð1Þ\nwhere w0¼(w0x,w0y) and v¼(vx,vy) are velocities reflecting the\ntilt and anisotropy of the cone, respectively (for details, see\nMethods). At the one-loop level, the self-energy correction leads\nto a renormalization of vbut does not affect w0(Supplementary\nFig. 4a)46. The RG flow of vis expressed as\n1\nvxdvx\ndl¼8\np2NZ2p\n0dj\n2p2 cos2jFgj/C0/C1\n;\n1\nvydvy\ndl¼8\np2NZ2p\n0dj\n2p2 sin2jFgj/C0/C1\n;ð2Þ\nwhere N¼4 is the number of fermion species, corresponding to\ntwo DPs and two spin projections, q¼q( cos j, sin j)i s\nmeasured from kD,l¼ln(L/q) is the momentum scale,\nL(¼0.667 Å/C01) is a momentum cutoff of the size of the\ninverse lattice constant33and is circular around the DP,\ngj¼2pe2N=ð16effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi\nv2\nxsin2jþv2\nycos2jq\nÞis the coupling, F(gj)\nhas the form Fgj/C0/C1\n¼ð /C0 p=2þgjþarccos gj=ffiffiffiffiffiffiffiffiffiffiffiffi ffi\n1/C0g2\njq\nÞ=gjand\neis the dielectric constant. (Note that equation (2) is obtained in\nthe leading order in 1/ Nassuming N441, which is valid both for\nthe weak and strong Coulomb interaction.)\nAssuming the four velocities given by the effective TB\ncalculation ( wTB\n0and vTB)25as initial velocities at q¼L,w e\nhave calculated the RG correction effects on wj\nsTðÞ(Fig. 3a–c)\nDj(E) (Fig. 4a–c) and the energy spectrum (Fig. 4d–f). (For the\njustifications of employing this TB model as well as the velocities\nwTB\n0andvTBin performing RG calculations, see Methods.) Here,\nwe note that the temperature is used as an explicit scale parameter\nin the calculation of wj\nsTðÞthat determines the RG flow. To get a\nreasonable agreement between the calculation and experiment, a\nphenomenological parameter uis introduced to adjust the\nvelocities at q¼Lsuch that w0\n0¼uwTB\n0andv0¼uvTB. The two\nparameters in the calculation, ( u,e), are then optimized from a\nleast-square fit to the experimental susceptibilities. Good\nagreements are obtained in the fit especially at the site Aand C\n(Fig. 3a–c), which lead ( u,e)E(0.35, 1). (Note that the fitting\nresults are sensitive to the choice of uwhile they are little\ndependent on ein the range eE1–30; for details, see Methods and\nSupplementary Figs 4–7). The calculation demonstrates that the\nnonlinear Tdependence of wj\nsTðÞbelow TETj\nflexcan be properly\nascribed to the logarithmic renormalization of vF. In Fig. 4a–c, the\ncalculated shape of Dj(E) is shown around the gapless point at EF.\nA strong suppression from the E-linear DOS is seen at low\nenergies due to the renormalization. Figure 4d–f, depicts the\ncorresponding energy spectrum around the DP (at kD), where the\ncolours indicate the magnitude of the site-spectral weight, nz\njqðÞ,\nin Fig. 1d–f, respectively. A nonlinear reshaping of the tilted cone\ninduced by the renormalization is clearly visible around the\ngapless point. It should be stressed that a good agreement isNATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666 ARTICLE\nNATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunications 5accomplished only when a small value of u(E0.35) is used. The\nfact that we have uo1 indicates a reduction of the initial\nvelocities or of the hopping amplitudes between the adjacentlattice sites. In conventional strongly correlated materials, the SR\npart of the Coulomb interaction is well known to induce this sort\nof hopping (or bandwidth) renormalization due to the frequency\ndependence of the self-energy53. We believe that the observed\nu-reduction effect in the initial velocities occurs because of this\nself-energy correction due to the SR Coulomb interaction, which\nis not considered implicitly in the original RG calculation.\nRemarkably, the calculation well reproduces the observed\ndifference in the thermal energy scale of the T-driven renorma-\nlization in wj\nsTðÞ,DEj(EkBTj\nflex), at the site Band ( A,C)a s\nindicated by thin arrows in Fig. 3a–c. This distinction stems from\nthe tilt of the Dirac cones and the resultant momentum\ndependence of the energy cutoff daround the DP,\ndj ;zðÞ ¼ ‘vFj;zðÞL, where vFj;zðÞ is the jdependent Fermi\nvelocity in the band z(Supplementary Fig. 1a,b). Namely, the\nenergy cutoff is large for the large- vFDFs, dominantly probed by\nthe site B, while it is small for the small- vFDFs, having a large\nweight on the site C(and A). Hence, the renormalization starts\nfrom a higher TinwB\ns(T) than in wC\ns(T) (and wA\ns(T)), producing\nthe observed energy scale difference DEB4EA,C, consistent\nwith the previous RG calculation of Isobe et al.46. It is also\nworth mentioning that tilted cones become more isotropic at\nlower energies near EFbecause of the non-uniform velocity\nrenormalization around each DP, as reflected in Fig. 4d–f. This is\nbecause the anisotropic term in the Hamiltonian is small\n(vx/vyE1) in a-I3(ref. 25), and we have | w0|oo|v| around the\nDP due to the RG flow (Supplementary Fig. 4a), leading the tilting\nterm ( w0) to be effectively negligible near EF.\nFrom all these, it is concluded that our RG analyses\nappropriately capture many of the essential parts of the\nexperimental results. They constitute experimental evidence for\nthe bandwidth renormalization (the u-reduction effect) due to the\nSR repulsion between electrons as well as the T-driven\nlogarithmic renormalization of vFby the LR part of the Coulomb\ninteraction. Nevertheless, we note that, at low temperatures, the\nagreement is less satisfactory for the site Bcompared with\nthe other sites (Fig. 3c), suggesting the presence of anothercorrelation effect. Indeed, we will clarify this point by a\nlattice-model simulation, as described below.\nEmergent ferrimagnetic spin polarization . The temperature\ndependence of w\nB\ns(T) in the experiment is appreciably stronger\nand more complex than what is predicted by the RG calculation\n(Fig. 3c). Indeed, the experimental wB\ns(T) exhibits an anomalous\nsign change at TE60 K and an upturn with a negative slope\nbelow TE40 K (inset of Fig. 3c), while the RG calculation shows\nmonotonic temperature dependence. The observation of wB\nso0i s\nin sharp contrast to wA\ns40 and wC\ns40 in the experiment (Fig. 5a),\nindicating an emergent ferrimagnetic spin polarization in which\nthe local magnetic field points antiparallel to the applied field at\nthe site Bwhile it is parallel to the field at all other sites (Fig. 6).\nTo further understand this sublattice-scale magnetism, we have\ninvestigated the Hubbard model with an onsite repulsive\n(Hubbard) interaction, U, at a mean-field level within the random\nphase approximation (RPA). For the RPA calculation, we have\nconsidered both the inter-band and intra-band contributions to\nthe spin susceptibility with a wave vector Q¼0(for details,\nsee Methods). Figure 5b presents the calculated temperature\ndependence of the RPA spin susceptibility at the site B. Using\nU¼0.14 eV, the RPA calculation (in particular the inter-band\nterm) clearly reproduces the observed negative spin susceptibility\nfor site B(wB\nso0) at similar temperatures. Moreover, the negative\nsusceptibility appears only at site Bin the calculation\n(Supplementary Fig. 3a–c) in good agreement with the experi-\nment (Fig. 5a and Supplementary Fig. 2). These facts show that0246/afii9851s (10–5 /afii9839B/kOe) /afii9851s (10–5 /afii9839B/kOe) /afii9851s (10–5 /afii9839B/kOe)0246\n0 50 100 1500246\nT (K)0 2 04 06 08 0–0.200.20.4\nT (K)b\nca\nTC\nj = C\nj = Bj = A (A′)13C NMR\n/afii9825-(BEDT-TTF)2I3P = 2.3 GPaflex\nTA(A′)\nflex\nTB\nflexj j j\nFigure 3 | Temperature driven velocity renormalization.\n(a–c) Temperature dependence of the calculated spin susceptibility in the\nRG approach (bold curves) for the non-equivalent site j¼C(a),A(¼A’)\n(b) and B(c) in the unit cell, plotted together with the experimental wj\nsTðÞ\n(symbols). Thin arrows indicate the inflection point, Tj\nflex. As the\nnon-interacting reference in the RG theory, we assumed the effective TB\nmodel of ref. 25. The susceptibilities are calculated as discussed in the text\nby employing the momentum cutoff L¼0.667 Å/C01and the optimum\nparameters ( u,e)¼(0.35, 1), determined from the fit to the data\n(Methods). The dotted lines ( wj\nspT) indicates the expected Tdependence\nfor gapless excitations around the DP in the absence of the RG correction68,\nwhere the bandwidth reduction effect is incorporated (through the\nparameter u). Thick bold arrows represent the suppression of wj\nsTðÞwith\nrespect to wj\nspTdue to the RG flow of the velocities, where the velocities\nflow towards larger ln L=qjjðÞ (longer wavelengths) on decreasing T. That is,\nTplays the role of the flow parameter in the experiment, and the RG flow\nmanifests itself typically in the energy range of DEj/C25kBTj\nflexaround EF. The\ninsets of c: the low- Tclose-up of wB\ns. Error bars stand for the s.e.m.ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666\n6 NATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunicationsthe ferrimagnetic polarization is induced by the onsite Hubbard\ninteraction. The fact that the negative polarization emerges solely\non the site Bmight be relevant to a superexchange-like interac-\ntion between the site Aand A’ (via B) (see Fig. 6). Density\nfunctional calculation24suggests a largest hopping amplitude on\nthis path (b2 in Supplementary Fig. 8), and X-ray and Ramanscattering measurements32,38point to the largest hole density\n(despite the small spin density) at the intermediate site Bin the\nunit cell. Then, if there is an antiferromagnetic (ferrimagnetic)\ncoupling between sites AandB(A’ and B), a large energy gain is\nexpected due to the kinetic energy of electrons, which favours the\nobserved pattern of the ferrimagnetic polarization.\n–5 0 50\n00.20.40.60.81.0Density of states (arb. units) Density of states (arb. units) Density of states (arb. units)–5 0 50.20.40.60.81.0\n–5 0 500.20.40.60.81.0\nE (meV)6\n04\n–22\n–4\n–6E (meV)\n0 0.02 –0.020.020–0.02j = B\nkD1\n06\n04\n–22\n–4\n–6E (meV)\n0 0.02 –0.020.020–0.02j = A (A′)\nkD6\n04\n–22\n–4\n–6E (meV)\n0 0.02 –0.02\nqx (Å–1)\nqx (Å–1)\nqx (Å–1)qy (Å–1)\nqy (Å–1)\nqy (Å–1)0.020–0.02j = C\nkD\ne\nfd\nb\nca\nRGLinear spectrum\nj= Bj = C\nj = A (A′)kBTC\nflex\nkBTB\nflexkBTA(A′)\nflex\nFigure 4 | LR Coulomb-induced titled Dirac cone reshaping. (a–c) The calculated DOS near the gapless point (at E¼EF¼0) plotted as a function of\nenergy Efor the non-equivalent site j¼C(a),A(¼A’) (b) and B(c) in the unit cell. The thin dotted and thick dashed curves are the DOS profiles for the\nlinear spectrum and the RG-corrected DOS, respectively, calculated for the momentum cutoff L¼0.667 Å/C01and optimum fitting parameters\n(u,e)¼(0.35, 1). Note that the bandwidth reduction effect, associated with u¼0.35 (o1), is taken into account in both curves. The thick bold parts in the\nRG-corrected DOS, highlighted around the gapless point, correspond to the energy range ( DEj/C25kBTj\nflex) where the T-driven RG flow is visible in the\nexperiment (also indicated by the horizontal left-right arrows). Vertical thick bold arrows represent the suppression of the DOS due to the LR Coulom b\ninteraction. ( d–f) Calculations of the reshaped tilted Dirac cone (around kD) induced by the RG flow of the velocities, derived from the same parameters as\nina–c. The label bar reflects the site-spectral weight nz\njqðÞaround the cone for the non-equivalent site j¼C(d),A(¼A’) (e) and B(f) in the unit cell\nplotted as a function of the wave vector q¼k/C0kD¼(qx,qy) (in Å/C01). The outer grey cone stands for the linear spectrum in the absence of the LR Coulomb\ninteraction, where the bandwidth reduction effect is considered (corresponding to the E-linear DOS in a–c). (See Methods for details.)NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666 ARTICLE\nNATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunications 7Generally speaking, RPA tends to overestimate the effect of\ncorrelations, as it does not consider the self-energy correction to\nthe energy bands54. In fact, the onsite interaction we use,\nU¼0.14 eV, is chosen smaller than what would be typically used\n(UE0.4 eV) (refs 25,26). This discrepancy brings another support\nthat the self-energy correction due to the SR interaction is of great\nimportance in this system, consistent with the bandwidth\nreduction effect we discussed in the RG calculation. Finally, we\nnote that RPA is unable to reproduce the observed nonlinear\nTdependence of wj\nsTðÞat low temperatures. This is because the\nself-energy correction due to the LR Coulomb interaction, which\nis the main origin of the nonlinear wj\nsTðÞ, is not taken into\naccount in RPA (for details, see Methods and Supplementary\nFig. 3d–f).\nDiscussion\nSo far, we have demonstrated three distinct Coulomb-interaction\nphenomena in the 2D massless DF phase of a-I3, which developsystematically at different temperature scales (or energy scales of\nthermal excitations), as summarised in Table 1. A bandwidth\nrenormalization (or the u-reduction of vF) occurs due to the SR\nCoulomb interaction which appears to exist from room\ntemperature down to lowest T. At temperatures TrTj\nflex(or in\nthe corresponding energy range around the gapless point at EF),\naT-driven logarithmic renormalization of vFand the resultant\nnon-uniform reshaping of the Dirac cones appear, due to the LRpart of the Coulomb interaction. With further decreasing T,\na ferrimagnetic spin polarization shows up in the unit cell because\nof the onsite Coulomb repulsion between electrons.\nFirst, we mention that the observed non-uniform reshaping of\ntitled Dirac cones in a-I\n3should affect other physical observables\nat low temperature or at low magnetic field. As the cones become\neffectively isotropic around each of DPs (Fig. 4d–f), the shape of\nthe cross-section of the cone is different at high energy and close\nto the gapless point, which should cause a T-dependent\nanisotropy of the in-plane electrical conductivity. Another\nexperiment that would be able to see the reshaping is infrared\nspectroscopic measurements, known as a powerful tool to reveal\nthe Landau level (LL) structure in graphene55. In a perpendicular\nmagnetic field ( H>) normal to the 2D plane, the massless DFs in\na-I3exhibit the LL spectrum, Ez;n¼z‘l/C01\nBffiffiffiffiffiffiffiffivxvyp1/C0O2/C0/C13=4ffiffiffiffiffiffiffiffi\n2njjp\n(refs 56,57), where z¼±distinguishes the electron and hole\nbands crossing at the gapless point, n(¼0,±1,±2,?)\nis the LL index, lB¼(‘/eH>)1/2is the magnetic length and\nO¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi\nw0x=vx ðÞ2þw0y=vy/C0/C12q\nis a tilting parameter. As EFlocates\nat the gapless point in a-I3and the n¼0 LL is half-filled,\none may be able to detect, for instance, the dipole\ntransition n¼0-n¼1 in the absorption line at the energy\nDE10¼‘l/C01\nBffiffiffiffiffiffiffiffiffiffi ffi2vxvyp1/C0O2/C0/C13=4. The velocities ( vx,vy) with a\nlogarithmic correction may appear in DE10at low field; vx(Evy)\nwill increase by a factor of four by changing ln( L/q) from 1 to 5\n(Supplementary Fig. 4a) and (1 /C0O2)3/4increases by a factor of\ntwo (from B0.5 toB1.0). One would expect to see this change in\nDE10at low temperatures where the LL broadening, mainly\ncaused by the thermal scattering of carriers in the n¼0L L\n(refs 27,28), becomes sufficiently small.\nTheoretically, the renormalization of the coupling constant of\nthe Coulomb interaction (the RG flow of vF) makes the system\nflow to a weak coupling regime at low energies (or at low Tin the\ncase of a T-driven RG flow as in our experiment)10. As organic\ncompounds are very clean and are little influenced by impurities,\nthis consequence of the RG flow implies that low- Telectron\nE\nqxkDIntra-band\n+ inter-band\n/afii9851B\nRPA < 0Intra-band j = B0.2 0.4 0.6-0.100.10.20.30.4\nj = Bj = A, C\nT/Tj\nFlex0\n0 50 100 150 200–0.100.10.20.30.4Spin susceptibility (arb. units)\nT (K)Hubbard + RPA \n(U = 0.14 eV)ba\n13C NMR\n/afii9825-(BEDT-TTF)2I3/afii9851j\ns(T)//afii9851j\ns(Tj\nFlex)P = 2.3 GPa\nFigure 5 | Emergent negative spin susceptibility at the site B.\n(a) Observed local electron spin susceptibility wj\nsTðÞnormalized to the value\nat the inflection point Tj\nflex,wj\nsTðÞ/wj\ns(Tj\nflex), plotted against the normalized\ntemperature, T/Tj\nflex, for the non-equivalent site j¼A(A’) (circles),\nB(triangles) and C(squares) in the unit cell. Dotted and dashed curves are\nguide to the eyes. ( b) Calculated Tdependence of the B-site-spin\nsusceptibility wB\nsin the RPA based on the Hubbard model. The total\nsusceptibility (solid curve) and the intra-band contribution (dashed curve)\nfor the onsite Hubbard interaction of U¼0.14 eV are shown. Inset:\nIntra/inter-band thermal excitations with the wave vector Q¼0in the\nDirac cone considered for the RPA calculation. The Fermi energy EFlocates\nat the band-crossing gapless point\nj = A\nB\nC\nba\nA′ (= A)BH\nFigure 6 | Ferrimagnetic spin polarization. Schematic illustration of the\nferrimagnetic spin polarization suggested at low temperatures (below 60 K)\nby our site-selective susceptibility measurements. Thick arrows represent\nthe direction of the local magnetic field on the non-equivalent site A(¼A’),\nBand Cin the unit cell, which is opposite to the external field direction\n(H||ain this figure) at the site Bwhile it is parallel at all other sites.ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666\n8 NATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunicationscorrelation effects, if any, would be induced by the SR part of the\nCoulomb interaction, as usually the case in conventional strongly\ncorrelated materials. In fact, the insulating phase possessing\ncharge order22,26,31–41emerges next to the 2D massless DF phase\non the P–Tphase diagram in a-I3(Fig. 1g), suggesting the\nvital role of the SR Coulomb interactions in the massless DF\nphase of this 3/4-filled system, in line with recent mean-field\ncalculations48. In the half-filled system graphene, strong electron\ncorrelations are predicted to stabilize Mott insulators and charge\ndensity waves10. However, the typical experimental conditions\nseem to locate away from these situations1,10,58, and no phase\ntransition has been reported yet. Under a strong magnetic field,\non the other hand, a gapped liquid phase is observed with a\nquantized Hall conductivity at fractional filling factors, stabilized\nby the SR part of the Coulomb interaction10. Similar physics may\noccur in thin films of a-I3as well, where the integer quantum Hall\neffects have recently been observed59.\nTo conclude, our NMR measurements combined with\ntheoretical calculations have demonstrated three T-dependent\nCoulomb interaction effects of 2D massless DFs (Table 1) in\npressurized a-I3(P4PC), having tilted Dirac cones and gapless\npoints fixed at EF. We found that the LR part of the Coulomb\ninteraction, which is unscreened around the gapless DPs, causes a\nT-driven renormalization of the velocity and induces a non-\nuniform reshaping of tilted Dirac cones. Quantitative analyses of\nthe cone reshaping based on the RG approach further necessitates\na large bandwidth reduction due to the SR electron correlation.\nMoreover, we showed that the onsite Coulomb repulsion gives\nrise to a ferrimagnetic spin polarization as unveiled by the\nnumerical calculations using the Hubbard model. These findings\ncan be distinguished from the case of weakly interacting 2D\nmassless DFs in graphene with vertical Dirac cones and are\nconsistent with the emergent correlated phase on the verge of the\nmassless DF phase in the P–Tphase diagram. Continuing this\nstudy to the vicinity of the phase transition at PC(E1.2 GPa)\nwould be of particular interest, which may connect the physics of\nthe massless DFs and conventional strongly correlated materials.\nMethods\nSample preparation .Single crystals of a-(BEDT-TTF) 2I3(a-I3) (ref. 60) with\nthe dimensions of 0.1 /C20.5/C22.0 mm3were synthesized from13C-enriched\nBEDT-TTF (ET) molecules using the conventional electrochemical method.To perform13C-NMR measurements, the central carbon atoms of BEDT-TTF\nmolecules connected by a double bond were 99% enriched by carbon-13 (13C)\nisotopes (inset of Fig. 1a) with a nuclear spin I¼1/2.\nPressurization scheme .A hydrostatic pressure of P¼2.3 GPa was applied to the\nsample using a BeCu/NiCrAl clamp-type pressure cell, with the Daphne 7373 oil as\na pressure medium. At this pressure, the oil locates close to the liquid–solid phasetransition point at room temperature61,62. To avoid applying uniaxial strains to the\nsample, the cell was kept at a sufficiently high temperature ( TE50/C176C) during\nthe pressurization. With decreasing temperature from T¼300–3 K, a pressure\nreduction of DPB0.1 GPa occurs inside the cell61,62. The inner pressure at\nthe lowest Twe measured (3 K) is, however, substantially higher than the\ntransition pressure to the charge-ordered phase at TE0(PCE1.2 GPa;\nsee Fig. 1g)22,30,39–41,63,64, indicating that the P-reduction will not change the\nphysics. Hence, we neglect this effect throughout the paper.13C-NMR measurements .13C-NMR measurements were performed in a-I3in a\nmagnetic field Hof 6 T applied parallel to the crystalline ab-plane (Fig. 1a). To get\nNMR signals, the standard spin-echo techniques were used with a commercially\navailable homodyne spectrometer. Spin-echo signals were recorded at a fixed radio\nfrequency after the conventional spin-echo pulse sequence of tp/2/C0t/C0tp(with\nt¼5–25msa n d tp/2/C0tp¼0.6–1.2 ms) and were converted into the NMR spectra\nvia Fourier transformation. The resonance frequency of the natural abundance13C\nnuclei in TMS (tetramethylsilane (CH 3)4Si) was used as the origin of the NMR\nshift. We note that the present work is targeted at lower Tand higher Pcompared\nwith the earlier NMR studies65,66, which is more suitable for exploring the nature\nthe low- T2D massless DF phase emerging at P4PCon the phase diagram\n(Fig. 1g)22,27–30,39.\nLine assignments of the NMR spectra .The details of line assignments for the\n13C-NMR spectra in this compound are given elsewhere37,65. Here, we shall\ndescribe only the essence. The13C-NMR spectra of a-I3show large temperature\ndependence (Fig. 2a and Supplementary Fig. 9) and field-angle ( c) dependence\n(Supplementary Fig. 10). They have eight13C lines that consist of two doublets\n(from the Band the Cmolecules) and one quartet (from the A(¼A’) molecule)\n(Fig. 2a). The doublet and the quartet are caused by the nuclear dipole–dipole\ninteraction in a ET molecule (between the two13C nuclear magnetic moments\naround the molecular centre; see the inset of Fig. 1a)37,51,65. The NMR total shift Sj\nfor a particular site jis determined from the centre-of-mass position of the\ncorresponding13C lines, which is expressed by a sum of the Knight shift Kj(¼/C22Ajwj\ns)\nand the chemical shift sjterms, SjT;cðÞ ¼ KjT;cðÞ þ sjcðÞ, as mentioned in the\ntext. (The orbital van-Vleck contribution is negligible in ET-based salts because of\nthe low lattice symmetry51.) Both KjT;cðÞ andsj(c) are highly anisotropic in this\nsystem37,65–67, causing clear c-dependence of the spectra. The anisotropy of the\nshift SjT;cðÞ is, however, largely different between the molecule A(A’) and\nmolecules BandC, reflecting the fishbone-like arrangement of ET molecules in the\ncrystalline ab-plane (Fig. 1b). Thus, by rotating the magnetic field Hin this plane,\none can assign the13C lines to the different sublattices37, with the aid of the X-ray\ndiffraction data under pressure33(Supplementary Fig. 10).\nConversion of the NMR shift to the spin susceptibility .Ina-I3, the\nT-dependence curve of the j-site total shift SjT;cðÞ shows a largely distinct feature\nfor different field orientations. For instance, the Tdependence of SAA0ðÞT;cðÞ is\nlarge at c¼60/C176showing a prominent decrease with decreasing T(Supplementary\nFig. 11a), whereas it is small at c¼120owith an increase on cooling\n(Supplementary Fig. 11b). This difference can be accounted for by the anisotropy of\ntheT-independent hyperfine coupling constant /C22AjcðÞ, where /C22AjcðÞcan be either\npositive or negative depending on the value of c, as we will describe below. First,\nwe note that the chemical shift sj(c)i sT-independent51as well as little affected by\nP(see Supplementary Discussion). Thus, the observed Tdependence of the total\nshift Sjcan be ascribed to the T-variation of the Knight shift Kj¼/C22Ajwj\ns. Here,\n/C22Ajis the hyperfine-coupling constant averaged for the two central13C nuclei in the\nmolecule j(Fig. 1a), which is c-dependent reflecting the anisotropy of the coupling\ntensor33,37,66,67. The principal values of the tensor are, however, weakly affected by\nthe change of TandP(Supplementary Figs 12 and 13). Moreover, the spin\nsusceptibility wj\nsis expected to be isotropic in this compound52. These points clearly\nindicate that Kjcan practically be expressed as Kj(T,c)¼/C22AjcðÞwj\nsTðÞ(for details,\nsee Supplementary Discussion).\nNotice that there are no excitations around the gapless DP in the ground state\nof massless DF systems. This means, at the lowest T, the Knight shift Kj(T,c)i s\nexpected to become vanishingly small68, and the total shift SjT;cðÞ resumes to the\nT-independent chemical shift sj(c) (Supplementary Fig. 12). For the site B, there is\na negative slope in the Tdependence of the total shift SjT;cðÞ below TE40 K,\nshowing a small increase of the shift (of B5 p.p.m.) towards lower T(inset of\nSupplementary Fig. 11b). This is to be associated to the ferrimagnetic spin\npolarization, which causes a local magnetic field that points opposite to the external\nfield only at the site B(Fig. 6)42. The effect is, however, negligible at the lowest\nTsince the thermally excited polarization vanishes as T-0 (Fig. 5b and\nSupplementary Fig. 3c). Hence, we fitted to the angular dependence of SjT;cðÞ for\nall sites at the lowest measured temperature ( T¼3 K) and assumed this fitted curveTable 1 | Interaction effects of Dirac fermions in a-(BEDT-TTF) 2I3.\nT(K) Long-range Coulomb Short-range Coulomb Emergent interaction effects\nDE/kBr300 — o-dependence of self-energy Bandwidth renormalization ( u-reduction)\nDE/kBrTj\nflex(E60–120) Self-energy effect due to V(q)p1/|q| — Logarithmic correction to vF\nDE/kBr60 — Onsite Hubbard U Ferrimagnetic spin polarization ( wB\nso0)\nBEDT-TTF, bis(ethylenedithio)-tetrathiafulvalene.\nHere, DE(¼kBT) indicates the energy range around the gapless point at EF, within which the inter-band thermal excitations are effective (see the inset of Fig. 2e). Tj\nflexstands for the peak temperature in\nthe first derivative of the electron spin susceptibility wj\ns(Fig. 2e) at the non-equivalent site j¼A(¼A’),BandCin the unit cell (Fig. 1b), oindicates the frequency and urepresents the phenomenological\nparameter introduced in the RG fit analyses (see the text).NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666 ARTICLE\nNATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunications 9to be the chemical shift value at each angle c. The total shift SjT;cðÞ for a\nparticular site is then converted to Kj(T,c) by subtracting this sj(c). The\nsubtraction is done at a field orientation where the total shift becomes close to themaximum in order to minimize the ambiguity of the chemical shift, namely,\nc¼60/C176for the site A(A’) and c¼120/C176for the site BandC(Supplementary Figs 11\nand 12). The value of the hyperfine coupling constant /C22A\njcðÞis calculated for these\nangles by employing the coupling tensors given at ambient pressure37by means of\nX-ray diffraction data reported at high pressure33. This yields /C22AAA0ðÞc¼60oðÞ ¼ 9.0,\n/C22ABc¼120oðÞ ¼ 6.4 and /C22ACc¼120oðÞ ¼ 9.9 in kOe mB/C01. In terms of these coupling\nconstants, the Knight shift Kj(T,c) is eventually converted to the local electron-spin\nsusceptibility wj\nsTðÞ(Fig. 2d).\nEffective tight-binding model for the tilted Dirac cone .In order to construct\nreasonable arguments for the fitting analyses to the observed wj\nsTðÞ(Fig. 2d), we\nhave introduced a four-band band-structure calculation of ref. 25. The low-energyeffective model based on this calculation shall be used as a non-interacting\nreference to the data analyses in particular in the RG calculation. (For details, see\nMethods: RG calculations). The rationales behind this choice are described here.\nIna-I\n3, it has been shown that the gapless DPs at ±kD(Fig. 1c) are fixed at EF\nfor a certain parameter region in the four-band TB parameter space with and\nwithout finite site potentials23,33,42–45. However, it is difficult to use bare TB\nparameters as adjustable variables in the fitting analyses of the low- Tpart of wj\nsTðÞ\n(in Fig. 2d). This is because the TB calculations lead to linear dispersion around the\nDP21,23, which causes wj\nspT, owing to the excitations around the gapless point at\nEF(ref. 68). Our experiment, on the other hand, exhibits nonlinear T-dependence\ninwj\nsTðÞat all sites below TETj\nflex(Fig. 2d,e) and a negative wB\ns(T) at low\nT(rTB\nflex/2E60 K; see the inset of Figs 3c and 5a). A model calculation based on\nsimple linear dispersion can hardly account for these features. Thus, instead of\nfitting the data with TB parameters, we will use them as a minimal non-interacting\nreference and perform more sophisticated RG calculations based on a continuum\nmodel derived from that reference.\nFor the minimal model in our study, we use the band-structure calculation of\nref. 25, which is practically based on a non-interacting TB model with adjustable\nsite-dependent potentials, associated to the four molecular sites in the unit cell,j¼A(1),A’(2), B(3) and C(4) (Fig. 1b). (This model shall be dubbed the effective\nTB model throughout this study.) Strictly speaking, this model takes into account\nthe electron–electron Coulomb interaction up to the nearest-neighbour terms and,at first glance, appears not to be a non-interacting model. However, as one works\nwithin a mean-field framework, the interaction merely ends up in additional site\npotentials in the expression of the Hamiltonian\n40,41. In this sense, ref. 25 can be\nalso considered as a TB model with adjustable site-dependent potentials from a\npractical point of view. Importantly, it is well known that the presence of thegapless point at E\nFis unaffected by modulations of this kind of site potentials\nwithin a range in this compound19,33,42,43. The chosen values of the site potentials\nin ref. 25, which are given by\nI1¼I2¼1:0964 ;I3¼3:8755 ;I4¼3:8277 in eV ðÞ ð 3Þ\nare within this range and are thus acceptable. Using these potentials, the\nHamiltonian of the effective TB model in ref. 25 can be eventually expressed by a4/C24 matrix, e\nij[i,j¼A(1),A’(2), B(3) and C(4)], whose Fourier transformed\nmatrix elements are given by\neijkðÞ ¼ tijkðÞ þ Iidij; ð4Þ\nwith the kinetic terms\nt11kðÞ ¼ 2ta10cosky; t12kðÞ ¼ ta2þ¼ ta3e/C0iky;\nt13kðÞ ¼ tb2þtb3eikx; t14kðÞ ¼ tb1þtb4eikx;\nt22kðÞ ¼ 2ta10cosky; t23kðÞ ¼ tb3eikyþtb2eikxþkyðÞ;\nt24kðÞ ¼ tb4þtb1eikx; t33kðÞ ¼ 2ta30cosky;\nt34kðÞ ¼ ta1þta1e/C0iky; t44kðÞ ¼ 2ta40cosky;\ntijkðÞ ¼ tjikðÞ/C2/C3/C3;ð5Þ\nThe hopping integrals are better determined by ab initio calculations such that\nthe resultant electronic bands become compatible with experimental observations\nin this system. For this, we employed the hopping integrals reported by the first-principle density-functional calculation at T¼8 K (ref. 24), as in ref. 25, which are\ngiven for the nearest neighbours by (in the unit of eV)\nt\nLT\na1¼/C0 0:0267 ;tLT\na2¼/C0 0:0511 ;tLT\na3¼0:0323 ;\ntLT\nb1¼0:1241 ;tLT\nb2¼0:1296 ;tLT\nb3¼0:0513 ;tLT\nb4¼0:0152 ;ð6Þ\nand for the next nearest neighbours as\ntLT\na10¼0:0119 ;tLT\na30¼0:0046 ;tLT\na40¼0:0060 : ð7Þ\n(For the definition of the integrals, see Supplementary Fig. 8.) The largest integrals,\ntLT\nb1andtLT\nb2, are known to vary about 15% by raising Tfrom 8 to 300 K (ref. 24),\nthough the variation is less than a few per cent below 100 K. As our fitting analysesprimarily focus on this low- Tregion, it is reasonable to omit the T-dependence and\nkeep using the hopping integrals estimated at T¼8K ,{ t\nLT\np;p¼a1/C0a4’}, at all Tas\nis done in ref. 25. By diagonalizing the Hamiltonian (equation (4)) in conjunctionwith the hopping integrals (equations (5)–(7)) and the site potentials\n(equation (2)), one obtains the four energy bands with tilted Dirac cones at EF,\nas shown in Fig. 1c.\nWe note that it is very important to use these hopping integrals in equations (3)\nand (4) to reproduce the observed sign change of the Hall coefficient RHfrom\nRH40t o RHo0 with decreasing T69. (For details, see refs 26,70.)\nFrom all these, we used the effective TB model in ref. 25 as our non-interacting\nreference to the RG analyses. It should be stressed that we do not mean toincorporate interaction effects at this level, and indeed the model we assumed is a\npurely non-interacting TB model with acceptable site-dependent potentials. Note\nthat these values of site potentials are realistic because they lead to a site-dependentcharge differentiation which is compatible with the observed X-ray and Raman\nscattering results in the conducting phase; see refs 25,32,34.\nGeneralized Weyl Hamiltonian and site-spectral weight\n.Around the\nband-crossing DPs, where the Fermi energy EF(put as E¼0 hereafter) is fixed in\na-I3due to the stoichiometry, the low-energy continuum model is shown to be\ngiven by the generalized Weyl Hamiltonian20,23,25,35\nH¼‘w0/C1qs0þvxqxsxþvyqysy/C0/C1\n; ð8Þ\nwhich describes the electronic states in the vicinity of one of the DPs. Here,\nw0¼(w0x,w0y)a n d v¼(vx,vy) are effective velocities describing the tilt and the\nanisotropy of the Dirac cone, respectively; s0is the 2 /C22 unit matrix; ( sx,sy)\nare the Pauli matrices; and q¼(qx,qy) is the 2D wave vector measured from the DP\natkD. Note that the twofold valley degeneracy associated to the two DPs at ±kD\n(Fig. 1c) will be omitted for simplicity, and we will hereafter focus on a single\nvalley (at kD). (When the valley degeneracy is to be included in some of the\nexpressions, we will specifically mention it.) The Hamiltonian (equation (8))\nis defined in a space spanned by the Luttinger–Kohn bases71:FLK1\nD/C12/C12/C11\nandFLK2\nD/C12/C12/C11\n.\nThese bases are the two degenerate Bloch states at kD, which are given by a\n(normalized) superposition of the highest occupied molecular orbital hj/C12/C12/C11\non each\nof the four different BEDT-TTFs (ETs) in the unit cell36\nFl\nD/C12/C12/C11\n¼X4\nj¼1al\njhj/C12/C12/C11\nl¼LK1 or LK2ðÞ ; ð9Þ\nwhere j¼A(1),A’(2), B(3) and C(4) represents the different sites (see Fig. 1b).\nDiagonalization of equation (8) yields the eigenvalue Ez(q) (equation (1))\nin terms of the two bands ( z¼±) and the eigenstates (Goerbig, M.O., private\ncommunication.)\ncz\nq/C12/C12/C12E\n¼1ffiffiffi\n2pzFLK1\nD/C12/C12/C11\nþeijqFLK2\nD/C12/C12/C11 /C0/C1\n¼1ffiffiffi\n2pX4\nj¼1zaLK1\nj/C16/C17\nþeijqaLK2\nj/C16/C17 no\nhj/C12/C12/C11\n; ð10Þ\nwhere tan jq¼vyqy/vxqx. We note that the two states FLK1 ;LK2\nD/C12/C12/C11\nin equation (10)\nhave an equal weight for any value of q, as in the two-band model of the graphene\nDirac cone, whereas the four states h1;2;3;4/C12/C12/C11\nin equssation 10 have not necessarily\nthe same weight25,72. To see this, we define a (normalized) q-dependent\nsite-spectral weight by taking a projection of cz\nq/C12/C12/C12E\nonto hj/C12/C12/C11\n,nz\njqðÞ ¼ hjcz\nq/C12/C12/C12DE/C12/C12/C12/C12/C12/C122\n(ref. 36) (Goerbig, M.O., private communication.), which reads\nnz\njqðÞ ¼1ffiffiffi\n2p zaLK1\nj/C16/C17\nþeijqaLK2\nj/C16/C17 no/C12/C12/C12/C12/C12/C12/C12/C122\n¼1\n2aLK1\nj/C12/C12/C12/C12/C12/C122\nþaLK2\nj/C12/C12/C12/C12/C12/C122\nþ2zaLK1\nj/C12/C12/C12/C12/C12/C12a\nLK2\nj/C12/C12/C12/C12/C12/C12cosj\nq/C0fj\n12/C16/C17/C26/C27\n;ð11Þ\nwhere fj\n12is the relative phase between aLK1\njandaLK2\nj.\nTaking the low-energy limit of the effective TB model in ref. 1\n(see equations (3)–(7)), one can derive the four effective velocities in the\ngeneralized Weyl Hamiltonian (equation (8)), which are given by\nwTB\n0¼/C0 5:06;0:750 ðÞ ;\nvTB¼6:70;6:86 ðÞ in 104ms/C01/C0/C1\n:ð12Þ\nUsing these velocities, the phase jqin equation (11) can be approximated as\njq/C25arctan qy=qx/C0/C1\n/C17jðÞ , where jis the angle between qand the kx-axis. It is\nshown from equation (11) that the site-spectral weight nz\njqðÞacquires an anti-phase\nrelation between the j¼Band the other sites, namely, fAA0ðÞ\n12¼fC\n12¼0 and fB\n12¼p.\nMoreover, al\nB/C12/C12/C12/C12/C240:8 and al\nC/C12/C12/C12/C12/C240:7 have an equal size for the bases l¼LK1 and\nl¼LK2, while one obtains aLK2\nAA0ðÞ/C12/C12/C12/C12/C12/C12/C29a\nLK1\nAA0ðÞ/C12/C12/C12/C12/C12/C12/C240 [or a\nLK1\nAA0ðÞ/C12/C12/C12/C12/C12/C12/C29a\nLK2\nAA0ðÞ/C12/C12/C12/C12/C12/C12/C240]. This\ncauses an oscillation of equation (11) as a function of jaround the DP, with a large\namplitude and an opposite phase on the j¼B-site and the j¼C-site, whereas the j\ndependence is small on the j¼A(A’) site (Supplementary Fig. 1c). As the cone is\ntilted in the k\nx-direction (inset of Supplementary Fig. 1a), the Fermi velocity\nbecomes highly anisotropic around the cone20,25,50(Supplementary Fig. 1a), andARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666\n10 NATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunicationscan be expressed as\nvFj;zðÞ ¼ w0xcosjþw0ysinjþzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi\nvxðÞ2cos2jþvxðÞ2sin2jq\n: ð13Þ\nThe most striking consequence of this anisotropy, in conjunction with\nequation (11), is that there is a large asymmetry in the site-spectral weight for the\nsiteA(A’),BandCaround the DP. Namely, the site Bpredominantly reflects the\nlarge- vFelectrons in the steep slope of the cone ( jEp); the site Cmostly probes the\nsmall- vFelectrons in the gentle slope ( jE0); and the site A(A’) probes the entire\nelectronic states on average around the DP (see Fig. 1d–f and Supplementary\nFig. 1a,c). This asymmetry of the site-spectral weight results in a clear difference in\nthe size of the j-site DOS, Dj(E,z), for the different sites— DC(E,z)4DA(A0)\n(E,z)4DB(E,z) (ref. 25)—where Dj(E,z) (per valley and ET molecule) is defined as\ntheqsummation of nz\njqðÞat a given energy in the band z, which is expressed as\nDjE;zðÞ ¼ 2VCZd2q\n2pðÞ2nz\njjðÞdE/C0EzqðÞ ðÞ : ð14Þ\nHere, VCis the 2D unit-cell volume in the conducting ab-plane. The contrasting\nfeatures of the BandCsite-spectral weights around the DP provide unique\nopportunities to probe the excitations of large- vFDirac electrons (in the steep\nslope) and small- vFDirac electrons (in the gentle slope) separately in terms of a\nsite-selective local measurement such as NMR.\nWe note that in the original effective TB calculation by Katayama et al.25, the\nvelocities are given in the unit of energy (meV), w0¼(w0x,w0y)¼(/C038.9, 4.8) and\nv¼(vx,vy)¼(51.5, 43.9), because both the primitive vectors and the reciprocal\nlattice vectors are set to have a unit length in their notation. To recover the\nordinary physical unit (length/time), one has to multiply the velocities either by\na/‘orb/‘, using the values of the lattice constants aandbat the current pressure\n(2.3 GPa), where ‘is the Planck constant divided by 2 p. By linearly extrapolating\nthe X-ray diffraction data obtained at 1.76 GPa (ref. 33) to 2.3 GPa, the lattice\nconstants are estimated as a¼8.567 Å and b¼10.282 Å. The velocities in\nequation (12) are obtained in terms of these lattice constants.\nRenormalization-group calculations .The observed nonlinear temperature\ndependence of the spin susceptibility below the inflection point Tj\nflexin Fig. 2d,e,\ncannot be understood within the non-interacting Dirac-fermion picture, as we\nmentioned above (see Methods: effective TB model). In this temperature range, the\nscreening effect should be weak reflecting the vanishing thermal excitations around\nthe gapless point at EF. For this kind of situation, it is well known from the RG\nstudy of graphene Dirac cone that the LR part of the unscreened Coulomb\ninteraction among electrons causes a logarithmic divergence of the Fermi velocity\nvFaround the DP7–14. A similar argument has been recently proposed for the tilted\nDirac cone in a-I3(ref. 46), in which a RG flow of the Fermi velocity is suggested as\na function of T. Hence, the most straightforward and reasonable way to understand\nthe low- Tnonlinear feature of Fig. 2d would be to attribute it to the T-driven\nrenormalization of vFdue to the LR Coulomb interaction.\nTo check this hypothesis, we have performed a RG calculation based on the\ngeneralized Weyl Hamiltonian (equation (8)) and tried to fit the data. A circular\nmomentum cutoff of L¼0.667 Å/C01around the DP ( q¼0) is introduced in the\nRG theory, which is of the size of the averaged inverse lattice constant, L¼2p/L\nwith L¼(aþb)/2¼9.425 Å at 2.3 GPa. For the initial values of the velocities at the\ncutoff momentum q¼L(that is, vandw0in equation (8)), we employ velocities\nderived from the effective TB model of ref. 25, wTB\n0andvTB(equation (12)), as\ndiscussed in the previous subsection (Methods: Generalized Weyl Hamiltonian and\nsite-spectral weight). In the one-loop order large- Nexpansion of the RG theory,\nv¼(vx,vy) are renormalized following equation (2) (given in the main text) and\ngrows logarithmically as functions of L/q(where qis measured from the DP),\nwhereas w0¼(w0x,w0y) are not renormalized (Supplementary Fig. 4a). We note\nthat equation (2) takes into account the screening effect of the Coulomb interaction\nincluding the polarization bubbles in the self-energy. It is applicable to any size of\nthe coupling gj¼2pe2N=ð16effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi\nv2\nxsin2jþv2\nycos2jq\nÞ, where eis the dielectric\nconstant and N¼4 is the number of fermion species corresponding to the two DPs\nin the Brillouin zone and two spin projections (that means the twofold valley\ndegeneracy is considered).\nReflecting the renormalization of v, the eigenenergy with the RG correction\nbecomes (with the band index z¼±)\ndEzqðÞ ¼ ‘w0/C1qþzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nvxqðÞ2q2\nxþvyqðÞ2q2\nyq /C16/C17\n: ð15Þ\nCorrespondingly, the local DOS and the electron-spin susceptibility for the site j\nare respectively given by the expressions\ndDjE;zðÞ ¼ 2VCZd2q\n2pðÞ2nz\njjðÞdE/C0dEzqðÞ/C16/C17\n; ð16Þ\ndwj\nsTðÞ ¼ mB@\n@HX\nz¼/C6N\n2VCZ1\n/C01dEZd2q\n2pðÞ2nz\njjðÞdE/C0dEzqðÞ/C16/C17\nfE/C0EZ\n2/C18/C19\n/C0fEþEZ\n2/C18/C19 /C26/C27\n;ð17Þwhere VC¼88.086 Å2is the 2D unit-cell volume in the conducting ab-plane\nestimated at 2.3 GPa, N¼4 is the number of fermion species (again, the twofold\nvalley degeneracy is included), EZ¼gmBHis the electron Zeeman energy and f(E)i s\nthe Fermi distribution. The integration with respect to qin equations (16) and (17)\nis done up to the momentum cutoff ( q¼L). Note that Tplays the role of the flow\nparameter in equation (17) (that is, a T-driven RG flow).\nIn Supplementary Figs 4–6, we present the calculated profiles of the velocities\n(Supplementary Fig. 4), the j-site DOS (Supplementary Fig. 5) and the j-site\nelectron-spin susceptibility (Supplementary Fig. 6) based on the RG\nequation (equation (2)) and equations (15)–(17). Here, to get a reasonable\nagreement with the experiment, we introduced a phenomenological parameter,u(r1), in the calculations which is defined by the expressions\nw\n0\n0¼uwTB\n0 and v0¼uvTB: ð18Þ\nThis parameter reflects a suppression of the velocity or a reduction of the hopping\namplitude tijbetween the lattice site iandjdue to the SR part of the Coulomb\ninteraction53, as mentioned in the main text. Then, the RG flow, which is\ndetermined by equation (2), is controlled by two parameters—the dielectric\nconstant eand the phenomenological parameter u.\nSupplementary Fig. 4b,c, presents the parameter dependence of the flow of the\nvelocities vxandvy. The dielectric constant eaffects the power of the flow function\nv¼v(L/q) (Supplementary Fig. 4b), while the parameter udetermines both the\ninitial values of the velocities and the power of the flow (Supplementary Fig. 4c).Supplementary Fig. 5b,c, depicts the eandudependence of the DOSdD\njE;zðÞ for\nthe site j¼A(A’). It is clearly seen that a prominent suppression of the DOS\ndevelops around E¼EF(¼0) by reducing the dielectric constant e(which\ncorresponds to an increase in the coupling gjat the cutoff momentum q¼L). The\nreduction of the parameter u, on the other hand, leads to an enhancement of the\nDOS because the DOS is linked to the inverse square of the velocities20.I n\nSupplementary Fig. 6b,c, we show the parameter dependence of the calculated\nelectron-spin susceptibility. As the susceptibility probes the kBT-average of the\nDOS near EF(¼0), the suppression of the DOS for small ecauses a reduction of\nthe susceptibility at low temperatures (Supplementary Fig. 6b). The enhancement\nof the susceptibility for a small value of ucan be also understood in the same\nfashion (Supplementary Fig. 6c).\nSupplementary Fig. 7a shows the result of the variance analyses of the T-driven\nRG-flow fit to the experimental spin susceptibilities for the site j¼A(A’) and C\n(Fig. 2d) based on equation (17). Here, the variance, plotted in the u/C0e/C0Var(wj\ns)\nspace, is defined by\nVarwsðÞ ¼X\nj¼AA0ðÞ ;C1\nnXn\ni¼1wj\nsTiðÞ /C0dwj\nsTiðÞ/C18/C192\n; ð19Þ\nwhere Tiis the experimental temperature points ( i¼1,2,?,n), and wj\ns(Ti) and\ndwj\nsTiðÞ are the measured and calculated susceptibilities at T¼Ti, respectively.\nNote that the variance is defined only for the results on the site A(A’) and C, since\nthe results on the site Bhas less good agreement with the RG-calculation (see\nSupplementary Fig. 7b–e). This is because of the emergent ferrimagnetic spin\npolarization—the negative susceptibility on the site Bat low temperatures (inset of\nFig. 3c)—which is due to the SR part of the Coulomb interaction, not considered in\nthis calculation. (For details, see Methods: Simulations with the Hubbard model.)\nThe calculated susceptibilities at selected values of uandeare shown for three\ndifferent sites j¼A(A’),BandCtogether with the experimental data in\nSupplementary Fig. 7b–e. Except for the site B, the agreement with the calculations\nand the experiments is pretty good for small values of uande. The variance\nVar(wj\ns) has a minimum around ( u,e)¼(0.35, 1) with little edependence up to\neE101.5(Supplementary Fig. 7b,c) and increases rapidly as one moves away from\nthis minimum especially when uis increased. Thus, we take these values of\n(u,e)¼(0.35, 1) as the optimal parameters hereafter. (Here, the result for e¼1i s\nchosen because the e-dependence is small and affects the result little.) The small\nvalue of u¼0.35(o1) is in agreement with the aforementioned reduction of the\nhopping amplitude due to the SR repulsion53and indicates the presence of\nmoderate electron correlations. (This point naturally supports our discussion basedon the Hubbard model in the next subsection to deal with the observed negative\nspin susceptibility on the site B.)\nLastly, it may be worthwhile to mention that the flow of the velocities\nv¼(v\nx,vy) with w0¼(w0x,w0y) staying constant in Supplementary Fig. 4a leads to\na situation where vx,vyc|w0x|,w0yis realized for a large value of the momentum\nscale L/q(c1) (that is, in the vicinity of the DP). This means that the tilting term\n(w0) becomes effectively negligible with respect to the anisotropy term ( v) at low\nenergy in the generalized Weyl Hamiltonian (equation (8)). Moreover, the values ofthe two velocities, v\nxandvy, remain very close to each other down to the vicinity of\nthe DP (for instance, ln( L/q)¼5 in Supplementary Fig. 4a corresponds to\nq¼0.0045 Å/C01in terms of L¼0.667 Å/C01). These points together suggest that the\nanisotropy of the Dirac cone becomes very small near the DP and the cone is\npractically isotropic at low energy, as one can see in the calculated cone inFig. 4d–f. This is can well account for the observed site dependence of the\nsusceptibility, the root of which is linked to the tilt and the anisotropy of the\ncone. The site dependence becomes vanishingly small at low temperaturesNATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666 ARTICLE\nNATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunications 11(Supplementary Fig. 2), which reflect the renormalization of the velocity that makes\nthe cone to be more and more isotropic at low energy.\nSimulations with the Hubbard model .In the previous subsection, we have shown\nthat the RG calculation based on the low-energy continuum model (equation (8))well reproduces the observed nonlinear T-dependence of wj\nsTðÞat the site A(A’)\nandC(Supplementary Fig. 7b–e) when the velocity suppression due to the SR\nCoulomb interaction is phenomenologically taken into account. However, the\nagreement is less good at the site B; in particular, the negative susceptibility wB\nso0\nbelow TE60 K cannot be reproduced at all, suggesting the presence of other\ninteraction effects.\nTo understand the origin of the observed wB\nso0, we have performed another\nsimulation of the susceptibility based on the RPA. For this, we start from the\nstandard Hubbard model with the onsite Hubbard interaction U, by extending\nearlier study50. The model Hamiltonian is given for the present system by the\nexpression\nH¼X\nia:jbðÞ ;stia:jbay\niasajbsþh:c:/C16/C17\nþX\njaUay\nja\"ay\nja#aja#aja\"; ð20Þ\nwhere aw\njasis the creation operator on the site j¼A(1),A’(2), B(3) and C(4) in the\nunit cell a(¼1,y,Nu.c.) with the spin s(¼m,k),Nu.c.is the total number of the\nunit cell, and tia:jbis the nearest-neighbour and next nearest-neighbour hopping\namplitude between the lattice site ( i,a) and ( j,b) (see Supplementary Fig. 8).\n(It should be noticed that this Hamiltonian (equation (20)) lacks the site-dependent\npotential term (equations (3) and (4)) we employed above. From qualitative points\nof view, the inclusion of these potentials does not alter the results much, and, forthe sake of simplicity, we shall omit this term in this subsection.)\nThe amplitude t\nia:jbat finite temperatures are estimated from the hopping\nintegrals given by the first-principle calculations24at 8 K, { tLT\np;p¼a1/C0a4’}\n(equations (6) and (7)), and at 300 K, { tHT\np;p¼a1/C0a4’}, as given in the\nfollowing (in eV)\ntHT\na1¼/C0 0:0101 ;tHT\na2¼/C0 0:0476 ;tHT\na3¼0:0093 ;\ntHT\nb1¼0:1081 ;tHT\nb2¼0:1109 ;tHT\nb3¼0:0551 ;tHT\nb4¼0:0151 ;\ntHT\na10¼0:0088 ;tHT\na30¼0:0019 ;tHT\na40¼0:0009 ;ð21Þ\nin combination with the interpolation formula given in ref. 50\ntpTðÞ ¼ tLT\npþtHT\np/C0tLT\np/C16/C17T/C08\n300/C08: ð22Þ\nWithin the mean-field approximation, the diagonalization of the Hubbard\nHamiltonian (equation (20)) yields\nX4\nj¼1~EijskðÞdjZskðÞ ¼ EZskðÞdiZskðÞ; ð23Þ\n~EijskðÞ ¼ EijkðÞ þ UN ishi dij; ð24Þ\nNjs/C10/C11\n¼1\nNu:c:X\nkX4\nZ¼1d/C3\njZ;/C0skðÞdjZ;/C0skðÞfEZ;/C0skðÞ /C0 m/C0/C1\n; ð25Þ\nwhere we define EijkðÞ ¼P\ndijtijeik/C1dij,dijis a vector connecting the nearest-\nneighbour lattice sites iandj,EZskðÞis the eigenvalue ( E1s4E2s4E3s4E4s),\ndiZs(k) is the corresponding eigenvector, f(E) is the Fermi distribution and mis the\nchemical potential. Note that the average electron number Njs/C10/C11\nis determined\nself-consistently from the conditionP\njsNjs/C10/C11\n¼6, reflecting the3\n4-filling of the band.\nIn the normal state, one has Nj\"/C10/C11\n¼Nj#/C10/C11\n; thus, the spin sis omitted hereafter.\nWe introduce the bare site-spin susceptibility matrix, ^wð0Þ, whose ( ij)-element is\ngiven by\nwð0Þ\nijQ;oðÞ ¼ /C01\nNu:c:X\nkX4\nZ;Z0¼1FZZ0\nijk;QðÞfEZkþQðÞ/C0/C1\n/C0fEZ0kðÞ/C0/C1\nEZkþQðÞ /C0 EZ0kðÞ /C0 ‘o/C0id;\nð26Þ\nin terms of a form factor\nFZZ0\nijk;QðÞ ¼ diZkþQðÞ d/C3\njZkþQðÞ djZ0kðÞd/C3\niZ0kðÞ; ð27Þ\nwhere id(d40) is an infinitesimally small imaginary part. Within the RPA\napproach, the spin fluctuations are estimated using the expression50\nwij;RPAQ;oðÞ ¼ ^wRPAðÞijQ;oðÞ ¼ ^I/C0^w0ðÞU^I/C16/C17/C01\n^wð0Þ/C20/C21\nijQ;oðÞ ; ð28Þ\n(where ^Iis the 4 /C24 unit matrix), and the total RPA j-site-spin susceptibility (for\nQ¼0ando¼0) is given by\nwj\nRPA¼X4\ni¼1^I/C0^w0ðÞU^I/C16/C17/C01\n^wð0Þ/C20/C21\nji0;0ðÞ : ð29ÞNow, we decompose the bare susceptibility wð0Þ\nijinto two parts (for the reason that\nwill become clear below): wð0Þ\nij¼wð0Þ;intra\nij þwð0Þ;inter\nij , where wð0Þ;intra\nij andwð0Þ;inter\nij\ncorrespond to the intra-band and inter-band contributions to the bare susceptibility\n(forQ¼0), respectively. The intra-band RPA susceptibility is then defined by\nwj;intra\nRPA¼X4\ni¼1^I/C0^w0ðÞ;intraU^I/C16/C17/C01\n^wð0Þ;intra/C20/C21\nji0;0ðÞ ; ð30Þ\nand the inter-band contribution to the total RPA susceptibility is expressed as\nwj;inter\nRPA¼wj\nRPA/C0wj;intra\nRPA : ð31Þ\nIn Supplementary Fig. 3a, the calculated temperature dependence of the intra-band\nRPA susceptibility wj;intra\nRPA is shown for U¼0.14 eV in comparison to the\nnon-interacting case ( U¼0) for the site j¼A(A’),BandC. It is seen that the\nintra-band susceptibility wj;intra\nRPA for a finite value of Ubecomes always larger than\nthe case for U¼0 at all temperature and all site j. The inter-band contribution, on\nthe other hand, is found to provide a site-dependent correction to the susceptibility\n(Supplementary Fig. 3b). Namely, the inter-band RPA susceptibility wj;inter\nRPA gives a\npositive contribution on the site A(A’) and C, whereas the contribution is negative\non the site B. The negative inter-band contribution on the site B(wB;inter\nRPAo0)\ndevelops with increasing Uand in turn leads to a negative total RPA susceptibility\n(wB\nRPAo0) above a critical Uvalue of UCE0.12 eV. The position of the minimum\nshifts towards higher energies with increasing U(Supplementary Fig. 3c). By taking\na value of U¼0.14 eV, the minimum of the total RPA susceptibility wB\nRPAappears at\naround TE50 K, which agrees well with the experiment (inset of Fig. 3c). These\nresults demonstrate that the SR part of the Coulomb interaction between electrons\ncauses a ferrimagnetic spin polarization at low temperature. This leads to a\nsituation where the site Bis subjected to a negative local magnetic field, pointing\nopposite to the external field, while the other sites ( A,A’ and C) feel a positive field,\nas schematically illustrated in Fig. 6.\nTo have an overall comparison of the RPA calculations with the experiment, the\nfirst derivative of the susceptibility is analysed for the case U¼0, the total RPA\nsusceptibility wj\nRPA(equation (29)) for U¼0.14 eV, and the observed spin\nsusceptibility (Fig. 2e), as depicted for the non-equivalent sites in Supplementary\nFig. 3d–f. The calculations capture the experimental features relatively well on the\nsiteA(A’) and Cabove the peak temperature ( TE50 K), whereas the calculation\ndoes not agree with the experiment at all on the site B. At low temperatures, on the\nother hand, the agreement between the calculation and experiment becomes worse\neven for the site A(A’) and C. That is, in the low temperature limit, the calculations\nshow a saturation both for U¼0 and the finite U, while the experiment exhibits a\nmonotonic decrease towards zero (Supplementary Fig. 3d and f). We believe thatthis disagreement arises because the present RPA calculation does not incorporate\ntheT-driven v\nF-renormalization effect due to the LR part of the Coulomb\ninteraction, as discussed in the previous subsection (see Methods: RG calculations).The v\nF-renormalization results in a super-linear temperature dependence in the\nspin susceptibility wj\ns(Supplementary Fig. 6), which causes a decrease of the first\nderivative of wj\nswith decreasing temperature as reflected in the experiment.\nFinally, we comment on the comparison of the experiment with an orthodox\nRPA fitting. This is done by assuming a simplified RPA (s-RPA) expression for thespin susceptibility, which is defined by\nwj\nsRPA¼wð0Þ\nj\n1/C0Ujwð0Þ\nj;j¼A;A0;BandC ðÞ ð 32Þ\nwhere w0ðÞ\njis the bare spin susceptibility and Ujis an adjustable parameter reflecting\nthe onsite Hubbard interaction. (Note that wð0Þ\njandwj\nsRPA correspond to the\ndiagonal term in equations (26) and (28), respectively, for Q¼0ando¼0 with\nthe Hubbard interaction Uin equation (20) replaced by the site-dependent\nparameter Uj.) Supplementary Fig. 14 presents the least-square fitting results to the\nexperiment using the s-RPA expression, which yields UA(A0)¼0.6,UB¼1.3 and\nUC¼0.4 (in eV). It is clearly seen that the observed nonlinear temperature\ndependence of the susceptibility cannot be reproduced by the s-RPA fit at all sites.In particular, there is an unphysical divergence in the calculation, which is linked to\nthe large U\njvalues used in the calculation that are too large compared with the\ntypical values applicable to this system24,26,31,35,40,41,48,50. It is thus concluded that\none has to consider the full-matrix elements of equation (28) in order to obtain a\nreasonable agreement with the experiment.\nData availability .The data that support the findings of this study are available\nfrom M.H. upon requests.\nReferences\n1. Wehling, T. O., Black-Schaffer, A. M. & Balatsky, A. V. Dirac materials. Adv.\nPhys. 63,1–76 (2014).\n2. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K.\nThe electronic properties of graphene. Rev. Mod. 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Interband effects of\nmagnetic field on Hall conductivity in the multi-layered massless Dirac fermion\nsystem a-(BEDT-TTF) 2I3.Phys. Rev. B 85,033401 (2012).\n70. Kobayashi, A., Suzumura, Y. & Fukuyama, H. Hall Effect and Orbital\nDiamagnetism in Zerogap State of Molecular Conductor a-(BEDT-TTF) 2I3.\nJ. Phys. Soc. Jpn. 77,064718 (2008).\n71. Luttinger, J. M. & Kohn, W. Motion of Electrons and Holes in perturbed\nperiodic fields. Phys. Rev 97,869–883 (1955).\n72. Goerbig, M. O. Electronic properties of graphene in a strong magnetic field.\nRev. Mod. Phys. 83,1193–1243 (2011).\nAcknowledgements\nWe gratefully acknowledge valuable discussions with Y. Suzumura, N. Nagaosa, H. Isobe,\nM. Imada, M. Ogata, H. Matsuura, H. Fukuyama, C. Hotta, S. Sugawara, T. Osada,\nT. Taniguchi, M. Potemski, M.-H. Julien and H. Mayaffre. In particular, we thankM. Horvatic ´and M. O. Goerbig for their thoughtful advices on the analyses and\ndedicated discussions. We also thank D. Liu for providing us unpublished results and forfruitful discussions. This work was supported by MEXT/JSPJ KAKENHI under Grant\nNoes 20110002, 21110519, 24654101, 25220709, 15K05166, 15H02108, JSPS Postdoctoral\nFellowship for Research Abroad (Grant No. 66, 2013) and MEXT Global COE Programat University of Tokyo (Global Center of Excellence for the Physical Sciences Frontier;Grant No. G04).\nAuthor contributions\nThe samples were prepared by M.T. The data were taken, analysed and interpretedby M.H. and K.I. with the help of K.M., C.B., D.B., G.M., A.K. and K.K. The\nrenormalization-group calculation was done by D.B. and analysed by M.H. with the help\nof D.B. and C.B. The simulations using the Hubbard model were carried out by G.M. andA.K. The project was supervised by K.K, and the manuscript was written by M.H. withK.M., D.B., A.K., C.B. and K.K.\nAdditional information\nSupplementary Information accompanies this paper at http://www.nature.com/\nnaturecommunications\nCompeting financial interests: The authors declare no competing financial interests.\nReprints and permission information is available online at http://npg.nature.com/\nreprintsandpermissions/\nHow to cite this article: Hirata, M. et al. Observation of an anisotropic Dirac cone\nreshaping and ferrimagnetic spin polarization in an organic conductor. Nat. Commun.\n7:12666 doi: 10.1038/ncomms12666 (2016).\nThis work is licensed under a Creative Commons Attribution 4.0\nInternational License. The images or other third party material in this\narticle are included in the article’s Creative Commons license, unless indicated otherwisein 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.\nTo view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/\nrThe Author(s) 2016ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12666\n14 NATURE COMMUNICATIONS | 7:12666 | DOI: 10.1038/ncomms12666 | www.nature.com/naturecommunications" }, { "title": "1607.03041v2.Low_loss_spin_wave_resonances_in_organic_based_ferrimagnet_vanadium_tetracyanoethylene_thin_films.pdf", "content": "Low loss spin wave resonances in organic -based ferrimagnet \nvanadium tetracyanoethylene thin films \nNa Zhu,1 Xufeng Zhang,1 I.H. Froning,2 Michael E. Flatté,3 E. Johnston -Halperin,2 \nand Hong X. Tang1 \n1Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA \n2Department of Physics, The Ohio State University, Columbus, Ohio 43210 -1117, USA \n3Optical Science and Technology Center and Depar tment of Physics and Astronomy, University of Iowa, Iowa City, Iowa \n52242, USA \n \nAbstract: \nWe experimentally de monstrate high quality factor spin wave resonances in an encapsulated thin film of the \norganic -based ferrimagnet vanadium tetracyanoethylene ( V[TCNE ]𝑥~2) coated on an a-plane sapphire substrate \nby low temperature chemical vapor depo sition. The thickness standing wave modes are observed in a broad \nfrequency range (1 GHz ~ 5 GHz) with high quality factor exceeding 3,200 in ambient air at room temperature, \nrivaling those of inorganic magnetic materials . The exchange constant of V[TCNE ]𝑥~2, a crucial material \nparameter for future study and device design of the V[TCNE ]𝑥~2, is extracted from the measurement with a \nvalue of (4.61±0.35)×10−16 m2. Our result establishes the feasibility of using organic -based materials for \nbuilding hybrid magnonic devices and circuits. \n \nSpin wave s, which are the collective excitation of the magn etization in magnetic materials , \nhave been attracting intensive attention recently due to potential applications in both \nfundamental research1–4 and device applica tions5–9 thank s to properties such as being ohmic \nloss free,10,11 long spin lifetime , and large -bandwidth tunability. In particular, yttrium iron \ngarnet (YIG, Y3Fe5O12) has been long considered as one of the most attractive magnon media, \nthanks to its magnetic, microwav e, mechanical and optical propertie s. As a result , YIG has \nbeen widely adopted to investigate the inte ractions among spin waves, micro waves12–19, \nacoustic waves20,21 and optical excitation s.22–26 However, high quality YIG films can only \nbeen grown on specific lattice -matched substrate s such as gadolinium gallium garnet (GGG , Ga3Gd5O12),27 so its integration with other substrates , such as silicon, is extremely difficult. \nMoreover, micro/nano patterning has long been an obstacle in YIG fabrication , which \nsignificantly limits the development and applications of YIG -based magnonic circuits. \n \nTo overcome these constraints , organic -based magnets are emerging as a n alternative solution. \nSuch materials can be grown as high quality thin films on various substrates ,28–30 have \nexcellent post fabrication capabilities for developing complex geometries , and require only \nlow temperature and ambient pressure deposition condition s.31 Particularly, vanadium \ntetracyanoethylene ( V[TCNE ]𝑥~2),32 an organic -based ferrimagnetic semiconductor (𝐸𝑔=\n0.5 eV, 𝜎=0.01 S/cm ),28 exhibits room temperature magnetic ordering ( 𝑇𝐶>600 K) with a \nnarrow ferromagnetic resonance linewidth (peak to peak linewidth of 1G) ,31 which is \ncomparable to that of YIG .33 These advantages suggest that V[TCNE ]𝑥~2 could be a \nsubstitute for YIG in developing high -quality magnonic circuits, especially for integration on \na chip . However, the saturation magnetization of V[TCNE ]𝑥~2 (95 G ) is over an order of \nmagnitude smaller compared with that of YIG (1850 G) , which will offer potential \napplications in l ow-current spin -transfer switching ,34 while also introduce the limitations in \nthe practical devices design in diverse areas such as high-density magnetic recording , power \ntransformer and electromagnetic interface (EMI) prevention components .35,36 \n \nAlthough magnetic excitations can take place in many different and complex forms, previous \nstudies on V[TCNE ]𝑥~2 have mainly focus ed on the characterization of its ferromagnetic \nresonance (FMR). The studies of magnetostatic waves and exchange interactions, which are \nvery comm on in YIG thin film devices, have thus far been overlooked. Magnetostatic waves, \nin contrast to the FMR mode in which all the spins precess uniformly, are propagating spin \nwaves that have non -zero wavevectors and non-uniform field distributions in the material.37 Unlike the FMR mode , which resonates at a single frequency, m agnetostatic waves can exist \nin a frequency range depending on their mode profile s. In particular , it has been demonstrated \nthat in ferromagnetic thin films magnetostatic waves can form a series of standing wave \nmodes due to spin pinning at the film surface s.38,39 These spin wave resonances have \npotential applications such as serving as high quality factor magnonic multi -mode resonators \nfor information storage and processing. \n \nIn this paper, spin wave resonances along the film thickness direction in a V[TCNE ]𝑥~2 thin \nfilm are investigated . Standing waves with different mode orders in both perpendicularly and \nin-plane magnetized thin films are recorded in a broad frequency range from 1 GHz to 5 GHz \nthat show very high quality ( Q) factors , exceeding Q = 3200, indicating high material quality \nand thickness uniformity of this organic -based ferrimagnetic thin film . The exc hange constant \nof V[TCNE ]𝑥~2 can be extracted from the free spectral range ( FSR) of the standing wave \nmodes, which is a crucial material parameter for the potential device applications of \nV[TCNE ]𝑥~2. Our findings advance the study on such magnetic polymers from the previous \nsimple FMR cha racterization , and establish the feasibility of their potential use for \ndeveloping functio nal spin wave devices. \nFIG. 1. (a) Schematic of the V[TCNE ]𝑥~2 magnonic waveguide used in this experiment. (b) An encapsulated V[TCNE ]𝑥~2 \nthin film on a sapphire substrate patterned with gold tr ansmission lines. (c) Simulated magnetic field distri bution of the \ncoplanar waveguide. The black arrows and colors indicate the magnetic field directions and amplitudes, respectively. (d) \nSchematic illustration of spin wave resonances along length, width and thickness direction s. (e) Standing wave modes with \nodd mode number in a 1-m-thick V[TCNE ]𝑥~2 thin film . \n \nThe V[TCNE] x~2 device used in our experiments is illustrated in Fig. 1(a), and consists of a \npair of gold coplanar waveguide s and a 1 -m-thick V[TCNE] x~2 film. The device is wire -\nbonded to a FR -4 printed circuit board for microwave measurement, which is not shown . The \ntwo 15 -mm-long coplanar transmission lines are designed with a 100 -m-wide center signal \nline and a 40-m-wide gap between the signal line a nd the ground, and a characteristic \nimpedance of 50 Ω. These two coplanar waveguides serve as the transducer for spin wave \nexcitation and detection , and are separated by 5 mm from each other . To fabricate these \ncoplanar waveguides, photolithography is first carried out on a 20×20×0.43 mm3 a-plane \n(100) sapphire substrate using bi -layer resist process ( S1813 photoresist on top of LOR 5A \nresist ). After exposing in EVG 62 0 Mask Aligner, the sample is developed in MF319. The \nprepatterned sample is then depo sited with 200 -nm-thick gold, and a following lift -off \nprocess eventually transfers the pattern to the gold laye r and forms the transducer chip , as \nshown in Fig. 1(b) . The microwave magnetic field distribution of the designed coplanar \nwaveguide is shown in Fig. 1(c ). \n \nThe V[TCNE ]𝑥~2 film is directly grown on the fabricated coplan ar circuit to a thickness of 1 \nm via low temperature ( 50℃ ) chemical vapor deposition in an argon atmosphere . It is \ndeposited into a 20×10 mm2 mesa bridging the two coplanar transmission lines using a \nshadow mask, and serves as the magnonic waveguide . TCNE and V(CO)6 are used in the \nratio of 10 :1 at evaporation temperature s of 50℃ and 10℃ , respectively, yielding a \ndeposition rate of 9.3 nm/min.40 Because of its air -sensitivity, the film is encapsulated using a \ntechnique that preserves the magnetic order for over one month under ambient condition .41 \nThe technique requires applying an epoxy (purchased from Ossila ) to the film surface . A \nglass cover slide is placed over the epoxy and pressed down firmly to spread the epoxy across \nthe film surface and edges to isolate the film from the air, and to provide mechanical \nprotection for the sample. In a final step, t he epoxy is cured by a white LED for over an hour. \n \nThe fabricated device is characterized by the microwave transmission measurement after \nwire-bonding to a FR -4 printed circuit board terminated with SMA connectors for microwave \ninput and output. The input signal is sent to the device from one port of the vector network \nanalyzer (VNA), and the output signal is amplified by a radio frequency (RF) amplifier to \ncompensate the insertion loss of the device , which we attribute to the low transducer \nefficien cy associated with the relatively sma ll spin density of the material . When a bias \nmagnetic field is applied to the device, magnetostatic waves can be excited in a given \nfrequency range and propagate in the V[TCNE ]𝑥~2 magnonic waveguide. Ideally, such transmission windows would show up in the transmission spectra as a series of peaks, which \ncorrespond to the discrete standing wave modes formed due to the confinement of the film \nboundaries. However, in our experiment, the electrical cross talk b etween th e two \ntransmission lines is significant due to the low spin density of the magnonic media , and gives \nrise to a transmission background which distort s the resonances into a Fano line shape , \nsometimes even Lorentzian dips, instea d of the ideal Lorentzian pea k line shape . \n \nDue to the finite size of ferromagnetic thin film s, it is possible that the spins near the film \nsurfaces are not able to precess due to the different anisotropy fields at the surface compared \nwith those in the bulk. In this condition, the reflection of propagating spin waves at surfaces \ncan form a series of standing wave modes along the thickness, length and width directions \nwith differen t FSR s, as shown in Fig. 1 (d). The magnetostatic resonances formed by volume \nwave s with non -zero in -plane wavevectors have been extensively studied and observed in \nYIG thin films and permalloy metallic thin films.42,43 For the volume waves in a tangentially \nmagnetized thin film when the direction of magnetic field in along length direction (axis x in \nFig. 1(a )), the dispersion relation is described by44 \n(1+2)+2|(1+2)12⁄|(−1+2+\n1+)12⁄\n×(1+)cot[|𝑘𝑦|𝑑(−1+2+\n1+)12⁄\n] \n+(1+𝜅)2(−1+2+𝜅\n1+𝜅)−𝜈2=0, (1) \nwhere =𝑘𝑥𝑘𝑦⁄ , 𝛺𝐻=𝐻04𝜋𝑀s ⁄ , 𝛺=𝜔4𝜋𝛾𝑀s ⁄ , 𝜅=𝛺𝐻(𝛺𝐻2−𝛺2) ⁄ , and 𝜈=\n𝛺(𝛺𝐻2−𝛺2) ⁄ . Here 𝑘𝑥 and 𝑘𝑦 are the wavevectors al ong length and width directions, d is \nfilm thickness, 𝐻0 is the effective magnetic field inside the film , and 4𝜋𝑀s is the saturation \nmagnetization , which is 95 G according to previous characterization of the V[TCNE ]x~2.31 𝛾 \nis gyromagnetic ratio with the value of 2.8 MHz/Oe from the previous study ,31 and 𝜔 is the resonance frequency. Based on the dimension of the deposited V[TCNE ]x~2 thin film, the \ncalcu lated FSRs along the length and width directions are around 20 kHz and 5 0 kHz, \nrespectively . As a result, the longitudinal and lateral standing wave modes are packed very \nclose together with FSRs much smaller than the typical ly ultra-narrow ferromagnetic \nresonance linewidth (~ 2 MHz ), and therefore are unresolvable in the spectrum. Similarly, for \nthe perpendicularly magnetized thin film, the frequency spacings of in -plane standing wave \nmodes can be calculated from the dispersion relation ,37 and t he theoretical FSRs are 12.5 kHz \nand 25.1 kHz for the modes along length and width directions , respectively, which are also \ntoo small to be resolved. \n \nBased on the discussion above, unde r the surface pinning condition, only spin wave \nresonances along the thickness direction are resovable due to the small film thickness , d. The \nresonances occur whenever the film thickness equals to an integral number of half \nwavelengths , but only odd modes can be excited by a uniform RF field ,38 as shown in Fig. \n1(e). The resonant frequencies are determined by \n 𝑓𝑛=𝑓0+𝑓mex(𝑛𝜋\n𝑑)2\n, (2) \nwhere 𝑓0 equals 𝛾(𝐻0−4𝜋𝑀 𝑠) and 𝛾√𝐻0(𝐻0+4𝜋𝑀s) for normal ly and tangentially \nmagnetized thin film, respectively.37 Here 𝐻0 is the applied bias magnetic field, 4𝜋𝑀 s is the \nsaturation magnetization, 𝑑 is film thickness, ex is the exchange constant , and 𝑓m= 4𝜋𝛾𝑀s. \nFIG. 2. Vector network analyzer transmission characterization of the V[TCNE ]𝑥~2 magnonic waveguide with magnitude \nresponse at different bias magnetic fields when the film is tangentially (a) and perpendicularly (b) magnetized . Inset: zoom -\nin microwave transmission spectra (symbols) and Fano line shape fitting (lines) of spin wave resonances under the uniform \nbias magnetic field. \n \nFigure s 2 (a) and (b ) show the normalized microwave transmission spectra when the film is \ntangentially an d normally magnetized , respectively . By varying the magnitude of the bias \nmagnetic f ield, spin wave resonances with F ano line shape are observed with frequencies \ndetermined by the magnitude of the applied magnetic field , which are attributed to the \nstanding wave modes along thickness direction , as discussed above . Clear fundamental (n = 1) \nmodes are observed in a large frequency range (1~5 GHz ). The additional features near the \nresonances with smaller extinction ratio are attributed to high order modes which have \nrelatively lower excitation and detection efficiencies. \n \nThe spin wave resonances have very narrow linewidth and large amplitude extinction ratios, \nwhich indicate the high quality of the V[TCNE] x~2 thin film . The Q factor of the transmission \npeaks is defined as 𝑓𝛿𝑓⁄ , where f is the center frequency and 𝛿𝑓 is the full width at half \nmaximum (FWHM). As sh own in the inset of Fig. 2(a), Fano line shape fitting is applied for \nthe resonance with center frequency at 2.45 GHz when the thin film is tangentially \nmagnetized . Q factor s as high as 3,267 are observed , corresponding to a linewidth 𝛿𝑓=0.75 \nMHz , comparable to that of FMR mode in YIG thin films .27,33 Similarly, the high Q factor (Q \n= 2671) is also achieved in the normal magnetization condition for the first order spin wave \nresonance centered at 4 GHz , as shown in the inset of Fig. 2(b) . \n \nThese high Q resonances which are comparable with those of YIG can be explained \nconsidering the typical damping mechanisms of spin -waves. In both YIG and V [TCNE ]x~2, \ndamping is caused by magnetic dipole coupling and therefore it is proportional to the \nsaturation magnetization; for example, surface imperfections permit decay of a spin wave \ninto two other spin waves (a magnetic Raman process).45,46 As the saturation magnetization \nof VTCNE is over an order of magnitude smaller than YIG, the limiting damping from these \nprocesses could be significantly smaller than in YIG. \n \nFIG. 3 . The f─H relation for both tangentially (a) and perpendicularly (b) magnetized thin films is fitted by Eq. (2) to extract \ngryomagnetic ratio and saturation magnetization. (c) Transmission spectrum of the in -plane magnetized thin film shows two \nresonances with different mode numbers. (d) Extracted exchange constant of the V[TCNE ]𝑥~2 thin film at different \nfrequencies, in both normal and ta ngential magnetization configuration s. The red dash line denotes the mean value of the \nextracted exchange constant. \n \nThe f -H relations can be extracted for both tangentially and normally magnetized thin films \nusing Eq. (2) , and are plotted in Figs. 3(a) an d (b). The gyromagnetic ratio 𝛾 is fitted with the \nvalues of 2.59 MHz/Oe and 2.73 MHz/Oe for tangential and normal magnetiz ation \nconfigurations , respectively . The fitted values in our experiments are close to the value \nreported previously (2.8 MHz/Oe). The fitted values f or the saturation magnetization for \ntangentially and normally magnetized thin film s are 56.15 G and 60.32 G, respectively. These \nfitted values of the saturation magnetization are slightly smaller compared with the previous \nstudies of V[TCNE ]𝑥~2thin film s which demonstrated 95 G saturation magnetization , \nperhaps resulting from the degradation of the spin density of this organic -based ferrimagnet \ndue to the ambient measurement condition.31 \n \nFig. 3( c) shows the spectrum of spin wave resonances at 819.58 Oe, where two clear \nresonances peaks with 5.625 MHz frequency spacing can be clearly resolved in this zoom -in \nspectrum . As we discussed above, only odd modes can be excited by a uniform RF field. \nTherefore, these two modes in the spectrum are attributed to the first and third order standing \nwave mode s along thickness dire ction. The nu mber of modes we can observe is mainly \nlimited by the narrow excitation bandwidth of the coplanar waveguide . These two modes are \nobserved in both normal an d tangential magnetization configura tions and exist at a wide \nfrequency range with frequency spacing around 5. 625 MHz. Based on Eq. (2), the exchan ge \nconstant of the V[TCNE ]𝑥~2 thin film can be extracted from the frequency spacing between \nthese two modes . The values of exchange constant extracted at different frequencies are plotted in Fig. 3 (d). The average value of calculated exchange constant is (4.61±0.35)×\n10−16 m2. This value is comparable to that of YIG (3×10−16 m2),37 demonstrating the wide \nvariety of potential applications of the V[TCNE ]𝑥~2 materials to be used as a substitute for \nYIG. \n \nIn summary, this work experimentally demonstrat es the spin wave resonances in an organic -\nbased ferrimagnet, V[TCNE ]𝑥~2. The standing wave mode s along the thickness direction are \nobserved wit h narrow linewi dth and Q factors more than 3,2 00, which rival those of YIG \ndespite the lack of crystalline structure in V[TCNE ]𝑥~2, and meanwhile offer more flexibilit y \nin circuit design and fabrication thank s to the benefits of the benign deposition conditions of \norganic materials . Due to the low spin density of the organic films, the free spectral ranges of \nin-plane confined modes are too small to be resolved in the current device configuration. \nFurther material and device development in patterning micro -scale waveguides may lead to \nthe observation of in -plane magnet ostatic wave modes. In our device fabrication processes, \nthe direct sample growth on gold patterned sapphire substrate s validate s the outstanding \nproperties of V[TCNE ]𝑥~2 materials such as low deposition temper ature, high rate, \nconformal coating on a wide variety of substrates and prepatterned circuit s can be realized for \nambient operation devices . Moreover, the exchange constant of V[TCNE ]𝑥~2 materials is \nobtained from the frequency spacing s between different orders of modes over a wide \nfrequency range, which is a critical material parameter for future device and material studies \nof organic -based magnonic circuits. Our work present s the intriguing potential of this \norganic -based magnetic material to be used in making high Q magnonic resonator s for hybrid \nhigh frequency electronic and spintronic circuits. \n This work is supported by DARPA/MTO MESO program and NSF Grant No. DMR -\n1507775 . 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Weichselbaumer,1A. Bauer,2J. Waizner,3\nH. Berger,4S. Maendl,1M. Garst,3, 5C. P\reiderer,2and D. Grundler6,\u0003\n1Physik Department E10, Technische Universit at M unchen, D-85748 Garching, Germany\n2Physik Department E51, Technische Universit at M unchen, D-85748 Garching, Germany\n3Institute for Theoretical Physics, Universit at zu K oln, D-50937 K oln, Germany\n4Institut de Physique de la Mati\u0012 ere Complexe, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne, 1015 Lausanne, Switzerland\n5Institut f ur Theoretische Physik, Technische Universit at Dresden, D-01062 Dresden, Germany\n6Institute of Materials and Laboratory of Nanoscale Magnetic Materials and Magnonics (LMGN),\n\u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), Station 12, 1015 Lausanne, Switzerland\n(Dated: October 2, 2018)\nChiral magnets with topologically nontrivial spin order such as Skyrmions have generated enor-\nmous interest in both fundamental and applied sciences. We report broadband microwave spec-\ntroscopy performed on the insulating chiral ferrimagnet Cu 2OSeO 3. For the damping of magnetiza-\ntion dynamics we \fnd a remarkably small Gilbert damping parameter of about 1 \u000210\u00004at 5 K. This\nvalue is only a factor of 4 larger than the one reported for the best insulating ferrimagnet yttrium\niron garnet. We detect a series of sharp resonances and attribute them to con\fned spin waves in\nthe mm-sized samples. Considering the small damping, insulating chiral magnets turn out to be\npromising candidates when exploring non-collinear spin structures for high frequency applications.\nPACS numbers: 76.50.+g, 74.25.Ha, 4.40.Az, 41.20.Jb\nThe development of future devices for microwave ap-\nplications, spintronics and magnonics [1{3] requires ma-\nterials with a low spin wave (magnon) damping. In-\nsulating compounds are advantageous over metals for\nhigh-frequency applications as they avoid damping via\nspin wave scattering at free charge carriers and eddy\ncurrents [4, 5]. Indeed, the ferrimagnetic insulator yt-\ntrium iron garnet (YIG) holds the benchmark with a\nGilbert damping parameter \u000bintr= 3\u000210\u00005at room\ntemperature [6, 7]. During the last years chiral mag-\nnets have attracted a lot of attention in fundamental\nresearch and stimulated new concepts for information\ntechnology [8, 9]. This material class hosts non-collinear\nspin structures such as spin helices and Skyrmions be-\nlow the critical temperature Tcand critical \feld Hc2\n[10{12]. Additionally, Dzyaloshinskii-Moriya interaction\n(DMI) is present that induces both the Skyrmion lattice\nphase and nonreciprocal microwave characteristics [13].\nLow damping magnets o\u000bering DMI would generate new\nprospects by particularly combining complex spin order\nwith long-distance magnon transport in high-frequency\napplications and magnonics [14, 15]. At low tempera-\ntures, they would further enrich the physics in magnon-\nphoton cavities that call for materials with small \u000bintrto\nachieve high-cooperative magnon-to-photon coupling in\nthe quantum limit [16{19].\nIn this work, we investigate the Gilbert damping in\nCu2OSeO 3, a prototypical insulator hosting Skyrmions\n[20{23]. This material is a local-moment ferrimagnet\nwithTc= 58 K and magnetoelectric coupling [24] that\ngives rise to dichroism for microwaves [25{27]. The\nmagnetization dynamics in Cu 2OSeO 3has already been\nexplored [13, 28, 29]. A detailed investigation on thedamping which is a key quality for magnonics and spin-\ntronics has not yet been presented however. To eval-\nuate\u000bintrwe explore the \feld polarized state (FP)\nwhere the two spin sublattices attain the ferrimagnetic\narrangement[21]. Using spectra obtained by two di\u000ber-\nent coplanar waveguides (CPWs), we extract a minimum\n\u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K, i.e. only about four times\nhigher than in YIG. We resolve numerous sharp reso-\nnances in our spectra and attribute them to modes that\nare con\fned modes across the macroscopic sample and\nallowed for by the low damping. Our \fndings substanti-\nate the relevance of insulating chiral magnets for future\napplications in magnonics and spintronics.\nFrom single crystals of Cu 2OSeO 3we prepared two\nbar-shaped samples exhibiting di\u000berent crystallographic\norientations. The samples had lateral dimensions of\n2:3\u00020:4\u00020:3 mm3. They were positioned on CPWs that\nprovided us with a dynamic magnetic \feld hinduced by\na sinusoidal current applied to the signal surrounded by\ntwo ground lines. We used two di\u000berent CPWs with ei-\nther a broad [30] or narrow signal line width of ws= 1 mm\nor 20\u0016m, respectively [31]. The central long axis of the\nrectangular Cu 2OSeO 3rods was positioned on the central\naxis of the CPWs. The static magnetic \feld Hwas ap-\nplied perpendicular to the substrate with Hkh100iand\nHkh111ifor sample S1 and S2, respectively. The direc-\ntion ofHde\fned the z-direction. The dynamic \feld com-\nponent h?Hprovided the relevant torque for excita-\ntion. Components hkHdid not induce precessional mo-\ntion in the FP state of Cu 2OSeO 3. We recorded spectra\nby a vector network analyzer using the magnitude of the\nscattering parameter S12. We subtracted a background\nspectrum recorded at 1 T to enhance the signal-to-noisearXiv:1705.03416v1 [cond-mat.str-el] 9 May 20172\nratio (SNR) yielding the displayed \u0001 jS12j. In Ref. [7],\nKlingler et al. have investigated the damping of the in-\nsulating ferrimagnet YIG and found that Gilbert param-\neters\u000bintrevaluated from both the uniform precessional\nmode and standing spin waves con\fned in the macro-\nscopic sample provided the same values. For Cu 2OSeO 3\nwe evaluated \u000bin two ways[32]. When extracting the\nlinewidth \u0001 Hfor di\u000berent resonance frequencies fr, the\nGilbert damping parameter \u000bintrwas assumed to vary\naccording to [33, 34]\n\u00160\r\u0001\u0001H= 4\u0019\u000bintr\u0001fr+\u00160\r\u0001\u0001H0; (1)\nwhere\ris the gyromagnetic factor and \u0001 H0the contri-\nbution due to inhomogeneous broadening. Equation (1)\nis valid when viscous Gilbert damping dominates over\nscattering within the magnetic subsystem [35]. When\nperforming frequency-swept measurements at di\u000berent\n\feldsH, the obtained linewidth \u0001 fwas considered to\nscale linearly with the resonance frequency as [36]\n\u0001f= 2\u000bintr\u0001fr+ \u0001f0; (2)\nwith the inhomogeneous broadening \u0001 f0. The conver-\nsion from Eq. (1) to Eq. (2) is valid when frscales linearly\nwithHandHis applied along a magnetic easy or hard\naxis of the material [37, 38]. In Fig. 1 (a) to (d) we show\nspectra recorded in the FP state of the material using the\ntwo di\u000berent CPWs. For the same applied \feld Hwe ob-\nserve peaks residing at higher frequency fforHkh100i\ncompared to Hkh111i. From the resonance frequencies,\nwe extract the cubic magnetocrystalline anisotropy con-\nstantK= (\u00000:6\u00060:1)\u0001103J/m3for Cu 2OSeO 3[31].\nThe magnetic anisotropy energy is found to be extremal\nforh100iandh111ire\recting easy and hard axes, respec-\ntively [31]. The saturation magnetization of Cu 2OSeO 3\namounted to \u00160Ms= 0:13 T at 5 K[22].\nFigure 1 summarizes spectra taken with two di\u000ber-\nent CPWs on two di\u000berent Cu 2OSeO 3crystals exhibit-\ning di\u000berent crystallographic orientation in the \feld H.\nFor the narrow CPW [Fig. 1 (a) and (c)], we observed a\nbroad peak superimposed by a series of resonances that\nall shifted to higher frequencies with increasing H. The\n\feld dependence excluded them from being noise or arti-\nfacts of the setup. Their number and relative intensities\nvaried from sample to sample and also upon remounting\nthe same sample in the cryostat (not shown). They disap-\npeared with increasing temperature Tbut the broad peak\nremained. For the broad CPW [Fig. 1 (b) and (d)], we\nmeasured pronounced peaks whose linewidths were sig-\nni\fcantly smaller compared to the broad peak detected\nwith the narrow CPW. We resolved resonances below\nthe large peaks [arrows in Fig. 1 (b)] that shifted with\nHand exhibited an almost \feld-independent frequency\no\u000bset from the main peaks that we will discuss later. It\nis instructive to \frst follow the orthodox approach and\nanalyze damping parameters from modes re\recting the\n69121518-0.4-0.20.0(d)Δ |S12|f\n (GHz)H\n || 〈111〉 \n69121518-6-30(c)Δ |S12| (10-2)f\n (GHz)\n-0.6-0.4-0.20.0H\n || 〈100〉 (b)broad CPWΔ |S12|\n-0.3-0.2-0.10.00\n.35 T(a)narrow CPWΔ |S12|0\n.25 T0\n.45 T0.55 TFIG. 1. (Color online) Spectra \u0001 jS12jobtained at T = 5 K\nfor di\u000berent Husing (a) a narrow and (b) broad CPW when\nHjjh100ion sample S1. Corresponding spectra taken on sam-\nple S2 for Hjjh111iare shown in (c) and (d), respectively.\nNote the strong and sharp resonances in (b) and (d) when us-\ning the broad CPW that provides a much more homogeneous\nexcitation \feld h. Arrows mark resonances that have a \feld-\nindependent o\u000bset with the corresponding main peaks and are\nattributed to standing spin waves. An exemplary Lorentz \ft\ncurve is shown in blue color in (b).\nexcitation characteristics of the CPW [29]. Second, we\nfollow Ref. [7] and analyze con\fned modes.\nLorentz curves (blue) were \ftted to the spectra\nrecorded with the broad CPW to determine resonance\nfrequencies and linewidths. Note that the corresponding\nlinewidths were larger by a factor ofp\n3 compared to the\nlinewidth \u0001 fthat is conventionally extracted from the\nimaginary part of the scattering parameters [39]. The\nextracted linewidths \u0001 fwere found to follow linear \fts\nbased on Eq. (2) at di\u000berent temperatures (details are\nshown in Ref. [31]). In Fig. 2 (a) we show a resonance\ncurve that was obtained as a function of Htaken with\nthe narrow CPW at 15 GHz. The curve does not show\nsharp features as Hwas varied in \fnite steps (symbols).\nThe linewidth \u0001 H(symbols) is plotted in Fig. 2 (b) for\ndi\u000berent resonance frequencies and temperatures. The\ndata are well described by linear \fts (lines) based on\nEq. (1). Note that the resonance peaks measured with\nthe broad CPW were extremely sharp. The sharpness\ndid not allow us to analyze the resonances as a function\nofH. We refrained from \ftting the broad peaks of Fig. 1\n(a) and (c) (narrow CPW) as they showed a clear asym-\nmetry attributed to the overlap of subresonances at \fnite\nwavevector k, as will be discussed below.\nIn Fig. 3 (a) and (b) we compare the parameter \u000bintr\nobtained from both di\u000berent CPWs (circles vs. stars) and\nthe two evaluation routes [40]. For Hkh100i[Fig. 3 (a)],\nbetween 5 and 20 K the lowest value for \u000bintramounts to\n(3.7\u00060.4)\u000210\u00003. This value is three times lower com-\npared to preliminary data presented in Ref. [29]. Beyond3\nFIG. 2. (Color online) (a) Lorentz curve (magenta line) \ftted\nto a resonance (symbols) measured at f= 15 GHz as a func-\ntion ofHat 5 K. (b) Frequency dependencies of linewidths\n\u0001H(symbols) for four di\u000berent T. We performed thep\n3-\ncorrection. The slopes of linear \fts (straight lines) following\nEq. 1 are considered to re\rect the intrinsic damping parame-\nters\u000bintr.\n04812H || 〈100〉 αintr (10-3)Δ H narrow CPWΔ\n f broad CPW\nH || 〈111〉 \n1020304050T\n (K)\n10203040500.00.20.40.60.8Δf0 (GHz)T\n (K)(b)( a)(\nd)( c)\nFIG. 3. (Color online) (a) and (b) Intrinsic damping param-\neters\u000bintrand inhomogeneous broadening \u0001 f0for two di\u000ber-\nent \feld directions (see labels) obtained from the slopes and\nintercepts at fr= 0 of linear \fts to the linewidth data (see\nFig. 2 (b) and Ref. [31]). Dashed lines are guides to the eyes.\n20 K the damping is found to increase. For Hkh111i\n[Fig. 3 (b)] we extract (0.6 \u00060.6)\u000210\u00003as the smallest\nvalue. Note that these values for \u000bintrstill contain an ex-\ntrinsic contribution and thus represent upper bounds for\nCu2OSeO 3, as we will show later. For the inhomogeneous\nbroadening \u0001 f0in Fig. 3 (c) and (d) the datasets are\nconsistent (we have used the relation \u0001 f0=\r\u0001H0=2\u0019\nto convert \u0001 H0into \u0001f0). We see that \u0001 f0increases\nwithTand is small for the broad CPW, independent\nof the crystallographic direction of H. For the narrow\nCPW the inhomogeneous broadening is largest at small\nTand then decreases by about 40 % up to about 50\nK. Note that a CPW broader than the sample is as-\nsumed to excite homogeneously at fFMR [41] transfer-\nring a wave vector k= 0 to the sample. Accordinglywe ascribe the intense resonances of Fig. 1 (b) and (d) to\nfFMR. UsingfFMR= 6 GHz and \u000bintr= 3:7\u000210\u00003at 5\nK [Fig. 3 (a)], we estimate a minimum relaxation time of\n\u001c= [2\u0019\u000bintrfr]\u00001= 6:6 ns.\nIn the following, we examine in detail the additional\nsharp resonances that we observed in spectra of Fig. 1.\nIn Fig. 1 (b) taken with the broad CPW for Hkh100i,\nwe identify sharp resonances that exhibit a characteris-\ntic frequency o\u000bset \u000efwith the main resonance at all\n\felds (black arrows). We illustrate this in Fig. 4(a) in\nthat we shift spectra of Fig. 1 (b) so that the positions of\ntheir main resonances overlap. The additional small res-\nonances (arrows) in Fig. 1 (b) are well below the uniform\nmode. This is characteristic for backward volume magne-\ntostatic spin waves (BVMSWs). Standing waves of such\nkind can develop if they are re\rected at least once at the\nbottom and top surfaces of the sample. The resulting\nstanding waves exhibit a wave vector k=n\u0019=d , with\norder number nand sample thickness d= 0:3 mm. The\nBVMSW dispersion relation f(k) of Ref. [13] provides a\ngroup velocity vg=\u0000300 km/s at k=\u0019=d[triangles in\nFig. 4 (b)]. Hence, the decay length ld=vg\u001camounts\nto 2 mm considering \u001c= 6:6 ns. This is larger than\ntwice the relevant lateral sizes, thereby allowing stand-\ning spin wave modes to form in the sample. Based on\nthe dispersion relation of Ref. [13], we calculated the fre-\nquency splitting \u000ef=fFMR\u0000f(n\u0019=d ) [open diamonds\nin Fig. 4 (b)] assuming n= 1 andt= 0:4 mm for the\nsample width tde\fned in Ref. [13]. Experimental val-\nues (\flled symbols) agree with the calculated ones (open\nsymbols) within about 60 MHz. In case of the narrow\nCPW, we observe even more sharp resonances [Fig. 1 (a)\nand (c)]. A set of resonances was reported previously\nin the \feld-polarized phase of Cu 2OSeO 3[26, 28, 42, 43].\nMaisuradze et al. assigned secondary peaks in thin plates\nof Cu 2OSeO 3to di\u000berent standing spin-wave modes [43]\nin agreement with our analysis outlined above.\n0.30.40.51.101.151.201.25-\n500-300-100100δf (GHz)(b)/s61549\n0H (T)v\ng (km/s)\n-10 -0.8-0.6-0.4-0.20.0f\n - f (0) (GHz)H || 〈100〉 (a)b\nroad CPWΔ |S12|δ\nf\nFIG. 4. (Color online) (a) Spectra of Fig. 1 (b) replotted as\nf\u0000fFMR(H) for di\u000berent Hsuch that all main peaks are at\nzero frequency and the \feld-independent frequency splitting\n\u000efbecomes visible. The numerous oscillations seen particu-\nlarly on the bottom most curve are artefacts from the cali-\nbration routine. (b) Experimentally evaluated (\flled circles)\nand theoretically predicted (diamonds) splitting \u000efusing dis-\npersion relations for a platelet. Calculated group velocity vg\natk=\u0019=(0:3 mm). Dashed lines are guides to the eyes.4\nThe inhomogeneous dynamic \feld hof the narrow\nCPW provides a much broader distribution of kcom-\npared to the broad CPW. This is consistent with the\nfact that the inhomogeneous broadening \u0001 f0is found to\nbe larger for the narrow CPW compared to the broad\none [Fig. 3 (c) and (c)]. Under these circumstances, the\nexcitation of more standing waves is expected. We at-\ntribute the series of sharp resonances in Fig. 1 (a) and\n(c) to such spin waves. In Fig. 5 (a) and (b) we highlight\nprominent and particularly narrow resonances with #1,\n#2 and #3 recorded with the narrow CPW. We trace\ntheir frequencies fras a function of HforHkh100iand\nHkh111i, respectively. They depend linearly on Hsug-\ngesting a Land\u0013 e factor g= 2:14 at 5 K.\nWe now concentrate on mode #1 for Hk h100iat\n5 K that is best resolved. We \ft a Lorentzian line-\nshape as shown in Fig. 5(c) for 0.85 T, and summarize\nthe corresponding linewidths \u0001 fin Fig. 5(d). The inset\nof Fig. 5(d) shows the e\u000bective damping \u000be\u000b= \u0001f=(2fr)\nevaluated directly from the linewidth as suggested in Ref.\n[29]. We \fnd that \u000be\u000bapproaches a value of about 3.5\n\u000210\u00004with increasing frequency. This value includes\nboth the intrinsic damping and inhomogeneous broad-\nening but is already a factor of 10 smaller compared to\n\u000bintrextracted from Fig. 3 (a). Note that Cu 2OSeO 3\nexhibiting 3.5\u000210\u00004outperforms the best metallic thin-\n\flm magnet [44]. To correct for inhomogeneous broad-\nening and determine the intrinsic Gilbert-type damping,\nwe apply a linear \ft to the linewidths \u0001 fin Fig. 5(d) at\nfr>10:6 GHz and obtain (9.9 \u00064.1)\u000210\u00005. Forfr\u0014\n10.6 GHz the resonance amplitudes of mode #1 were\nsmall reducing the con\fdence of the \ftting procedure.\nFurthermore, at low frequencies, we expect anisotropy to\nmodify the extracted damping, similar to the results in\nRef. [45]. For these reasons, the two points at low frwere\nleft out for the linear \ft providing (9.9 \u00064.1)\u000210\u00005.\nWe \fnd \u0001fand the damping parameters of Fig. 3 to\nincrease with T. It does not scale linearly for Hkh100i\n[31]. A deviation from linear scaling was reported for\nYIG single crystals as well and accounted for by the con-\n\ruence of a low- kmagnon with a phonon or thermally\nexcited magnon [5]. In the case of Hkh111i(cf. Fig. 3\n(b)) we obtain a clear discrepancy between results from\nthe two evaluation routes and CPWs used. We relate\nthis observation to a misalignment of Hwith the hard\naxish111i. The misalignment motivates a \feld-dragging\ncontribution [38] that can explain the discrepancy. For\nthis reason, we concentrated our standing wave analysis\non the case Hkh100i. We now comment on our spectra\ntaken with the broad CPW that do not show the very\nsmall linewidth attributed to the con\fned spin waves.\nThe sharp mode #1 yields \u0001 f= 15:3 MHz near 16 GHz\n[Fig. 5 (d)]. At 5 K the dominant peak measured at 0.55 T\nwith the broad CPW provides however \u0001 f= 129 MHz.\n\u0001fobtained by the broad CPW is thus increased by a\nfactor of eight and explains the relatively large Gilbert\nFIG. 5. (Color online) (a)-(b) Resonance frequency as a func-\ntion of \feld Hof selected sharp modes labelled #1 to #3 (see\ninsets) for Hkh100iandHkh111iat T = 5 K. (c) Exemplary\nLorentz \ft of sharp mode #1 for Hkh100iat 0.85 T. (d) Ex-\ntracted linewidth \u0001f as a function of resonance frequency fr\nalong with the linear \ft performed to determine the intrinsic\ndamping\u000bintrin Cu 2OSeO 3. Inset: Comparison among the\nextrinsic and intrinsic damping contribution. The red dotted\nlines mark the error margins of \u000bintr= (9:9\u00064:1)\u000210\u00005.\ndamping parameter in Fig. 3 (a) and (b). We con\frmed\nthis larger value on a third sample with Hkh100iand ob-\ntained (3.1\u00060.3)\u000210\u00003[31] using the broad CPW. The\ndiscrepancy with the damping parameter extracted from\nthe sharp modes of Fig. 5 might be due to the remaining\ninhomogeneity of hover the thickness of the sample lead-\ning to an uncertainty in the wave vector in z-direction.\nFor a standing spin wave such an inhomogeneity does\nnot play a role as the boundary conditions discretize k.\nAccordingly, Klingler et al. extract the smallest damp-\ning parameter of 2 :7(5)\u000210\u00005reported so far for the\nferrimagnet YIG when analyzing con\fned magnetostatic\nmodes [7].\nTo summarize, we investigated the spin dynamics in\nthe \feld-polarized phase of the insulating chiral mag-\nnet Cu 2OSeO 3. We detected numerous sharp reso-\nnances that we attribute to standing spin waves. Their\ne\u000bective damping parameter is small and amounts to\n3:5\u000210\u00004. A quantitative estimate of the intrinsic\nGilbert damping parameter extracted from the con\fned\nmodes provides even \u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K. The\nsmall damping makes an insulating ferrimagnet exhibit-\ning Dzyaloshinskii-Moriya interaction a promising can-\ndidate for exploitation of complex spin structures and\nrelated nonreciprocity in magnonics and spintronics.\nWe thank S. Mayr for assistance with sample prepa-\nration. Financial support through DFG TRR80, DFG\n1143, DFG FOR960, and ERC Advanced Grant 291079\n(TOPFIT) is gratefully acknowledged.5\n\u0003Electronic mail: dirk.grundler@ep\r.ch\n[1] I. Zutic and H. Dery, Nat. Mater. 10, 647 (2011).\n[2] M. Krawczyk and D. Grundler, J. Phys.: Condens. Mat-\nter26, 123202 (2014).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. 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Phys. 85(1999)." }, { "title": "2110.14713v2.Low_temperature_competing_magnetic_energy_scales_in_the_topological_ferrimagnet_TbMn6Sn6.pdf", "content": "Low temperature competing magnetic energy scales in the topological ferrimagnet\nTbMn 6Sn6\nS. X. M. Riberolles,1Tyler J. Slade,1, 2D. L. Abernathy,3G. E. Granroth,3Bing\nLi,1, 2Y. Lee,1P. C. Can\feld,1, 2B. G. Ueland,1Liqin Ke,1and R. J. McQueeney1, 2\n1Ames Laboratory, Ames, IA, 50011, USA\n2Department of Physics and Astronomy, Iowa State University, Ames, IA, 50011, USA\n3Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA\n(Dated: June 28, 2022)\nTbMn 6Sn6is a metallic ferrimagnet displaying signatures of both topological electrons and topo-\nlogical magnons arising from ferromagnetism and spin-orbit coupling within its Mn kagome layers.\nInelastic neutron scattering measurements \fnd strong ferromagnetic (FM) interactions within the\nMn kagome layer and reveal a magnetic bandwidth of \u0018230 meV. The low-energy magnetic ex-\ncitations are characterized by strong FM Mn-Mn and antiferromagnetic (AFM) Mn-Tb interlayer\nmagnetic couplings. We observe weaker, competing long-range FM and AFM Mn-Mn interlayer\ninteractions similar to those driving helical magnetism in the YMn 6Sn6system. Combined with\ndensity-functional theory calculations, we \fnd that competing Mn-Mn interlayer magnetic interac-\ntions occur in all RMn6Sn6compounds with R= Y, Gd \u0000Lu, resulting in magnetic instabilities\nand tunability when Mn- Rinteractions are weak. In the case of TbMn 6Sn6, strong AFM Mn-Tb\ncoupling ensures a highly stable three-dimensional ferrimagnetic network.\nI. INTRODUCTION\nThe potential technological applications of magnetic\ntopological insulators and Weyl semimetals has generated\nnew research directions aimed at understanding the cou-\npling between magnetism and topological fermions. This\nhas brought renewed interest in magnetic kagome met-\nals, such as Mn 3Ge [1, 2], Fe 3Sn2[3, 4], Co 3Sn2S2[5{7]\nand FeSn [8], where both magnetism and topological elec-\ntronic band crossings are hosted in the kagome layer. In-\nteresting topological responses, such as large anomalous\nHall conductivity, are tied to the underlying magnetic or-\nder that can be impacted by both geometrical frustration\nand Dzyaloshinskii-Moriya (DM) interactions. In princi-\nple, these materials may host topological magnons in the\npresence of DM interactions [9], opening up even more in-\nteresting avenues for the study of topological phenomena\nin metallic kagome systems.\nThe hexagonal RMn6Sn6(R166) compounds ( R=\nrare-earth) consist of alternating Mn kagome and Rtrian-\ngular layers. Contemporary studies of R166 compounds\nhave focused on the interplay between their complex\nmagnetism and topological electronic kagome band cross-\nings [10{17]. R166 materials display a variety of magnetic\nstructures, including antiferromagnetic (AFM), ferrimag-\nnetic, and complex helical ordering, that are dependent\non the nature of the host Rion [10, 18{25]. In addition,\nunique temperature and \feld-driven magnetic instabili-\nties found in R166 compounds [14, 15, 26, 27] promise to\nopen new avenues in topological state control and switch-\ning.\nInR166, the intralayer Mn-Mn interactions are\nstrongly ferromagnetic (FM) and magnetic complexity\narises from a combination of competing Mn and Rmag-\nnetic anisotropies (for moment-bearing R-ions) and com-\npeting interlayer magnetic interactions [28{31]. R166compounds with non-moment-bearing rare-earths, such\nas Y166, are easy-plane AFMs where competing FM and\nAFM coupling between FM Mn layers drives transitions\nfrom collinear to complex helical magnetic phases dis-\nplaying net chirality and a topological Hall response in\napplied magnetic \felds [10, 14, 15, 20]. For moment-\nbearing rare-earths, the magnetism is strongly a\u000bected\nby rare-earth anisotropy and coupling between Mn and\nRlayers. In Tb166, strong uniaxial anisotropy of the\nTb ion and AFM Mn-Tb coupling favors unique uniax-\nial collinear ferrimagnetic state that has realized Chern-\ngapped topological fermions with a quantized magneto-\ntransport response [16]. Surprisingly, Tb166 possesses a\nspin reorientation transition from easy-axis to easy-plane\nferrimagnetism [21, 22, 26, 27]. These discoveries demon-\nstrate great potential for novel topological phenomena to\nbe discovered by exploring other R166 materials via rare-\nearth engineering [32] or by the application of symmetry-\nbreaking external \felds.\nTo access this potential, we must address several open\nquestions regarding the fundamental nature of the mag-\nnetism within R166 compounds. For example, is the Mn\nmagnetism of an itinerant or local-moment nature and\nare the Mn-Mn interactions transferrable across the R166\nmaterials? What is the variability of R-Mn interactions\nandRanisotropy across the series? Also, given recent re-\nports on the connection between thermally driven mag-\nnetic \ructuations and quantum transport in Y166 [10]\nand Tb166 [13], what is the role of magnetic \ructuations\nin the emergent topological properties through the R166\nfamily?\nHere, we address the magnetic interactions in Tb166\nin detail using inelastic neutron scattering (INS) and\ndensity-functional theory (DFT) calculations. Using\nINS, we observe a hierarchy of competing interlayer Mn-\nMn interactions in Tb166 similar to those used to explain\nthe complex temperature- and \feld-driven helical mag-arXiv:2110.14713v2 [cond-mat.str-el] 24 Jun 20222\nnetism observed in Y166 [10, 14, 15, 17, 20]. We \fnd that\nstrong uniaxial Tb magnetic anisotropy and AFM cou-\npling between Mn and Tb layers generates a rigid three-\ndimensional ferrimagnetic lattice. A clean spin gap of 6.5\nmeV suppresses collective spin \ructuations at tempera-\ntures relevant for quantum transport ( <20 K). Thus,\nit is likely that the main avenue available for tuning the\ntopological band states in Tb166 is by controlling the spin\nreorientation transition. Results of our DFT calculations\nlargely agree with the sign, magnitude and overall hier-\narchy of interlayer couplings found experimentally after\nthe introduction of on-site Coulomb repulsion (DFT+U).\nThe INS data also show that FM intralayer Mn-Mn in-\nteractions in both Tb166 and Y166 (Ref. [17]) are compa-\nrably strong and push the overall magnon bandwidth up\nto\u0018230 meV. However, increasingly broad lineshapes for\nTb166 do not allow the observation of magnetic excita-\ntions above\u0018125 meV. Unlike reports of a K-point gap\ncaused by DM interactions in the magnon spectrum of\nY166 [17], this severe line broadening in Tb166 obscures\nany evidence of a topological magnon gap. This suggests\nthat, despite our quantitative modeling of the spin-wave\nspectrum presented here, there is still much to be learned\nabout the itinerant character of Mn magnetism and the\nrole of spin-orbit interactions in R166 materials.\nII. EXPERIMENTAL DETAILS\nSingle crystals of Tb166 were grown from excess Sn\nusing the \rux method. A nominal (TbMn 6)5Sn95mo-\nlar ratio of elemental Tb (Ames Laboratory 99.9%), Mn\n(Research Organic/Inorganic Chemical Corp, 99.995%),\nand Sn (Alfa Aesar, 99.99%) was weighed and loaded\ninto the growth side of a 5 mL fritted alumina crucible\nset [33]. The crucibles were \rame sealed under vacuum\ninside an 18 mm diameter fused silica ampule with a\nsmall amount of silica wool placed above and below the\ncrucibles to serve as cushioning, and heated to 1180\u000eC\nin 12 hours. After dwelling at 1180\u000eC for 3 hours, the\nfurnace was quickly cooled in 3 hours to 775\u000eC and then\nslowly cooled over 300 hours to 575\u000eC. Upon reaching\nthe \fnal temperature, the tube was rapidly removed from\nthe furnace, inverted into a metal centrifuge, and the ex-\ncess \rux decanted. The crucibles were opened to reveal\nlarge (up to 300 mg), shiny, hexagonal crystal plates (see\nFig. 1(a)).\nLow temperature magnetization was measured using a\nQuantum Design Magnetic Property Measurement Sys-\ntem (MPMS 3), SQUID magnetometer ( T= 1:8\u0000300\nK,Hmax=70 kOe). A Tb166 single crystal sample was\nmounted on a plastic disc and the \feld was applied along\nc. Prior to measuring the sample, the blank disc was\nmeasured and used for a background subtraction. Fig-\nure 1(a) shows low temperature magnetization measured\nat 2 K with Hkcthat accurately reproduce the pre-\nviously reported hysteresis loop displaying a saturated\nmagnetization of \u00194\u0016B/f.u. [34].Tb166 crystallizes in the HfFe 6Ge6-type structure with\nhexagonal space group P6/mmm (No. 191) and Mn, Sn1,\nSn2, Sn3 and Tb ions, respectively, sitting at the 6i, 2e,\n2d, 2c and 1b Wycko\u000b positions [35], see Fig. 1(c-d).\nFrom a Rietveld analysis of XRD data collected at 300 K\n(see Fig. 1(b)), we obtain re\fned values of 5.53317(6)\nand 9.0233(1) \u0017A for lattice parameters aandc, as well as\natomic coordinates z Mn=0.2539(2) and z Sn1=0.1624(2),\nin close agreement with previous reports [21, 22]. Be-\nlow 423 K, both the Mn and Tb layers simultaneously\ndevelop FM order, but couple antiferromagnetically, re-\nsulting in an overall ferrimagnetic order. All magnetic\nmoments initially lie in the basal plane, but remarkably,\nupon cooling between 350 K and 305 K a spin reorien-\ntation takes place, resulting in the ground state collinear\nferrimagnetic arrangement of Mn and Tb moments along\nthec-axis [21, 22] shown in Fig. 1(c) and 1(d).\nINS measurements were performed on the Wide\nAngular-Range Chopper Spectrometer (ARCS) located\nat the Spallation Neutron Source at Oak Ridge National\nLaboratory [36]. An array of \fve crystals with a total\nmass of 495.6 mg was co-aligned with the ( H;0;L) scat-\ntering plane set horizontally, and attached to the cold\nhead of a closed-cycle-refrigerator. Data were collected\nat the base temperature of 7 K using incident energies\nofEi= 30, 75, 160 and 250 meV [elastic resolutions\nare listed in Table I in the Supplemental Material (SM)\n[37]]. ForEi= 30, 160 and 250 meV, the sample was\nrotated around 180 degrees in one degree increments for\nfull coverage of q,Espace, where q(E) is the momen-\ntum (energy) transfer, respectively. For Ei= 75 meV,\nthe rotation increment was reduced to half a degree.\nThe INS data were reduced to qandE, symmetrized\nto improve statistics, and cuts made for further analy-\nsis using Mantid [38]. The neutron scattering data are\ndescribed using the momentum transfer in hexagonal re-\nciprocal lattice units, q(H;K;L ) =2\u0019\na2p\n3(H^a\u0003+K^b\u0003) +\n2\u0019\ncL^c. The INS data are presented in terms of the or-\nthogonal vectors (1 ;0;0), (\u00001;2;0), and (0,0,1), as shown\nin Fig. 1(e). Special K- and M-points in the Brillouin\nzone are found at ( H;K;L ) = (1\n3;1\n3;0) and (1\n2,0,0) and\nsymmetry-related points, respectively. The INS data are\ndisplayed as intensities that are proportional to the spin-\nspin correlation function S(q;E). To improve statistics,\nthe data have been symmetrized with respect to the crys-\ntallographic space group P6/mmm .\nWe \frst examined the elastic scattering from our co-\naligned crystals [shown in Fig. 1(g)] and compared the\ndata to simulations of the nuclear and magnetic scat-\ntering [shown in Fig. 1(f)]. Using the Bilbao crystal-\nlographic server, we \fnd that below 250 K the mag-\nnetic structure adopts the high-symmetry magnetic space\ngroup P6/mm'm' (No. 191.240) where both magnetic\nsublattices are restricted to have their ordered moments\nlying along the c-axis [39]. The ordered magnetic mo-\nment at 4.5 K are reported as 2.17 and 9.0 \u0016Bfor Mn\nand Tb, respectively [26]. Using these values and the\nP6/mm'm' symmetry, we simulated the corresponding3\nnuclear and magnetic neutron di\u000braction patterns for the\n(0;K;L ) plane using mag2pol [40]. The good agree-\nment obtained between simulated and experimentally\nmeasured patterns con\frms the high quality of our sam-\nples as well as the previously reported low temperature\nferrimagnetic ground state in Tb166.\nIII. MINIMAL HEISENBERG MODEL FOR\nTHE SPIN EXCITATIONS\nBefore describing the INS data, we \frst discuss a min-\nimal description of the magnetic interactions in Tb166\nand the key features of the resultant spin excitations.\nKagome layers are known for unusual magnetic behavior\ndue to geometric frustration and the role of spin-orbit\ncoupling via the DM interaction. All known hexago-\nnalR166 compounds possess FM kagome layers with an\neasy-plane Mn magnetic anisotropy which minimizes the\nrole of intralayer geometric frustration [41]. However,\nthe competition between Mn-Mn FM and AFM inter-\nlayer magnetic interactions is known to cause magnetic\ninstabilities in Y166 that lead to complex helical phases\n[10, 17, 20].\nForR166 compounds with magnetic rare-earth ions,\ntwo additional factors control the magnetic behavior.\nThe \frst is strong AFM coupling between the Rand\nMn sublattices that can result in tightly bound Mn- R-\nMn collinear ferrimagnetic trilayers. The second factor\nis the single-ion anisotropy of the rare-earth ion. For fer-\nrimagnetic Gd166, the weak anisotropy of the spin-only\nGd3+ion combined with easy-plane Mn anisotropy and\nGd\u0000Mn AFM coupling results in antiparallel ordered Gd\nand Mn moments lying in the basal layer [21]. On the\nother hand, R= Tb\u0000Ho ions possess uniaxial anisotropy\nthat competes with the Mn easy-plane anisotropy. This\ncompetition, along with higher-order contributions to\ntheRanisotropy[28, 30], drives spin reorientation tran-\nsitions where the ordered Mn and Rmoments rotate\nin unison [21, 22]. As mentioned above, Tb166 adopts\nan out-of-plane uniaxial ferrimagnetic ground state [see\nFig. 1(c)], with Mn and Tb moments collectively rotating\nto fully lie in the basal plane above Tsr= 350 K.R= Dy\nand Ho are similar ferrimagnets with spin reorientation\ntransitions, but the weaker R-ion anisotropy results in a\nground state easy-axis that is tilted away from the c-axis\n[21, 22]. Close to Tsr, the competing Rand Mn single-ion\nanisotropies drive \frst-order magnetization processes in\napplied magnetic \felds [26, 28].\nGiven the already interesting role of competing inter-\nlayer interactions in Y166 and competing anisotropies\ninR= Tb\u0000Ho, it remains to consider their com-\nbined role in R166 with magnetic rare-earths. We\nde\fne a general Heisenberg model with the Hamilto-\nnianH=Hintra+Hinter+Haniso+HDMthat consists\nof isotropic intralayer and interlayer pairwise exchange,\nsingle-ion anisotropy, and DM interactions.\nIn our minimal description, each Mn kagome layerpossesses strong nearest-neighbor (NN) FM exchange\n(J <0) which determines the large overall magnon band-\nwidth.\nHintra=JX\nhi0 is the AFM coupling between neighbor-\ning Mn and Tb layers, with Tb having a spin angular\nmomentum of S= 3. We label interactions between Mn\nlayers by a layer index k(JMM\nk). Due to the Tb layer,\nadjacent Mn layers above and below a given Mn layer are\ninequivalent. Our data indicate that the FM coupling be-\ntween next-nearest neighbor (NNN) Mn-Mn layers sepa-\nrated by a Sn 4block (JMM\n2) is stronger than the coupling\nbetween NN Mn-Mn layers separated by a TbSn 2block\n(JMM\n1), in agreement with analysis of neutron di\u000braction\ndata [20, 31].\nBy itself,JMM\n2forms strongly coupled FM Mn-Mn bi-\nlayers and generates a bilayer splitting !B= 2sjJMM\n2j\nof the single-layer dispersion into odd and even modes,\nas shown in Fig. 2(a). The K-point splits into two (odd\nand even) topological magnon crossings that remain un-\ngapped in the absence of DM interactions.\nThe strong AFM interaction JMTgenerates a ferri-\nmagnetic exchange \feld with energy scale !F= 2(6s\u0000\nS)JMT.!Fincreases the odd-even splitting and gives\nrise to a new branch of Tb character with a spin gap of\n\u0001Tb=!Fat the \u0000-point, as shown in Fig. 2(b).\nThe introduction of uniaxial single-ion anisotropy for\nboth Tb and Mn ( KTandKM) is given by\nHaniso =KMX\ni(sz\ni)2+KTX\ni(Sz\ni)2(3)\nwhere the sums are over each sublattice. Whereas Mn is\nexpected to have a weak easy-plane anisotropy ( KM&0),\nTb has a large uniaxial anisotropy at low temperatures\n(KT<0). WithKM= 0,KTgenerates a spin gap4\nFIG. 1. (a) Single-crystal magnetization data for Tb166 recorded at 2 K with Happlied along c. The inset shows a typical single-crystal\nsample of Tb166. (b) Powder x-ray di\u000braction measurements of Tb166 collected at room temperature and \ftted using Rietveld re\fnement\nanalysis. (c) Ferrimagnetic ground state structure of TbMn 6Sn6. Key interlayer interactions are shown with heavy black arrows. (d)\nMagnetic interactions within a single Mn-Sn kagome layer. (e) 2D hexagonal Brillouin zone showing conventional reciprocal lattice vectors\na\u0003andb\u0003and special points, \u0000 (black), K (blue) and M (red). Inelastic neutron scattering data are discussed in terms of the orthogonal\nvectors (1,0) and (-1,2). (f) Simulated (0, K,L) elastic single crystal neutron scattering intensity containing both nuclear and magnetic\ncomponents for Tb166 below 250 K. The reciprocal space is here set in the conventional way. Antiparallel magnetic moments of 9.0(Tb)\nand 2.17(Mn) \u0016Bare set along c. (g) Tb166 elastic single crystal neutron scattering data collected on ARCS in the (0, K,L) scattering\nplane at 7 K.\nFIG. 2. (a) Monolayer kagome spin wave dispersion with energy\nin units of sjJj(orange dots) and Mn-Mn bilayer dispersion with\nJMM\n2= 0:5jJj(blue lines). The latter shows the bilayer splitting\nof odd and even modes by !B= 2sjJMM\n2j. (b) Low-energy dis-\npersion when Mn bilayers are coupled through Tb with S= 3s\nandJMT=\u00000:04J(blue lines). The odd bilayer mode and Tb\nmode (dashed line) are shifted by the ferrimagnetic exchange \feld,\n!F= 2(6s\u0000S)JMT, as shown. Red lines include uniaxial Tb\nsingle-ion anisotropy with KT= 0:07JandKM= 0 that intro-\nduces a spin gap in the even mode (\u0001) and increases the Tb mode\nspin gap (\u0001 Tb). (c) Interlayer dispersion of low-energy branches\nwith identical bilayer splitting, JMM\n1+JMM\n2= 0:5Jfor cases where\nJMM\n1=JMM\n2(blue lines), JMM\n1= 0 (red lines), and JMM\n2= 0\n(gray lines). (d) Interlayer dispersion of low-energy branches when\nJMM\n1=JMM\n2and the coupling between Mn layers in adjacent unit\ncells,JMM\n3, is either ferromagnetic (red lines), antiferromagnetic\n(blue lines) or zero (gray dashed lines).\n\u0001\u0019p\n2sSKTJMTfor the even branch and increases\n\u0001Tbsuch that \u0001 Tb\u0000\u0001 = 2SKT+!F, as shown in\nFig. 2(b).\nWe now consider the e\u000bect of JMM\n1. WhenJMM\n1= 0,the interlayer dispersion of the low-energy branches is\nmainly controlled by JMT. AsJMM\n1is increased, mod-\nels indicate that the bilayer splitting becomes !B=\n2sjJMM\n1+JMM\n2jand the interlayer bandwidth of odd and\neven modes sharply increases and reaches a maximum\nwhenJMM\n1=JMM\n2, as shown in Fig. 2(c). The limit\nwhereJMM\n2= 0 corresponds to isolated trilayer Mn-Tb-\nMn blocks where the interlayer bandwidth is zero.\nTo better describe the experimental data, an interac-\ntion between like Mn layers in adjacent unit cells, JMM\n3,\nis introduced as well. As shown in Fig. 2(d), JMM\n3op-\npositely a\u000bects the interlayer odd and even bandwidths\nwhile preserving the A-point gap at q= (0;0;1=2). For\nexample, when JMM\n3is AFM, the bandwidth of the odd\nmode increases and the even mode decreases.\nFinally, the presence of DM interactions is principally\nassociated with gapping at the Dirac points at K and\nhas recently been reported in Y166 [17]. However, as de-\nscribed below, we \fnd no clear evidence for a K-point gap\nin Tb166, due to the presence of strong damping. There-\nfore, it is not necessary to introduce DM interactions to\nmodel our data (HDM= 0).\nIV. INTERLAYER DISPERSIONS\nHaving outlined the various expectations for the spin\nwave dispersion in Tb166, we now describe the features of\nthe INS data. Figure 3(a) shows a slice through the Ei=\n30 meV data along the ( H;0;0) and (0;0;L) directions\nthrough the (0,0,2) \u0000-point. The lowest energy mode\nis the even branch, which displays a clean spin gap of\n\u0001 = 6.5 meV as shown by the resolution-limited peak in\nthe energy cut through the \u0000-point at (0,0,2) [Fig. 3(b)].5\nFIG. 3. (a) Slices of the neutron intensity showing the disper-\nsion through the (0,0,2) \u0000-point along ( H;0;0) and (0;0;L) for\ndata taken with Ei= 30 meV. Pink lines correspond to the model\ndispersion relation obtained from \fts described below. Gray verti-\ncal lines identify Brillouin zone centers (solid) and zone boundary\npoints (dashed), as labeled on the top axis. (b) Energy spectrum\nthrough (0,0,2) averaged over qranges of \u0001 H= \u0001K=\u00060:035\nand \u0001L= 0:1 rlu. The red line is a Gaussian \ft that indicates\na resolution-limited peak corresponding to a spin gap of \u0001 = 6 :5\nmeV.\nAlong (0;0;L), the even branch has limited interlayer\ndispersion, reaching only 14 meV at the A-point, whereas\nthe intralayer dispersion of the even branch along ( H,0,0)\nextends to much higher energies.\nWe also glimpse a narrow band of excitations near \u001825\nmeV in Fig. 3(a) that corresponds to the Tb mode. Fig-\nure 4 shows the Tb mode dispersion along ( H;0;0) and\n(0;0;L) more clearly using Ei= 75 meV and focusing\non Brillouin zones where the structure factor of the even\nbranch is close to zero ( L=oddorH=odd).\nThe odd branch is observed in slices of the data taken\nwith higher incident energies of 75 and 160 meV, as\nshown in Fig. 5. The even and odd branches have struc-\nture factors that are maximized in Brillouin zones with\nL=even andL=odd, respectively. Fig. 5(a) and the\nconstant energy cuts in Fig. 5(b) show that the interlayer\nodd branch disperses from roughly 60 meV at the \u0000-point\ndown to 40 meV at the A-point. Constant- qenergy cuts\nat (0,0,3) and (0,0,4) in Fig. 5(c) also demonstrate a \u0000-\npoint energy of\u001860 meV for the odd branch. Considering\nthe spin gap, this allows for an estimate of an odd-even\nsplitting of !B+!F\u001955 meV. Figs. 5(a) \u0000(c) show that\nthe odd branch is signi\fcantly weaker and broader than\nthe resolution-limited low-energy even and Tb branches,\nbut has a much larger interlayer bandwidth.\nVarious data cuts similar to those shown in Figs. 3 \u00005\nwere used to produce a list of dispersion points, !i(q),\nfor even, odd, and Tb interlayer branches in various Bril-\nlouin zones. In this list, we also include the energies\nof the intralayer Tb modes along ( H;0) [Fig. 4(a)] and\n(\u0000K;2K) whose dispersions are sensitive to JMTand\nKT. We used this list of 100 observables to \ft the ex-\nperimental dispersion to the reduced Heisenberg model\nH=Hinter+Haniso using SpinW [42]. The Mn and Tb\nFIG. 4. Slices of the intensity along ( H;0;3) (left) and (1 ;0;L)\n(right) with Ei= 75 meV showing the intralayer and interlayer\ndispersion of the Tb mode, respectively. The two slices employed\nreciprocal space averaging of \u0001 L=\u00060:1 and \u0001H=\u00060:1 rlu,\nrespectively, with \u0001 K=\u00060:058 rlu used in both slices. Pink lines\ncorrespond to the dispersion relation obtained from \fts described\nbelow. Gray vertical lines identify Brillouin zone centers (solid)\nand zone boundary points (dashed), as labeled on the top axis.\nspin values are \fxed to s= 1 andS= 3, respectively.\nForHaniso, the spin reorientation transition of Tb166\nand the general magnetic structures of other R166 com-\npounds suggest that Mn has weak easy-plane anisotropy\n(KM&0). However, \fxing KM= 0 results in a \ft-\nted spin gap that is much lower than experimental val-\nues. We assume that this discrepancy is caused by addi-\ntional contributions to the magnetic anisotropy, such as\nexchange anisotropy, that are not included in our model.\nThe introduction of KM<0 to our \ftting (as an e\u000bective\nuniaxial Mn anisotropy) dramatically improves the \ftted\nspin gap. We note that alternative \ftting schemes with\nKM= 0 and anisotropic JMTinteractions give similar\n\ftting results when JMT\nzz\u00191:30JMT\nxx.\nForHinter, the observed odd-even splitting of \u001855 meV\nis determined primarily by jJMM\n1+JMM\n2jand the A-point\ngap of\u001825 meV byjJMM\n1\u0000JMM\n2j. However, the de-\ntermination of the signs and relative strength of JMM\n1\nandJMM\n2requires careful \ftting of the interlayer disper-\nsions. We ran 41 di\u000berent \ftting iterations starting with\nequal values of JMM\n1andJMM\n2. All \ftting sessions \fnd\nJMM\n1+JMM\n2\u0019\u000024 meV with two local minima where\nJMM\n2=JMM\n1\u00194 or 1/3. Both interactions are FM. The\ncase where JMM\n2=JMM\n1\u00194 turns out to be the global\nminimum with a reduced \u001f2= 0:8 which is lower than\n\u001f2= 1:0 for the other case. The \fts \fnd that JMM\n2is the\ndominant interlayer interaction, con\frming the expecta-6\nFIG. 5. (a) Slices of the intensity dispersion along (0 ;0;L) with\nEi= 75 meV show the odd branch between 40 \u000060 meV. Pink\nlines correspond to the model dispersion relation obtained from\n\fts described below. Gray vertical lines identify Brillouin zone\ncenters (solid) and zone boundary points (dashed), as labeled on\nthe top axis. (b) Constant energy cuts along (0 ;0;L) atEi= 75\nmeV (lower panel) and 160 meV (upper panel) summed over \u0001 H=\n\u00060:1, \u0001K=\u00060:058 rlu and \u0001 E=\u00062:5 meV. Gaussian \fts reveal\nthe dispersion of the odd branch. (c) Constant- qcuts at (0,0,3) and\n(0,0,4) summed over \u0001 H=\u00060:1, \u0001K=\u00060:058, and \u0001 L= 0:25\nrlu showing even, Tb, and odd modes at the \u0000-point.\ntion based on neutron di\u000braction studies of the double-\n\rat spiral AFM structure of Y166 [10, 17, 20].\nIn the overall \fts to Hinter, we \fnd that an AFM JMM\n3\nmust be introduced to account for the di\u000berent band-\nwidths of even (\u001810 meV) and odd ( \u001820 meV) inter-\nlayer dispersions, as shown in Figs. 2(d) and 5(a). An\nAFMJMM\n3will compete with FM JMM\n1and could lead to\na destabilization of the ferrimagnetic stacking sequence.\nHowever, calculations of the classical stability of the fer-\nrimagnetic state described below suggest that JMM\n3is\nnot strong enough to create such an instability in Tb166.\nSimilar competing interactions have been proposed for\nY166, but with AFM JMM\n1and FMJMM\n3[10, 14]. This\ncannot be the case for Tb166, since the odd branch would\nhave a minimum in the dispersion at \u0000, which is not ob-\nserved experimentally.\nFitting the spin wave dispersions produced the set of\ninterlayer exchange parameters in Table I where error\nbars correspond to the variances obtained over all \ftting\niterations. Further details of the \ftting procedure are\ndescribed in the SM [37]. Within our model, the \ft pa-\nrameters predict an additional four modes (two odd and\ntwo even) at higher energies. These modes are not clearly\nobserved in the current experiment, as discussed below.\nV. INTRALAYER DISPERSIONS\nThe intralayer dispersions are steeper than the inter-\nlayer modes and can extend well beyond 100 meV. The\nodd and even modes can be isolated in the INS data\nbased on their structure factors which are maximized in\nBrillouin zones with L=oddandL=even, respectively.\nSlices from the Ei=75 and 160 meV data correspondingTABLE I. Heisenberg parameters for TbMn 6Sn6as obtained\nfrom \fts to the neutron data.\nCoupling Energy (meV) description\nJ -28.8 (2) intralayer FM\nJMT1.42 (6) interlayer AFM\nJMM\n1 -4.4 (4) interlayer FM\nJMM\n2 -19.2 (2) interlayer FM\nJMM\n3 1.8 (2) interlayer AFM\nKM-1.30 (6) uniaxial anisotropy\nKT-1.70 (12) uniaxial anisotropy\n!B \u001847 bilayer splitting\n!F \u00188 ferrimagnetic exchange\nFIG. 6. (a) Slices of the data highlighting the dispersion of the even\nmode along the ( H,0,0) and (\u0000K,2K,0) directions in the (0,0,4)\nzone withEi= 160 meV (lower panel) and Ei= 250 meV (upper\npanel). (b) Slices of the data highlighting the dispersion of the odd\nmode along the ( H,0,0) and (\u0000K,2K,0) directions in the (0,0,3)\nzone withEi= 160 meV. For (a) and (b), the data are averaged\nover \u0001L=\u00060:5 and either \u0001 H=\u00060:1 or \u0001K=\u00060:058. In\nall panels, pink lines correspond to model dispersions with L= 0\n(solid lines) and L= 0:5 (dashed lines).\nto even modes with L= 4 and odd modes with L= 3 are\nshown in Figs. 6(a) and 6(b), respectively. To gain bet-\nter statistics, the data are averaged over \u0001 L=\u00060:5 rlu\nwhich broadens features by e\u000bectively averaging over the\ninterlayer bandwidth. For L= 4, the even mode has a\nM-point energy of \u001970 meV. For L= 3, the odd mode is\nmore strongly broadened by interlayer interactions than\nthe even mode, but we clearly observe the even-odd mode\nsplitting of\u001955 meV.\nWe obtained the intralayer exchange parameters de-\n\fned inHintra by \ftting various cuts of the lowest odd\nand even branches similar to those shown in Figs. 3 and 6.\nDuring the \ft, all parameters of HinterandHaniso were7\nFIG. 7. Slices of the Ei= 250 meV data after averaging over\n\u0001L=\u00067 showing the dispersion along the (a) ( \u0000K,2K,0), (b)\n(H,0) and (c) (2 K,1=2\u0000K) directions. For all panels, the data are\nadditionally averaged over either \u0001 H=\u00060:1 or \u0001K=\u00060:058.\n(d)-(f) Model calculations of the neutron intensities with the same\nreciprocal space averaging of the data as in (a)-(c) and convolved\nwith a Gaussian energy FWHM of 12 meV. In all panels, pink\nlines correspond to model dispersions with L= 0 (solid lines) and\nL= 0:5 (dashed lines).\n\fxed to the values in Table I. Ultimately, we achieved\nsatisfactory agreement with the data with only one pa-\nrameter corresponding to the nearest-neighbor Mn-Mn\nintralayer FM interaction with J=\u000028:8(2) meV. The\nmain reason for this simple result is that the dispersive\nfeatures quickly deteriorate at higher energies by becom-\ning very broad and weak.\nFigure 7(a)-(c) shows intralayer dispersion data after\nsumming over a large range of \u0001 L=\u00067 rlu. This im-\nproves statistics and allows higher-energy features to be\nobserved, but it mixes odd and even modes and averages\nover the interlayer dispersions. Excitations are observed\nup to\u0018125 meV which includes evidence for the top of\nthe odd branch near the M-point at \u0018115 meV [Fig. 7(b)]\nand the bottom of the fourth branch (even) at the M-\npoint near 70 meV [Fig. 7(c)]. These data are compared\nto model calculations in Figs. 7(d)-(f) that average over\nthe same reciprocal space ranges. From the model, the\nK-point Dirac crossing of the even mode is predicted to\noccur near 90 meV. However, we are not able to resolve\nany K-point gapping in the INS data.\nVI. FIRST-PRINCIPLES CALCULATIONS OF\nTHE INTRINSIC MAGNETIC PROPERTIES\nDFT calculations were carried out to investigate the\nintrinsic magnetic properties in Tb166, which includes\nmagnetization, the interlayer exchange couplings, and\nmagnetocrystalline anisotropy (MA). The strongly cor-\nrelated Tb-4 fstates were treated in both the DFT+ U\nmethod and the so-called open-core approach. We also\nexplored the e\u000bects on the exchange couplings of addi-\ntional electron repulsion for Mn-3 dorbitals in DFT+ U.\n(a)\n (b)\n (c)\nFIG. 8. Intrinsic magnetic properties calculated in Tb166\nand compared to the experimental values. (a) On-site Mn spin\nmagnetic moment ms\nMnand (b) interlayer exchange parame-\nters as functions of Hubbard Uapplied on Mn 3 d-states. Hub-\nbardUon Mn-3dis included using the around-the-mean-\feld\ndouble-counting scheme in DFT+ U. (c) Variation of energy\nas a function of spin-quantization axis rotation. \u0012= 0 °corre-\nsponds to the out-of-plane spin orientation parallel to the c-\naxis. Fit1 and Fit2 correspond to \fttings to the expressions of\nE(\u0012) =K1sin2\u0012near\u0012= 0 andE(\u0012) =K1sin2\u0012+K2sin4\u0012\nover the full \u0012range, respectively.\nDetails of these calculations can be found in the SM [37].\nResults are shown in Fig. 8.\nWe \frst investigate the spin and orbital magnetic mo-\nments in Tb166. Tb-4 fare treated within DFT+ Uusing\nthe fully-localized-limit (FLL) double-counting scheme,\nand spin-orbit-coupling (SOC) is included using the sec-\nond variation method. The calculated spin and or-\nbital magnetic moments of Tb are mTb\ns= 6:26\u0016Band\nmTb\nl= 2:96\u0016B, respectively, consistent with Hund's\nrules. The calculated total magnetic moments of Tb and\nMn,mTb= 9.23\u0016BandmMn=2.42\u0016B, respectively,\nagree with the low-temperature experimental results of\nmTb= 9:0\u0016BandmMn= 2:17\u0016B[26].\nThe four interlayer isotropic exchange couplings dis-\ncussed above are calculated by mapping the total en-\nergies of \fve collinear spin con\fgurations (see SM [37])\nintoHinter de\fned in Eqn. (2). Mn and Tb spin de-\nrived from the spin magnetic moment, sMn=mMn\ns=2\nandSTb=mTb\ns=2, are used in the mapping procedure.\nThe overall ferrimagnetic structure is stabilized by JMM\n2\nandJMT. In all our calculations, we found that the Mn-\nTb coupling JMTis AFM and is a strong contributor to\nthe overall magnetic energy when considering the high\nTb spin and multiplicity of 12 neighboring Mn atoms.\nThe dominant interlayer Mn-Mn coupling, the FM JMM\n2,\nis also con\frmed in DFT, although its amplitude is over-\nestimated by\u001850%. On the other hand, we found AFM\nJMM\n1and FMJMM\n3. All three calculated JMM\nkhave the\nsame sign as the values calculated for Y166 [14], and their\namplitudes are also comparable [10]. However, for the\nweaker couplings JMM\n1andJMM\n3, the signs of calculated\nvalues disagree with those deduced from INS.\nTo resolve this discrepancy, we consider the electron\ncorrelation e\u000bects of Mn-3 dorbitals on exchange cou-8\nplings in DFT+ U. We note that various Uvalues have\nbeen applied on Mn-3 dorbitals in the previous studies\nof R166. For example, Tb166 bandstructure was cal-\nculated in plain DFT ( U= 0) while U= 4 eV was\nused in DFT+DMFT to explain the band structures of\nY166 measured by ARPES [11]. Especially, the Ude-\npendence of interlayer Mn-Mn couplings in Y166 has al-\nready been investigated with U= 0{3:5 eV using the\nFLL double-counting scheme in DFT+ U. However, as\nshown in Ref. [10], the FLL scheme quickly overestimates\nthe Mn magnetic moment with \fnite U. Thus, instead,\nhere we use the around-the-mean-\feld double-counting\nscheme [43], which is usually believed to be more suit-\nable for less-strongly-correlated metallic systems. Unlike\nthe FLL scheme, we found that the mMn\nsremains close to\nexperimental value with Uvalues of 0{2 eV, as shown in\nFig. 8(a). Compared to magnetization, the variation of\nthe exchange parameters is much more pronounced, al-\nthough the experimental state has the lowest energy for\nU= 0{2 eV. Figure 8(b) shows JMM\ni(i= 1;2;3) and\nJMTcalculated using various Uvalues, compared to ex-\nperiment. Remarkably, both JMM\n1andJMM\n3can change\ntheir signs with increasing U. WithU= 1:5{1:8 eV, the\nsigns of all interlayer Jvalues become consistent with\nthose deduced from INS. Thus, while DFT gives a reason-\nable description of the dominant magnetic interactions in\nTb166, including Mn-3 delectron correlations can further\nimprove the description of JMM\n1andJMM\n3.\nElectron correlations can have profound e\u000bects on\nmagnetic interactions and spin excitations [44], especially\nin more localized systems. The most recent Mn-based\nexamples include the extensively studied layered topo-\nlogical materials, MnBi 2Te4[45] and MnSb 2Te4[46, 47],\nwhere a sizable U= 4{5 eV on Mn- dorbitals was\nneeded to correctly describe the magnetic interactions\nin DFT+Uwhile the plain DFT fails to predict the cor-\nrect magnetic ground state. Although the widely-used\nDFT+Umethod provides the simplistic Hubbard cor-\nrection beyond DFT, the choice of the correlated orbitals\nand the associated value of the Hubbard Uparameter is\nnot well-de\fned for metallic systems like Tb166. More-\nover, the non-local exchange-correlation potentials can\nalso be important, and a simple Uparameter may not be\nsu\u000ecient [48] to best describe the electronic structures.\nFuture experimental and theoretical works may be help-\nful to further clarify the electron correlation role in Tb166\nand determine the best Uparameter.\nThe MA energy (MAE) is also investigated by calcu-\nlating the total energies of the ferrimagnetic state as a\nfunction of spin-quantization direction, which is shown in\nFig. 8(c). In agreement with the ground state structure\nof Tb166, the MAE displays strong uniaxial anisotropy\nwith a minimum energy at \u0012= 0 (easy-axis) relative to\nthec-axis. Moreover, the non-monotonic dependence of\nEon\u0012is consistent with substantial higher-order MAE\nconstants. Over the full range of \u0012, we \ft MA energy\n(see Fit2 in Fig. 8(c)) to the expression\nE(\u0012) =K1sin2\u0012+K2sin4\u0012: (4)The resulted large ratio of K2=K1=\u00001:25 is su\u000ecient\nto drive the spin reorientation transition [28]. To bet-\nter compare with the single-ion anisotropy deduced from\nlow-temperature INS, we also \ft MA energy (see Fit1\nin Fig. 8(c)) with E(\u0012) =K1sin2\u0012near\u0012= 0, which\ncorresponds to the ground state anisotropy. This pro-\nvidesK1\u001943 meV/f.u. and can be compared to our\nexperimental value [see Eqn.(3) and Table I] according to\nKtot=\u0000(KTS2+ 6KMs2) = 23:1 meV/f.u. Thus, DFT\noverestimates the MAE by \u001885%, which is a reasonable\nagreement considering that an accurate ab initio descrip-\ntion of MA is generally challenging, especially in complex\n4fintermetallics. The Tb-4 fcontributions dominate the\neasy-axis MA in Tb166 as the Mn sublattice contribution\nis one-order of magnitude smaller and easy-plane.\nVII. DISCUSSION\nThe INS data for Tb166 provide a minimal set of ex-\nchange and anisotropy parameters that are largely con-\nsistent with our DFT results and indirect estimations\nof these energy scales from magnetization and neutron\ndi\u000braction data (see e.g. Refs. [20, 28, 31]). The key\nconclusions are: (1) large intralayer FM interactions be-\ntween Mn ions, (2) interlayer interactions that are dom-\ninated by FM coupling between Mn layers spaced by Sn\nlayers (JMM\n2) and AFM coupling between Mn and Tb\nlayers, (3) the presence of competing, weaker AFM and\nFM Mn-Mn interlayer couplings, and (4) a net uniaxial\nmagnetic anisotropy.\nWith respect to (2) and (3), we consider the overall\nstability of the ferrimagnetic structure of Tb166 by ex-\namining the classical magnetic energies of collinear layer\nstackings given by\nE= 6JMT[(s1+s2)\u0001Sa+(s3+s4)\u0001Sb]+3JMM\n1(s1\u0001s2+s3\u0001s4)\n+ 3JMM\n2(s1\u0001s4+s2\u0001s3) + 6JMM\n3(s1\u0001s3+s2\u0001s4):(5)\nHere, the numbers label successive Mn layers and letters\nlabel Tb layers for a six-layer stack. The ground state\nferrimagnetic structure has an energy of\nEferri=\u000024sSJMT\u00006s2(JMM\n1+JMM\n2\u00002JMM\n3):(6)\nThe next higher-energy state corresponds to AFM up-\ndown-down-up (UDDU) Mn layer stacking. For uniaxial\nanisotropy, the classical UDDU state will decouple the\nMn and the Tb layers and\nEUDDU =\u00006s2(\u0000JMM\n1+JMM\n2+ 2JMM\n3): (7)\nThe parameters in Table I provide Eferri=\u0000220 meV\nandEUDDU =\u0000110 meV, indicating that the high sta-\nbility of the ferrimagnetic ground state arises from JMT.\nIn the absence of JMT(as for Y166), the collinear fer-\nromagnetic, ferrimagnetic and UDDU states are nearly\ndegenerate since JMM\n1\u0019 \u00002JMM\n3. This suggests that9\nsimilar competition between these interlayer interactions\ndrives complex helical ordering observed in Y166.\nBased on these comparisons, it is interesting to con-\nsider the transferability of exchange interactions in\nTb166 with other R166 compounds. INS investigations\nof Y166 in Ref. [17] report a NN intralayer exchange that\nis nearly identical to Tb166. While the interlayer inter-\nactions in Y166 are not studied in detail in Ref. [17], the\nbilayer splitting energy is reported as jJMM\n1+JMM\n2j\u001924\nmeV, which is the same as Tb166. This suggests that\nJMM\n1andJMM\n2interactions are both FM and have similar\nstrengths in Y166 and Tb166. One caveat is that addi-\ntional intralayer and interlayer interactions are also \ft in\nRef. [17]. Interestingly, our DFT calculations support an\nAFMJMM\n1and FMJMM\n3, and vice versa, with the result\ndepending on the choice of the correlation parameter U.\nOverall, these comparisons give some con\fdence that the\nMn-Mn magnetic interactions in R166 compounds share\na remarkable similarity: the JMM\n2is FM and dominates\nthe interlayer Mn-Mn coupling, while JMM\n1andJMM\n3are\nmuch weaker and competing. The variation of Rion and\nslight changes in structure will likely a\u000bect the overall\nbalance ofJMM\n1andJMM\n3.\nThere is little data reporting the magnitude of the Mn-\nRcoupling in other R166 compounds. For Gd166, the\nenergy scale for the Gd mode is reported to be \u001824 meV\nfrom powder INS data [49] which is very similar to the\nTb mode energy observed here. However, given the ab-\nsence of Gd single-ion anisotropy, simulations (see SM\n[37]) show that this energy corresponds to the top of the\nGd mode at\u0019!F+ 2JSGd= 12sJMnGd, allowing an es-\ntimate ofJMnGd\u00192 meV. The energy of the Tb mode is\nlifted appreciably by anisotropy, 2 SKT= 10 meV. Thus,\nour reported JMTis about 30% smaller than JMnGd, a\nresult that is roughly consistent with a decrease of 4 f-5d\noverlap due to lanthanide contraction [50]. Extrapolat-\ning to Ho166 and Er166 should result in weaker ferrimag-\nnetism. For Er166, this weakening results in the observed\ndecoupling of the Mn and Er sublattice magnetic order-\ning at high temperatures [19].\nThe magnetic anisotropies of Tb166 determined from\nINS may present some inconsistencies with our under-\nstanding of R166 compounds. At low temperatures,\nTb166 is dominated by the large uniaxial anisotropy of\nthe Tb ion, a result that is consistent with our INS data\nand DFT results. However, the INS data cannot be mod-\neled with an easy-plane Mn anisotropy parameter since\nthe spin gap becomes too small. Instead, we obtain the\nbest \ftting results by assuming that Mn also has uniaxial\nsingle-ion anisotropy. This is inconsistent with INS data\nfrom Y166 that \fnds a rather large value of 5 meV for the\nMn easy-plane single-ion anisotropy parameter, although\nthe spin gap itself is not reported [17]. It is very possi-\nble that both Mn-Mn and Mn-Tb exchange anisotropy\ncontributes to the spin gap as well. First-principles cal-\nculations \fnd signi\fcant exchange anisotropy of the in-\ntralayer coupling in Y166 [10]. In Tb166, the Tb mag-\nnetic anisotropy is temperature dependent, and our MAEcalculations in the ground state are consistent with the\nexpected conditions for the spin reorientation transition\nthat occurs at 350 K. It will be interesting to study the\nspin excitations in this temperature regime to learn more\nabout the unusual magnetic anisotropy of R166 com-\npounds.\nFinally, we would like to discuss brie\ry the role that\nmagnetic instabilities and \ructuations play in the band\ntopology of Tb166. The magnetic stacking of FM Mn\nand Tb layers in Tb166 is very stable to competing in-\nterlayer interactions due to the large Tb-Mn coupling.\nThus, the only avenue available for tuning of topologi-\ncal band states in Tb166 is by controlling the magnetic\nanisotropy and, consequently, the spin reorientation tran-\nsition. This will a\u000bect the size of the Chern gap, which\nis maximized for the uniaxial moment con\fguration. On\nthe approach to the spin reorientation at elevated tem-\nperatures, we might ask whether magnetic \ructuations\nplay any role in quantum transport. Recent muon spec-\ntroscopy results report a correlation between quantum\ntransport in Tb166 and the suppression of slow ( \u0018MHz)\nmagnetic \ructuations that appear below 120 K [13]. The\norigin of these slow magnetic \ructuations is a mystery,\nbut our INS data indicate that they do not arise from\ncollective spin wave modes which are gapped out on a\nTHz scale.\nVIII. SUMMARY\nINS data for Tb166 provide a minimal set of exchange\nand anisotropy parameters that are largely consistent\nwith indirect estimations of these energy scales provided\nby magnetization data and neutron di\u000braction, as well as\nby our DFT calculations. The key conclusions are: (1)\nlarge intralayer FM interactions between Mn ions, (2) in-\nterlayer interactions that are dominated by FM coupling\nbetween Mn layers spaced by Sn layers ( JMM\n2) and AFM\ncoupling between Mn and Tb layers, (3) the presence of\nweaker FM and AFM Mn-Mn interlayer couplings, and\n(4) an overall uniaxial magnetic anisotropy. These results\nsuggest that the magnetism of R166 compounds, with a\nvariety of magnetic ground states and high-temperature\nor high-\feld instabilities, may be understood with trans-\nferable set of magnetic interactions. A complete under-\nstanding of these interactions and their evolution through\nthe R166 family could allow for the prediction of addi-\ntional topological responses accessible via tuning of the\nmagnetism using external applied \felds or rare-earth en-\ngineering protocols.\nIX. ACKNOWLEDGMENTS\nRJM, LK, YL, BGU, BL and SXMR's work at the\nAmes Laboratory is supported by the U.S. Department\nof Energy (USDOE), O\u000ece of Basic Energy Sciences, Di-\nvision of Materials Sciences and Engineering. TJS and10\nPC are supported by the Center for the Advancement of\nTopological Semimetals (CATS), an Energy Frontier Re-\nsearch Center funded by the USDOE O\u000ece of Science,\nO\u000ece of Basic Energy Sciences, through the Ames Lab-\noratory. Ames Laboratory is operated for the USDOE\nby Iowa State University under Contract No. DE-AC02-\n07CH11358. TJS is also partially funded by the Gordon\nand Betty Moore Foundation (Grant No. GBMF4411).\nA portion of this research used resources at the SpallationNeutron Source, which is a USDOE O\u000ece of Science User\nFacility operated by the Oak Ridge National Laboratory.\nL.K. is supported by the U.S. DOE, O\u000ece of Science, Of-\n\fce of Basic Energy Sciences, Materials Sciences and En-\ngineering Division, and Early Career Research Program.\nA portion of this research used resources of the National\nEnergy Research Scienti\fc Computing Center (NERSC),\na U.S. DOE O\u000ece of Science User Facility operated un-\nder Contract No. DE-AC02-05CH11231\n[1] N. Kiyohara, T. Tomita, and S. Nakatsuji, \"Giant\nAnomalous Hall E\u000bect in the Chiral Antiferromagnet\nMn3Ge\", Phys. Rev. Applied 5, 064009 (2016).\n[2] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel,\nA. C. Komarek, C. Shekhar, N. Kumar, W. 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J.\nMcQueeney, \"Defect-driven ferrimagnetism and hidden\nmagnetization in MnBi 2Te4\", Phys. Rev. B 103, 184429\n(2021).\n[47] S. X. M. Riberolles, Q. Zhang, E. Gordon, N. P. Butch,\nL. Ke, J. Q. Yan, and R. J. McQueeney, \"Evolution\nof magnetic interactions in Sb-substituted MnBi 2Te4\",\nPhys. Rev. B 104, 064401 (2021).\n[48] Y. Lee, T. Kotani, and L. Ke, \\Role of nonlocality in\nexchange correlation for magnetic two-dimensional van\nder Waals materials\", Phys. Rev. B 101, 241409 (2020).\n[49] P. Tils, M. Loewenhaupt, K. H. J. Buschow, and R. S.\nEccleston, \"Intersublattice exchange coupling in Gd-Mn\ncompounds studied by INS\", J. Alloys Compd. 279, 123-\n126 (1998).\n[50] Y. Lee, R. Skomski, X. Wang, P. P. Orth, A. K. Pathak,\nB. N. Harmon, R. J. McQueeney, I. I. Mazin, and Liqin\nKe, \" Interplay between magnetism and band topology in\nKagome magnets RMn6Sn6\", arXiv: 2201.11265 (2022)." }, { "title": "2201.03781v1.Nonreciprocal_dynamics_of_ferrimagnetic_bimerons.pdf", "content": "Nonreciprocal dynamics of ferrimagnetic bimerons\nLaichuan Shen,1Jing Xia,2Zehan Chen,3, 4Xiaoguang Li,5Xichao Zhang,6\nOleg A. Tretiakov,7Qiming Shao,3, 4, 8,\u0003Guoping Zhao,2Xiaoxi Liu,6Motohiko Ezawa,9,yand Yan Zhou1,z\n1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China\n2College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n3Department of Electronic and Computer Engineering, The Hong Kong University\nof Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China\n4Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China\n5Center for Advanced Material Diagnostic Technology, College of Engineering\nPhysics, Shenzhen Technology University, Shenzhen 518118, China\n6Department of Electrical and Computer Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan\n7School of Physics, The University of New South Wales, Sydney 2052, Australia\n8Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic\nTechnology, The Hong Kong University of Science and Technology, Hong Kong, China\n9Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan\n(Dated: January 12, 2022)\nMagnetic bimerons are topologically nontrivial spin textures in in-plane easy-axis magnets, which can be used\nas particle-like information carriers. Here, we report a theoretical study on the nonreciprocal dynamics of\nasymmetrical ferrimagnetic (FiM) bimerons induced by spin currents. The FiM bimerons have the ability to\nmove at a speed of kilometers per second and do not show the skyrmion Hall effect at the angular momentum\ncompensation point. Our micromagnetic simulations and analytical results demonstrate that spin currents are\nable to induce the nonreciprocal transport and a drift motion of the FiM bimeron even if the system is at the\nangular momentum compensation point. By analyzing the current-induced effective fields, we find that the\nnonreciprocal transport is attributed to the asymmetry of the bimeron structure. Our results are useful for\nunderstanding the physics of bimerons in ferrimagnets and may provide guidelines for building bimeron-based\nspintronic devices.\nIntroduction. Reciprocity is a fundamental principle in\nmany fields, such as in mechanics and thermodynamics [1].\nHowever, when a certain symmetry of the system is bro-\nken, the reciprocal relation may be violated and nonrecipro-\ncal phenomena appear [1–3]. Nonreciprocal transport which\nplays an important role in various application devices, such\nas in the diode [4–7] and shift register [8], has been reported\nfor (quasi-)particles, for instance, electrons [9], phonons [4],\nphotons [10] and magnons [11]. For topological solitons,\ne.g. skyrmions [12], they also show such a transport in the\nasymmetrical racetrack which requires sophisticated repro-\ncessing [13–16]. However, nonreciprocal transport attributed\nto the intrinsic characteristics of topological solitons still re-\nmains to be discovered.\nDifferent types of topological spin textures have been in-\nvestigated for a few decades, such as domain walls [17, 18],\nskyrmions [12, 19, 20] and bimerons [21–31], which emerge\nin ferromagnetic (FM) [17, 19, 25], ferrimagnetic (FiM) [32–\n35] and antiferromagnetic (AFM) [18, 36, 37] materials. In\nparticular, FM skyrmions are promising as nonvolatile infor-\nmation carriers to serve the future memory and logic com-\nputing devices [38–44]. However, the FM skyrmion shows\nthe transverse drift during its motion due to the existence of\na nonzero Magnus force, which may lead to the annihila-\ntion of the fast-moving skyrmion at the sample edge. This\n\u0003Email: eeqshao@ust.hk\nyEmail: ezawa@ap.t.u-tokyo.ac.jp\nzEmail: zhouyan@cuhk.edu.cnphenomenon is referred to as the skyrmion Hall effect [45–\n47]. Compared to the FM skyrmion, the AFM skyrmion is\nfree from the skyrmion Hall effect, as the compensated lat-\ntice structures of antiferromagnets lead to a perfect cancel-\nlation of the Magnus force [48, 49]. However, the compen-\nsated magnetic moments in antiferromagnets give rise to the\ndifficulties in detecting AFM spin textures [50]. Recently,\nFiM materials have received great attention, since the AFM\nspin dynamics is realized in ferrimagnets at the angular mo-\nmentum compensation point [33, 51] and unlike the antiferro-\nmagnet, even for compensated ferrimagnet, we can detect the\nmagnetization of one sublattice using magnetotransport mea-\nsurements, such as anomalous Hall effect or tunnel magne-\ntoresistance. On the other hand, a magnetic bimeron consist-\ning of two merons is considered as the topological counter-\npart of a magnetic skyrmion in in-plane magnets and is stabi-\nlized in various magnetic materials [22, 24, 26–31]. Recent\nreports show that two-dimensional CrCl 3[52, 53] and van der\nWaals LaCl/In 2Se3heterostructures [54] are promising can-\ndidates for hosting bimerons. Additionally, the bimeron is\na stable solution in ferromagnets [25, 26, 55], antiferromag-\nnets [28, 30, 37] and frustrated magnets [29, 56]. Although a\nbimeron is topologically equivalent to a skyrmion, the former\nhas richer dynamics [30, 55].\nIn this work, based on the Landau-Lifshitz-Gilbert (LLG)\nequation [57], we theoretically study the current-induced dy-\nnamics of FiM bimerons with intrinsic asymmetrical shape.\nNumerical and analytical results demonstrate that the FiM\nbimeron driven by opposite currents could exhibit different\nspeeds, that is, it shows the nonreciprocal dynamics. By an-arXiv:2201.03781v1 [cond-mat.mes-hall] 11 Jan 20222\nalyzing the current-induced effective fields, it is found that\nsuch a nonreciprocal behavior is attributed to the asymmetry\nof the bimeron structure. In addition to the nonreciprocal dy-\nnamics, a drift motion of the FiM bimeron may be induced\nby the current even if the system is at the angular momentum\ncompensation point.\nProposal of nonreciprocal transport of FiM bimerons. We\nconsider a FiM film with two sublattice magnetization M1and\nM2[Fig. 1(a)], and the interfacial Dzyaloshinskii-Moriya in-\nteraction (DMI) [38, 40] is introduced, which can be induced\nat the magnetic layer/heavy metal interface. To form FiM\nbimerons, the ferrimagnets with in-plane magnetic anisotropy\n(such as DyCo 5[58]) are promising materials. Here we focus\non the study of a FiM film with in-plane easy-axis anisotropy,\nin which the asymmetrical bimeron is a stable solution, sim-\nilar to the cases of FM [55] and AFM [30] bimerons. For\nthe interfacial DMI shown in Fig. 1(a), it is usually responsi-\nble for stabilizing the skyrmion with rotational symmetry (the\naxis of rotation is parallel to the polar axis that is perpendicu-\nlar to thex-yplane) [59]. For the FiM system we consider, the\nin-plane magnetic anisotropy forces the magnetization to tilt\naway from the polar axis, which violates the rotational sym-\nmetry dominated by DMI. As reported in Refs. [27, 59], in\nthe tilted magnetic phases (the magnetization of a homoge-\nneously magnetized state is tilted away from the polar axis),\nthe rotationally symmetrical spin texture is an incompatible\nform and the asymmetrical spin textures appear. Figure 1(b)\nshows the spin structure of a FiM bimeron, and the compo-\nnents of its reduced magnetization si=Mi=jMijand Néel\nvector n= (si\u0000sj)=2are presented in the Fig. 1(c). The\nNéel vector nin real space for a FiM bimeron is plotted in\nFig. 1(d), showing that although the size of the left meron\nis different from that of the right meron, the bimeron’s spin\nstructure still has mirror symmetry about the x-zplane.\nAdditionally, we derive a closed equation for the Néel vec-\ntorn[60, 61] (see Supplemental Material [62] for details),\n\u00160\u001a2(1+\u000b2)\n2\u0015n\u0002n+\u001b_n=\u0000n\u0002f\u0003\nn+\u000b\u001an\u0002_n+2\u000fn\u0002p\u0002n.\n\u001a=MS1\n\r1+MS2\n\r2and\u001b=MS1\n\r1\u0000MS2\n\r2are the staggered and\nnet spin densities, respectively [61]. MSi,\ri,\u00160,\u0015and\u000b\nare the saturation magnetization, gyromagnetic ratio, vacuum\npermeability constant, homogeneous exchange constant and\ndamping constant, respectively. f\u0003\nnand\u000frelate to the ef-\nfective field and current density, respectively (Supplemental\nMaterial [62]). In the above equation, only the damping-like\nspin torque is considered, while the field-like spin torque is\nnot included. The effect of field-like spin torque on the FiM\nbimeron has been discussed in Supplemental Material [62].\nThe above equation indicates that the current-induced effec-\ntive field relates to the cross product of the Néel vector nand\npolarization vector p. Such current-induced effective fields\nare of interest to us in the following symmetry analysis. As\nmentioned earlier, the bimeron’s spin structure is symmetric\nabout thex-zplane, so that the Néel vector nis canceled in\ntheydirection and the ycomponent of nwill not contribute\nto the nonreciprocal dynamics. Thus, we only need to pay\nattention to the components of nin thex-zplane and their\ncorresponding current-induced effective fields ( n\u0002p). As\nshown in Figs. 1(e)-1(j), we sketch two vectors nLandnRto represent the x-z-plane components of the Néel vector n\nfor two merons, respectively. Based on the symmetry con-\nsideration, nLandnRare symmetric about the xaxis for a\nbimeron with a symmetrical shape (see Fig. S1 of Supple-\nmental Material [62]), while for the bimeron studied here it\nhas an asymmetrical shape, resulting in the breaking of this\nsymmetry. Figure 1(e) shows the results of n\u0002pforp=ex,\nwhere the cross product operation causes the current-induced\neffective fields to be perpendicular to the x-zplane. When we\nchange the sign of the current, which is equivalent to chang-\ning the direction of p,i.e.,p=ex!\u0000ex, the corresponding\ncurrent-induced effective fields are still perpendicular to the\nx-zplane [Fig. 1(f)]. By comparing Fig. 1(e) with Fig. 1(f),\nwe see that for p=exand\u0000ex, the current-induced ef-\nfective fields are symmetric about the x-zplane, so that the\nbimeron does not exhibit the nonreciprocal motion behavior.\nSimilar mirror symmetry is observed for the case of p=\u0006ez\n[Figs. 1(i) and 1(j)], so there is no nonreciprocal phenomenon.\nHowever, for p=ey[Fig. 1(g)] and\u0000ey[Fig. 1(h)], n\u0002p\nis in thex-zplane, and obviously nL\u0002eyis not mirror-\nsymmetrical to nL\u0002(\u0000ey), so that the opposite currents have\ndifferent effects on the meron on the left in Fig. 1(d) [this re-\nsult also applies to the meron on the right in Fig. 1(d)]. For\na bimeron with a symmetrical shape, as mentioned above,\nnLandnRare symmetric about the xaxis, indicating that\nthe effect of the positive current on the left meron (the right\nmeron) is equivalent to that of the negative current on the right\nmeron (the left meron). Therefore, although opposite currents\nhave different effects on each meron, the structural symmetry\ncauses the opposite currents to have the same effects on the\nwhole, so that the bimeron with a symmetrical shape will not\nshow nonreciprocal transport, which has been confirmed in\nSupplemental Material [62]. For the FiM bimeron studied in\nthis work, it has an asymmetrical shape [Fig. 1(c)], resulting in\nthe presence of nonreciprocal phenomena. Namely, nonrecip-\nrocal transport is attributed to the asymmetry of the bimeron\nstructure.\nCurrent-induced nonreciprocal transport and drift motion\nof FiM bimerons. To verify the above analysis, we have sim-\nulated the magnetization dynamics of FiM bimerons and ob-\ntained the bimeron speeds, as shown in Figs. 2(a)-2(c). We\nindeed observe that only when the polarization vector pis\nalong theydirection (perpendicular to the symmetry plane\nof the bimeron’s spin structure), the FiM bimeron driven by\nopposite currents has different speeds, that is, it exhibits the\nnonreciprocal transport [Fig. 2(b)]. As expected by the above\nsymmetry analysis, for the cases where pis along the xorz\ndirections [Figs. 2(a) and 2(c)], such a nonreciprocal trans-\nport does not appear. Here we employ the LLG equation\nwith the damping-like spin torque [63, 64] to simulate the\ndynamics of FiM bimerons, and the simulation details are\ngiven in Supplemental Material [62]. We also simulate the\ncreation of FiM bimerons (see Figs. S4 and S5 of Supplemen-\ntal Material [62]) and the creation process is given in Supple-\nmental Movie 1-3. Moreover, based on the definition of the\nguiding center ri=\u00001=(4\u0019Q)R\n[in\u0001(@xn\u0002@yn)]dS[65],\nwe obtain the time evolution of riand the bimeron veloc-\nityvi= _ri(see Fig. S8 of Supplemental Material [62]).3\nFIG. 1. (a) Schematic of the studied model, where M1andM2denote two sublattice magnetization of ferrimagnet. The interface-induced\nDMI and in-plane easy-axis magnetic anisotropy Kare considered. (b) The real-space spin structure of a FiM bimeron. (c) The components\nof reduced magnetization sand the Néel vector nfor a FiM bimeron with a positive topological charge Q. (d) The Néel vector nin real space.\nThe color represents the zcomponents of n, andnzof two merons has opposite signs. (e)-(j) The sketch of two vectors nLandnR, and the\ncross product of the vector nL,Rand polarization vector p.\nFIG. 2. (a)-(c) The FiM bimeron speeds as functions of the current density jforp=ex,\u0000ey, and\u0000ez. (d)-(f) The angle between the actual\nand desired motion directions for p=ex,\u0000ey, and\u0000ez. Here, the FiM bimeron has a positive Q, and the numerical and analytical results\nare obtained by solving LLG equation and Eq. (1), respectively. (g) The alternating current pulse adopted in our simulation. (h)-(j) The time\nevolution of the guiding center ( rx,ry) for different polarization vectors p.\u000b= 0:05,MS1= 1:1MS2and\r1= 1:1\r2.\nQ=\u00001=(4\u0019)R\ndS[n\u0001(@xn\u0002@yn)]is the topological\ncharge [22, 49].\nWe now discuss the current-induced drift motion of FiM\nbimerons. Figures 2(a)-2(c) present the bimeron speeds in\nthe desired motion direction (it is in y,xandydirections for\np=ex,\u0000eyand\u0000ez, respectively). The FiM bimeron at\nthe angular momentum compensation point has an ability to\nmove with a speed of about km s\u00001, similar to the AFM spin\ntextures [28]. However, spin currents may induce a drift speed\nwhich is perpendicular to the desired motion direction, even\nif the FiM system is at the angular momentum compensation\npoint ( i.e.,\u001b=MS1\n\r1\u0000MS2\n\r2= 0). As shown in Figs. 2(d) and2(f) where p=exand\u0000ezrespectively, the angle between\nthe actual and desired motion directions is not zero, that is,\nthe FiM bimeron shows a drift motion, and such an angle in-\ncreases with the applied currents j. For the case of p=\u0000ey,\nthe drift motion can be safely disregarded [Fig. 2(e)].\nTo explain the simulation results, we derived the Thiele\nequation (see Supplemental Material [62] for details) [61, 66–\n68], from which we obtain the steady motion speeds,\n\u0012\nvx\nvy\u0013\n=1\n\u0011\u0012\n\u000bLyy\u0000G\u0000\u000bLxy\nG\u0000\u000bLxy\u000bLxx\u0013\u0012\nFx\nFy\u0013\n;(1)\nwhere\u0011=\u000b2(LxxLyy\u0000L2\nxy)+G2.Lij=\u00160\u001atzR\ndS(@in\u00014\nFIG. 3. The numerical (symbols) and analytical (curves) velocities\nfor a FiM bimeron in systems with different values of MS1=M S2,\nwhere three polarization vectors (a) p=ex, (b)p=\u0000eyand (c)\np=\u0000ezare considered. Here, j= 5 MA cm\u00002,MS2= 376 kA\nm\u00001,\r1= 1:1\r2, and other parameters are the same as those used\nin Fig. 2. The numerical and analytical results are obtained from the\nLLG equation and Eq. (1), respectively.\n@jn)andG= 4\u0019Q\u0016 0tz\u001bwith the layer thickness tz.Fi\ndenotes the driving force induced by the damping-like spin\ntorque (its expression is given in Supplemental Material [62]).\nIfG= 0, Eq. (1) indicates that the bimeron speed is inversely\nproportional to the damping (see Fig. S9 of Supplemental Ma-\nterial [62]).\nTo verify the above analytical formula, we simulate the\nmotion of FiM bimerons and calculate the bimeron veloci-\nties for different values of MS1=M S2. Figure 3 shows the\ncomparison of the numerical and analytical velocities, where\nthe analytical velocities for all polarization vectors pare in\ngood agreement with the numerical results. Moreover, Fig. 3\nshows that one of the velocity components ( vx,vy) is sym-\nmetric about MS1=M S2= 1:1, while the other component is\nantisymmetric. From Eq. (1), we see that one of the veloc-\nity components is proportional to 1=(C+G2)with a con-\nstantC,i.e.,vi/1=(C+G2), while the other component\nvj/G=(C+G2), whereG/(MS1=M S2\u0000\r1=\r2)be-\ncause we fixed the values of MS2and\r1in the simulation.\nForvi/1=(C+G2)it presents a symmetric curve, while for\nvj/G=(C+G2), an antisymmetric curve is obtained.\nAssuming that the main driving force is in the ydirection\n(Fy6= 0) and the system is at the angular momentum com-\npensation point ( G= 0), from Eq. (1), the drift speed is ob-\ntained,vx=\u000b(LyyFx\u0000LxyFy)=\u0011. Note thatLyy(orLxx)\nis always nonzero for spin textures. According to the aboveformula of the drift speed, we find that there are two factors\nwhich cause the drift motion even if G= 0. The first factor\nis the presence of a nonzero Lxy[27] and the second factor\nis that an additional force perpendicular to the desired motion\ndirection is induced by the applied currents. In order to ver-\nify the above analysis, we calculated the numerical values of\nLandF(Figs. S10 and S12 of Supplemental Material [62]),\nand then substituting them into Eq. (1) gives the analytical\ndrift speeds which are consistent with the numerical results, as\nshown in Figs. 2(d)-2(f). Moreover, the numerical values of L\nandFconfirm that the drift motion presented in Figs. 2(d) and\n2(f) is due to the presence of a nonzero Lxyand an additional\nforce [they originate from the deformation of the bimeron’s\nspin structure after the spin currents are applied (Fig. S13\nof Supplemental Material [62])]. The drift speed due to the\nnonzeroLxyis greater than that due to the presence of an ad-\nditional force for the case of Fig. 2(d), while the drift speed in\nFig. 2(f) is dominated by the additional force.\nSince FiM bimerons exhibit the nonreciprocal transport, an\nalternating current pulse presented in Fig. 2(g) induces the\nbimeron to show a ratchet motion [69–73] if we take p=\u0000ey\n[Fig. 2(i)]. Thus, FiM bimerons are ideal information carriers\nin AC racetrack storage devices [71]. For p=exand\u0000ez,\nthe bimeron does not show the nonreciprocal motion in the y\ndirection and the final value of ryis zero [Figs. 2(h) and 2(j)],\nwhile due to the presence of the drift motion [Figs. 2(d)\nand 2(f)], the final values of rxare not equal to zero.\nTo quantify the nonreciprocal transport of FiM bimerons,\nthe speed difference \u0001vi=jvi(+j)j\u0000jvi(\u0000j)jis defined.\nIn Figs. 4(b)-4(d), \u0001viis calculated as a function of the cur-\nrent density j, the damping \u000band the ratio of MS1andMS2.\nAccording to the fitting results shown in Fig. 4(b), the rela-\ntionship between \u0001vxandjis well described by this func-\ntion\u0001vx=k2j2+k4j4, where we take p=\u0000eyand\nG= 0. To understand the results shown in Fig. 4(b), let\nus return to Eq. (1). For G= 0, Eq. (1) is simplified as\nvx=Fx=(\u000bLxx), whereLxxandFxare related to the\nbimeron’s spin structure so their values are affected by the ap-\nplied currents. As mentioned earlier, opposite currents have\ndifferent effects on the FiM bimeron with an asymmetrical\nshape, so that the value of jFx=Lxxjfor a positive current is\ndifferent from that for a negative current. Thus, the relation\nbetweenFx=Lxxandjmust include even terms in addition\nto odd terms, so the speed vxis written as a general polyno-\nmial form,vx=P\nl=1kljl=2with coefficient kl. Consider-\ning the first two terms of such a polynomial, the fitting results\nalmost match the numerical simulations [Fig. 4(a)] (if more\nhigh-order terms are considered, the gap between the fitting\nresults and numerical simulations will be narrowed). From\nthe above speed vx, we obtain the speed difference that only\ncontains even terms, \u0001vx=k2j2+k4j4+\u0001\u0001\u0001(the magnitude\nofk2,k4and\u0001\u0001\u0001is directly related to the strength of nonrecip-\nrocal transport). In addition, taking different damping \u000b,\u0001vx\nis calculated and summarized in Fig. 4(c), showing that \u0001vx\nis inversely proportional to \u000b, as indicated by this equation\nvx=Fx=(\u000bLxx). Moreover, by changing the value of MS1,\n\u0001vxand\u0001vyfor different MS1=M S2are obtained, as shown\nin Fig. 4(d), where \u0001vxreaches its maximum value at the an-5\nFIG. 4. (a) The FiM bimeron speeds vxas functions of the current\ndensityj. The symbols are obtained from the numerical simulations\nand the curves are the fitting results. For the case where the fitting\nfunction isvx=P2\nl=1kljl=2,k1= 50:158andk2=\u00000:375[the\nunit ofklis m s\u00001(MA cm\u00002)\u0000l]. Forvx=P4\nl=1kljl=2,k1=\n41:726,k2=\u00000:274,k3= 3:04\u000210\u00003andk4=\u00003:085\u000210\u00005.\nThe adopted parameters are the same as those used in Fig. 2(b). (b)-\n(d) The speed differences \u0001vias functions of the current density\nj, the damping constant \u000band the ratio of MS1andMS2. The nu-\nmerical and analytical results are obtained by solving LLG equation\nand Eq. (1), respectively. In panel (b), the green dashed and red solid\ncurves are the fitting results of \u0001vx=k2j2(withk2=\u00000:375) and\n\u0001vx=k2j2+k4j4(withk2=\u00000:274andk4=\u00003:085\u000210\u00005),\nrespectively. The default parameters are p=\u0000ey,j=\u000625MA\ncm\u00002,\u000b= 0:05,MS1= 1:1MS2and\r1= 1:1\r2.\ngular momentum compensation point ( MS1=M S2= 1:1), and\n\u0001vxand\u0001vyalmost are symmetric and antisymmetric about\nMS1=M S2= 1:1, respectively.\nGeneralization of nonreciprocal transport to magnetic\nskyrmions. In the above sections, we discussed the nonrecip-\nrocal transport of the FiM bimeron with a positive topological\nchargeQ. Such a nonreciprocal transport is also observed for\nthe FiM bimeron with a negative Q(see Fig. S17 of Sup-\nplemental Material [62]). The results of nonreciprocal trans-\nport of FiM bimerons can be extended to other types of spin\ntextures with broken symmetry. For general topological soli-\ntons, e.g. skyrmions, they have a symmetrical structure so\nnonreciprocal transport does not appear, while the bimerons\nunder investigation have intrinsic asymmetrical shape, result-\ning in the presence of nonreciprocal dynamics. Therefore, in\norder to attain the nonreciprocal transport, a break in struc-\ntural symmetry of spin textures is required. As shown in Fig.\nS18 of Supplemental Material [62], when an in-plane mag-\nnetic field is utilized to break the rotational symmetry of a FM\nskyrmion, the skyrmion driven by opposite currents exhibits\nnonreciprocal transport. Compared to the intrinsic asymme-\ntry of bimerons ( k2\u0019\u00000:3, Fig. 4), this externally inducedasymmetry of skyrmions gives rise to a weak nonreciprocity\n(k2\u0019\u00000:01, Fig. S18 of Supplemental Material [62]).\nConclusions. We have analytically and numerically studied\nthe drift and nonreciprocal motions of FiM bimerons driven\nby spin currents. Our results demonstrate that due to the defor-\nmation of the bimeron’s spin structure, spin currents may in-\nduce a drift speed which is perpendicular to the desired motion\ndirection, even if the FiM system is at the angular momentum\ncompensation point. Moreover, the symmetry analysis shows\nthat since the FiM bimeron studied here has an asymmetrical\nshape, the bimeron driven by opposite currents exhibits non-\nreciprocal transport. Our analysis of nonreciprocal transport\nof FiM bimerons is applicable to other types of spin textures\nwith broken symmetry and our results are useful for building\nbimeron-based spintronic devices, such as bimeron diode and\nAC racetrack memory.\nThis study is supported by Guangdong Spe-\ncial Support Project (Grant No. 2019BT02X030),\nShenzhen Fundamental Research Fund (Grant No.\nJCYJ20210324120213037), Shenzhen Peacock Group\nPlan (Grant No. KQTD20180413181702403), Pearl River\nRecruitment Program of Talents (Grant No. 2017GC010293)\nand National Natural Science Foundation of China (Grant\nNos. 11974298 and 61961136006). J.X. acknowledges the\nsupport by the National Natural Science Foundation of China\n(Grant No. 12104327). X.Li acknowledges the support by the\nGuangdong Basic and Applied Basic Research Foundation\n(Grant No. 2019A1515111110). X.Z. was an International\nResearch Fellow of Japan Society for the Promotion of Sci-\nence (JSPS). X.Z. was supported by JSPS KAKENHI (Grant\nNo. JP20F20363). O.A.T. acknowledges the support by the\nAustralian Research Council (Grant No. DP200101027),\nthe Cooperative Research Project Program at the Research\nInstitute of Electrical Communication, Tohoku University\n(Japan), and by the NCMAS 2021 grant. Q.S. acknowledges\nfunding support from the Shenzhen-Hong Kong-Macau\nScience and Technology Program (Category C, Grant No.\nSGDX2020110309460000), Research Grant Council-Early\nCareer Scheme (Grant No. 26200520), and the Research\nFund of Guangdong-Hong Kong-Macao Joint Laboratory for\nIntelligent Micro-Nano Optoelectronic Technology (Grant\nNo. 2020B1212030010). G.Z. acknowledges the support by\nthe National Natural Science Foundation of China (Grant\nNos. 51771127, 51571126, and 51772004), the Scientific\nResearch Fund of Sichuan Provincial Education Department\n(Grant Nos. 18TD0010 and 16CZ0006). 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They\nshare features with ferrimagnets in that opposing spin orientations exist in two or more sublattices.\nSpin transfer torque and spin pumping are essential ingredients in antiferromagnetic and ferrimag-\nnet spintronics. This paper develops an out-of-equilibrium quantum theory of the spin dynamics of\ncollinear magnets containing many spins coupled to normal metal reservoirs. At equilibrium, the\nspins are parallel or antiparallel to the easy axis. The theory, therefore, covers collinear antiferro-\nmagnets and ferrimagnets. We focus on the resulting semi-classical spin dynamics. The dissipation\nin the spin dynamics is enhanced due to spin-pumping. Spin accumulations in the normal metals\ninduce deterministic spin-transfer torques on the magnet. Additionally, each electron’s discrete spin\nangular momentum causes stochastic fluctuating torques on the antiferromagnet or ferrimagnet. We\nderive these fluctuating torques. The fluctuation-dissipation theorem holds at high temperatures,\nincluding the effects of spin-pumping. At low temperatures, we derive shot noise contributions to\nthe fluctuations.\nI. INTRODUCTION\nSpin transfer torque (STT) and spin pumping (SP)\nare essential ingredients in the generation and detection\nof spin currents and are central components in modern\nspintronics research and devices [1]. The use of mag-\nnetic insulators enables signal propagation without mov-\ning charges and could provide low-dissipation and ultra-\nfast memory devices [2]. Initially, much of spintronic\nresearch focused on the study of STT [3–5] and SP [6–\n8] in ferromagnets (FMs). Subsequently, this included\nalso works on fluctuations [9–11] and pumped magnon\ncondensates [12–14].\nUnlike FMs, whose macroscopically apparent magnetic\nproperties have been known for thousands of years, anti-\nferromagnets (AFMs) carry zero net magnetic moments\nand were elusive for some time. Even after their dis-\ncovery, AFMs were believed to have few potential ap-\nplications [15] and were disregarded in the early days\nof spintronics research. Recent theoretical and experi-\nmental findings have highlighted the potential of using\nAFMs in spintronics applications, thus starting the field\nof antiferromagnetic spintronics. Key discoveries were\nthe robustness of AFMs to external magnetic perturba-\ntions and the high resonance frequency of antiferromag-\nnetic material [16, 17]. The prediction [18] and subse-\nquent experimental detection [19] of an STT in AFMs\nsparked a massive interest in using AFMs as the active\ncomponent in spintronics devices [16, 20]. Moreover, it\nwas predicted that contrary to what was believed, anti-\nferromagnets are as efficient in pumping spin currents as\nFMs [21]. This effect was later experimentally detected in\nthe easy-axis AFM MnF 2[22]. These discoveries opened\nup the possibility of utilizing AFMs in spintronic applica-\ntions, enabling the possible fabrication of stray-field-free\ndevices operating in the THz-regime [16, 23], allowing for\nmuch faster device operation than in FMs.In recent years, the spin dynamics in AFMs have been\nexplored extensively, including the effects of disorder [24],\ngeneration of spin-Hall voltages [25], and the proper-\nties of antiferromagnetic skyrmions [26]. The spin dy-\nnamics of ferrimagnetic materials have also been stud-\nied [27]. Phenomenological models of intra- and cross-\nlattice torques were introduced in [28]. Ref. 29 fur-\nther discusses the competition between intra and cross-\nsublattice spin pumping in specific models of antiferro-\nmagnets.\nAs in antiferromagnets, ferrimagnets have opposing\nmagnetic moments. However, these moments have differ-\nent magnitudes, resulting in a net magnetization. These\nfeatures result in rich spin dynamics ranging from be-\nhavior reminiscent of antiferromagnets to ferromagnets.\nA prime example of a ferrimagnet is yttrium-iron-garnet\n(YIG). The low-energy magnon modes in YIG resemble\nmodes in ferromagnets.\nIn the study of non-equilibrium effects, the Keldysh\npath integral approach to non-equilibrium quantum field\ntheory is a powerful tool in the study of non-equilibrium\nsystems beyond linear response [30, 31]. Although most\nof the research on STT and SP utilized a semiclassical\napproach, some works have used the Keldysh framework\nin the study of spin dynamics in FMs out of equilib-\nrium [10, 11, 32–34]. Moreover, the Keldysh method was\nrecently used to formalize a fully quantum mechanical\ntheory of STT and SP, including the effects of quan-\ntum fluctuations [35]. These fluctuations have become\nincreasingly relevant with the development of new devices\noperating in the low-temperature regime. Nevertheless,\napplying the Keldysh method to derive microscopic re-\nlations for SP, STT, and fluctuating torques in an AFM\nsystem is lacking.\nIn this paper, we extend the approach of Ref. 35,\nwhich examined a ferromagnet in the macrospin approx-\nimation coupled to normal metals featuring spin andarXiv:2402.04719v1 [cond-mat.mes-hall] 7 Feb 20242\ncharge accumulation, to a similar system but instead\nfeaturing a collinear magnet with many individual spins\ncoupled at different sublattices to normal metals. Our\nstudy thus covers antiferromagnets, ferrimagnets, and\nferromagnets. We derive the spin dynamics using a\nfully quantum mechanical Keldysh non-equilibrium ap-\nproach. We find expressions for the spin transfer torque,\nspin-pumping-induced Gilbert damping, and fluctuat-\ning fields, including low-temperature shot-noise contri-\nbutions. The Gilbert damping and fluctuations contain\nboth inter-lattice and intra-lattice terms. Using Onsager\nreciprocal relations, we relate the spin pumping and spin\ntransfer coefficients. Our results enhance the knowledge\nof the microscopic expressions of STT and SP and fluctu-\nating torques in antiferromagnets and ferrimagnets cou-\npled to normal metals in the low-energy regime, where\nquantum fluctuations become essential.\nThe subsequent sections of this paper are structured\nas follows. In Sec. II, we introduce the model employed\nfor the itinerant electrons in the normal metals, the lo-\ncalized magnetic moments in the antiferromagnetic or\nferrimagnet, and the electron-magnon coupling between\nthem. We then present the key findings of this paper\nin Sec. III, including microscopic definitions of the spin\ntransfer torque, spin pumping, and fluctuating torques\nin many spin magnets, being an antiferromagnetic, fer-\nrimagnet, or ferromagnet. The derivation of an effective\nmagnon action, achieved by integrating fermionic degrees\nof freedom resulting from the interaction with normal\nmetals, is detailed in Sec. IV. The evaluation of this ef-\nfective action is then provided in Sec. V. Finally, Sec. VI\nconcludes the paper.\nII. MODEL\nWe consider a bipartite collinear magnet coupled to\nan arbitrary number of normal metal reservoirs. The\nmagnet can represent an antiferromagnet, a ferrimagnet,\nor a ferromagnet. The total Hamiltonian is\nˆH=ˆHe+ˆHem+ˆHm (1)\nin terms of the Hamiltonian describing the electrons in\nthe normal metal ˆHe, the Hamiltonian describing the in-\nteraction between the electrons and the magnet ˆHem, and\nthe Hamiltonian of the magnet ˆHm.\nThe Hamiltonian of the electrons combined with the\nHamiltonian representing the interaction between the\nelectrons and the magnet is\nˆHe+ˆHem=Z\ndrˆψ†\"\nHe+ℏ−1X\niuiσ·ˆSi#\nˆψ , (2)\nwhere ˆψ†= ( ˆψ†\n↑,ˆψ†\n↓) is the spatially dependent 2-\ncomponent itinerant electron field operator, and σis the\nvector of Pauli matrices in the 2 ×2 spin space. In the\nHamiltonian (2), ui(r) represents the spatially depen-\ndent exchange interaction between the localized spin atsiteiand the itinerant electrons. This interaction is lo-\ncalized around spin iinside the magnet. The sum over\nthe localized spins iconsists of a sum over sites in sublat-\nticeAand sublattice B, i.e.,P\ni. . .→P\na. . .+P\nb. . ..\nThe localized spin operator ˆSihas a total spin angular\nmomentum Si=ℏp\nsi(si+ 1) where siis the (unitless)\nspin quantum number of the localized spin, such that\nˆSi2=ℏ2si(si+ 1). For large si, the difference between\nSi/ℏandsiis a first-order correction, and we can ap-\nproximate Si≈ℏsi.\nThe spin-independent part of the single-particle elec-\ntron Hamiltonian is\nHe=−ℏ2\n2m∇2+Vc, (3)\nwhere Vcis the spatially dependent charge potential.\nIn the classical limit of the magnet, the spins at sublat-\nticeAare along a certain direction and the spins at sub-\nlattice Bare along the opposite direction in the ground\nstate. We will consider the semiclassical spin dynam-\nics near the instantaneous classical direction of the spins\nthat we let be along the zdirection and adiabatically\nadjust the evolution of the small deviation [10, 11, 35].\nIn the following, it is constructive to expand the in-\nteraction term to the second order in the magnet cre-\nation/annihilation operators using a Holstein-Primakoff\ntransformation,\nˆHem=ˆH0+ˆH1+ˆH2, (4)\nwhere ˆH0is the interaction with the classical magnetic\nground state and ˆH1(ˆH2) is the interaction term to the\nfirst (second) order. The classical ground state contribu-\ntion to the interaction is then\nˆH0=Z\ndrˆψ†Vsσzˆψ , (5)\nwhere the magnitude of the spatially dependent spin po-\ntential experienced by the itinerant electrons is\nVs(r) =X\nasaua(r)−X\nbsbub(r), (6)\nand oscillates rapidly with the staggered field.\nIn the macrospin approximation,P\niSican be treated\nas a giant spin in ferromagnets. Then, ui(r) becomes\nthe effective exchange interaction. Ref. 35 shows how the\nelectronic Hamiltonian ˆHecombined with the electron-\nmagnon Hamiltonian to zeroth order ˆH0become partic-\nularly transparent in ferromagnet-normal metal systems\nin terms of the scattering states of the itinerant electrons\nfor the macrospin dynamics. We generalize this approach\nto magnet-normal metal systems with individual local-\nized spins. In this picture, the electronic Hamiltonian\nremains simple, as in Ref. 35:\nˆHe+ˆH0=X\nsαϵαˆc†\nsαˆcsα, (7)3\nwhere ˆ csαannihilates an electron with spin s(s=↑or\ns=↓). The quantum number α=κnϵcaptures the\nleadκ, the transverse waveguide mode n, and the elec-\ntron energy ϵ. The electron energy consists of a trans-\nverse contribution ϵnand a longitudinal contribution\nϵ(k) =k2/2m, where kis the longitudinal momentum,\nsuch that ϵ=ϵn+ϵ(k). The eigenenergy is spin degener-\nate, since the leads are paramagnetic. Furthermore, we\nconsider identical leads such that the eigenenergy is in-\ndependent of the lead index. The system setup is shown\nL R N N AF\nFIG. 1. An antiferromagnet (AF) with conductors (N) on\neither side connected to a right (R) and a left (L) lead.\nin Fig. 1 for the case of two leads. In Eq. (7) and similar\nexpressions to follow, the sum over the scattering states\nimplies thatP\nαXsα=P\nκnR∞\nϵndϵXsκn(ϵ). In the scat-\ntering approach, the field operator is\nˆψ=X\nsαˆcsαψsα, (8)\nwhere ψsα(r) is the wave function of a scattering state\nof spin sand quantum number α.\nThe Hamiltonian of the antiferromagnet is [ idenotes\na site at sublattice A(i=a) orB(i=b)]\nHm=ℏ−2X\nijJijˆSi·ˆSj−Kℏ−2X\ni\u0010\nˆSi·ez\u00112\n+γµ0X\naHA\na·ˆSa+γµ0X\nbHB\nb·ˆSb,(9)\nwhere Jijis the symmetric exchange interaction, K > 0\nis the easy-axis anisotropy energy, and γ=g∗µB/ℏis\nthe (absolute value of) the effective gyromagnetic ratio,\nwhere g∗is the effective Land´ e g-factor and µBis the\nBohr magneton. In Eq. (9), HA,B\niis the external mag-\nnetic field in units of Am−1at lattice site i={a, b}, and\nµ0is the vacuum permeability, which appears because\nwe are employing SI units. In reality, HA=HBin the\npresence of a uniform external magnetic field. However,\nto illustrate and understand the physics, we allow the\nexternal fields at sublattices AandBto differ, and to\ndepend on the lattice site.\nWe consider the low-energy excitations from the semi-\nclassical ground state of the staggered spin orientation.\nTo this end, we carry out a Holstein-Primakoff expansion\nto the second order in magnon excitations at each sub-\nlattice AandBdescribed via the annihilation operators\nˆaaandˆbbas detailed in Appendix A. Introducing theraising/lowering fields as H±=Hx±iHy, the magnon\nHamiltonian becomes\nHm=E0+X\naEA\naˆa†\naˆaa+X\nbEB\nbˆb†\nbˆbb\n+ 2X\naa′Jaa′√sasa′ˆa†\naˆaa′+ 2X\nbb′Jbb′√sbsb′ˆb†\nbˆbb′\n+ 2X\nabJab√sasb[ˆaaˆbb+ ˆa†\naˆb†\nb]\n+γµ0ℏX\narsa\n2[HA\na−ˆaa+HA\na+ˆa†\na]\n+γµ0ℏX\nbrsb\n2[HB\nb−ˆb†\nb+HB\nb+ˆbb], (10)\nwhere the classical ground state energy E0is\nE0=X\naa′sasa′Jaa′+X\nbb′sbsb′Jbb′−2X\nabsasbJab\n−2KX\nis2\ni+ℏµ0X\nasaHA\naz−ℏµ0X\nbsbHB\nbz,(11)\nand is disregarded in the following.\nEA(B)\na(b)= 2X\nb(a)sb(a)Jab−X\na′(b′)sa′(b′)Ja(b)a′(b′)\n+ 2sa(b)K∓ℏγµ0HA(B)\na(b)z(12)\nis the energy of a local excitation, where the upper sign\nholds for sites on sublattice Aand the lower sign holds\nfor sites on sublattice B.\nIn the scattering basis of the electronic states, the cor-\nrections to the antiferromagnetic ground state electron-\nmagnon interaction to quadratic order in the magnet op-\nerators becomes ˆHem−ˆH0=ˆH1+ˆH2. The first-order\ncontribution of electron-magnon interaction is\nˆH1=X\naαβr\n2\nsah\nˆaaˆc†\n↓αWαβ\na↓↑ˆc↑β+ ˆa†\naˆc†\n↑αWαβ\na↑↓ˆc↓βi\n+X\nbαβr\n2\nsbh\nˆb†\nbˆc†\n↓αWαβ\nb↓↑ˆc↑β+ˆbbˆc†\n↑αWαβ\nb↑↓ˆc↓βi\n,(13)\nand describes the spin-flip scattering of the itinerant elec-\ntrons associated with creating or annihilating localized\nmagnons. The dimensionless matrix Wiis governed by\nthe exchange potential ui(r) and the scattering states\nwave functions ψsα:\nWαβ\niss‘=Z\ndrψ∗\nsα(r)siui(r)ψs‘β(r), (14)\nand is Hermitian, Wαβ\ni↑↓= [Wβα\ni↓↑]∗. The electron-magnon\ninteraction that is second order in the magnon operators4\nis\nˆH2=−X\naαβˆa†\naˆaa\nsah\nˆc†\n↑αWαβ\na↑↑ˆc↑β−ˆc†\n↓αWαβ\na↓↓ˆc↓βi\n+X\nbαβˆb†\nbˆbb\nsbh\nˆc†\n↑αWαβ\nb↑↑ˆc↑β−ˆc†\n↓αWαβ\nb↓↓ˆc↓βi\n,(15)\nwhere the matrix elements are defined in Eq. (14). We\nnote that our electron-magnon-interaction is isotropic in\nspin space and will give rise to zeroth-, first-, and second-\norder magnon terms in the Hamiltonian, i.e. ˆH0,ˆH1and\nˆH2, respectively. This is in contrast to the model used\nin Refs. 12 and 32, where only the first-order term ˆH1is\nconsidered.\nFinally, in normal metal reservoirs, the occupation of\nthe state is\n⟨c†\ns′αcsβ⟩=δαβnss′α, (16)\nwhere the 2 ×2 out-of-equilibrium distribution is\nnss′α=1\n2[fκ↑(ϵα)) +fκ↓(ϵα))]δss′\n+1\n2[fκ↑(ϵα))−fκ↓(ϵα))]uκ·σss′, (17)\nallowing for a (lead-dependent) spin accumulation in the\ndirection of the unit vector uk.f↑andf↓are general\ndistribution functions for spin-up and spin-down parti-\ncles, which generally differ for elastic or inelastic trans-\nport [35]. In equilibrium, the distribution function only\ndepends on energy,\nfeq\nκ↑(ϵ) =feq\nκ↓(ϵ) =f(ϵ−µ0), (18)\nwhere fis the equilibrium Fermi-Dirac distribution and\nµ0is the equilibrium chemical potential.\nIn inelastic transport, the spin- and charge accumula-\ntions µCandµScorrespond to chemical potential in a\n(spin-dependent) Fermi-Dirac function,\nfin\nκ↑(ϵ) =f(ϵ−µ0−µC\nκ−µS\nκ/2) (19a)\nfin\nκ↓(ϵ) =f(ϵ−µ0−µC\nκ+µS\nκ/2). (19b)\nFor notational simplicity, we define the chemical poten-\ntials\nµκ↑=µ0+µC\nκ+µS\nκ/2 (20a)\nµκ↓=µ0+µC\nκ−µS\nκ/2. (20b)\nIn the limit of small charge and spin accumulations com-\npared to the Fermi level, it can be derived that\nµC\nκ+µS\nκ\n2=Z\ndϵ\u0002\nfin\nκ↑(ϵ)−f(ϵ)\u0003\n. (21)\nIn the elastic regime, the distribution function cannot\ngenerally be described as a Fermi-Dirac function. Thedistribution function is instead given as a linear combi-\nnation of Fermi-Dirac functions in the connected reser-\nvoirs [35],\nfel\nsκ(ϵ) =X\nlRsκlf(ϵ−µl), (22)\nwhere the index lruns over the reservoirs, and Rsκlis the\nlead and spin-dependent transport coefficient for reser-\nvoirl. The transport coefficients satisfy\nX\nlRsκl= 1. (23)\nIn the elastic transport regime, it is advantageous to de-\nfine the effective charge and spin accumulations through\nµC\nκ+µS\nκ\n2=Z\ndϵ\u0002\nfel\nκ↑(ϵ)−f(ϵ)\u0003\n. (24)\nThe elastic and inelastic transport regime results in dif-\nferent results for the fluctuations in the magnetization\ndynamics of the magnet.\nHaving specified the model for the system in consider-\nation, we proceed by presenting the main results of the\npaper.\nIII. MAIN RESULTS: EQUATIONS OF MOTION\nThis section presents the main results of our work.\nOur primary result is the derivation of a Lan-\ndau–Lifshitz–Gilbert–Slonczewski (LLGS) equation for\nthe localized (normalized) spins mi=Si/Siin a gen-\neral magnet coupled to normal metal reservoirs,\n∂tmi=τb\ni+τf\ni+τsp\ni+τstt\ni (25)\nvalid for low-energy excitations when the equilibrium\nmagnetization is parallel (antiparallel) to the z-axis. The\nbulk antiferromagnet torque τb\nifor a site i={a, b}arises\nfrom contributions of anisotropy, exchange coupling, and\nexternal fields, and reads\nτb\ni=−z×\u0000\nℏ−1Eimi+γµ0Hi\u0001\n. (26)\nwhere Eiis the energy of a local excitation and Hiis\nthe applied field. Hence, the bulk torque remains unaf-\nfected by the presence of normal metal reservoirs and the\nassociated spin- and charge accumulations.\nThe spin transfer torque τstt\niis induced by spin accu-\nmulation in the normal metals, and can be expressed as\nfollows:\nτstt\ni=ℏ−1X\nκ\u0002\nβI\niκz×µS\nκ−βR\niκz×(z×µS\nκ)\u0003\n.(27)\nIn Eq. (27), the superscripts ” R,I” denote the real and\nimaginary part. The site and lead-dependent coefficients5\nβiκare expressed in terms of the microscopic scatter-\ning matrix elements defined in Eq. (14) evaluated at the\nFermi energy:\nβiκ=−2i\nsiX\nnWκnκn\ni↑↓, (28)\nand can be calculated numerically for any particular sys-\ntem configuration.\nThe spin pumping torque τsp\nicontains contributions\nfrom both sublattices and is given by\nτsp\ni=X\nj\u0002\nαR\nijz×∂tmj+αI\nijz×(z×∂tmj)\u0003\n,(29)\nwhere jruns over all sites and αijis expressed in the low-\nenergy limit using the scattering matrix elements evalu-\nated at the Fermi energy,\nαij=2π√sisjX\nκλnmWκnλm\ni↓↑Wλmκn\nj↑↓, (30)\nandαR(I)denotes the real (imaginary) part of the matrix.\nUsing the Onsager reciprocal relations in Appendix B, we\nfind that the spin transfer torque and spin pumping are\nrelated in the case of the most relevant case of a single\nreservoir,\nX\njαij=βi. (31)\nFinally, the fluctuating torque τf\niis expressed in terms\nof a fluctuating transverse field Hf\ni,\nτf\ni=−γµ0z×Hf\ni. (32)\nThe fluctuating field exhibit interlattice and intralattice\ncorrelators ⟨HµiHνj⟩, where µ, ν={x, y}:\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nxj⟩= ImΣK\nij+ 4Im ˜Σ↑↓ij (33a)\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nyj⟩=−ReΣK\nij−4Re˜Σ↑↓ij (33b)\n2√sisjγ2µ2\n0⟨Hf\nyiHf\nyj⟩= ImΣK\nij−4Im˜Σ↑↓ij, (33c)\nwhere the time arguments tandt′of the fields and the\nrelative time argument ( t−t′) of the self energies are\nomitted for simplicity. The self-energy Σ is due to charge\nand longitudinal spin accumulations in the normal metals\nand is nonzero even in equilibrium. It is conveniently\nwritten as a product of a frequency-dependent quantity\nπ(ω) and a site and scattering states dependent quantity\nσij[35]:\nΣK\nij(ω) =i\nℏX\nκλσijκλπκλ(ω), (34)\nwith\nπκλ(ω) =−2Z\ndϵ[2n↑↑κ(ϵ)n↓↓λ(ϵ+ℏω)\n−n↑↑κ(ϵ)−n↓↓λ(ϵ+ℏω)] (35a)\nσijκλ=2π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↑↓. (35b)Conversely, the self-energy matrices ˜Σ↑↓are due to trans-\nverse spin accumulation in the normal metals and, as a\nresult, vanish in equilibrium. Analogous to the decom-\nposition in Eq. (34), we write\n˜ΣK\n↑↓ij=−i\nℏX\nκλ˜σ↑↓ijκλ˜πκλ(ω), (36)\nwhere\n˜π↑↓(ω) =−4Z\ndϵn↑↓κ(ϵ)n↑↓λ(ϵ+ℏω) (37a)\n˜σ↑↓ijκλ=−π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↓↑, (37b)\nThe noise matrices π(ω) and ˜ π(ω) are similar to what\nwas found in Ref. 35, and are calculated in the equi-\nlibrium, elastic, and inelastic transport regime in Sec.\nV B. Crucially, the shot noise differs on various sites, due\nto the site-dependence of σand ˜σ. At equilibrium, the\nfluctuation-dissipation theorem holds, e.g.\n2siγ2µ2\n0⟨Hf\nµiHf\nνi⟩=δµναii4kBTξ\u0012ℏω\n2kBT\u0013\n, (38)\nwhere ξ(x) =xcothx.\nIn the next section, we discuss the Keldysh action of\nthe model presented in this section and derive an effec-\ntive action by integrating out the fermionic degrees of\nfreedom.\nIV. KELDYSH THEORY AND EFFECTIVE\nACTION\nIn this section, we derive the semiclassical spin dynam-\nics by using an out-of-equilibrium path integral formal-\nism [30]. We introduce the closed contour action Sand\nthe partition function Z,\nZ=Z\nD[¯aa¯bb¯c↑c↑¯c↓c↓]eiS/ℏ. (39)\nThe action Sconsists of contributions from the localized\nmagnetic excitations aandb, and the spin-up c↑and\nspin-down electrons c↓from the scattering states. We\nwill integrate out the fermion operators and get an effec-\ntive action for the magnetic excitations aandb, which\nincludes effective transverse and longitudinal fields that\narise from the charge and spin accumulations in the nor-\nmal metals.\nWe follow Ref. 30 and replace the fields in Eq. (39)\nwith ” ±” fields residing on the forward and backward\npart of the Schwinger-Keldysh contour. The action in\nthe±basis is given in Appendix C. These fields are not\nindependent of each other and can be Keldysh rotated\ninto a new basis that takes into account the coupling\nbetween them. The rotated fields have the advantage\nof suggesting a transparent physical interpretation, cor-\nresponding to the semiclassical equations and quantum\ncorrections.6\nA. Keldysh action\nFor magnons, the classical ( cl) and quantum ( q) fields\nare defined linear combinations of the ±-fields, as de-\nscribed in detail in Appendix C. In Keldysh space, it is\nconvenient to also introduce the matrices\nγq=\u0012\n0 1\n1 0\u0013\nγcl=\u0012\n1 0\n0 1\u0013\n. (40)\nThe Keldysh rotated magnon action becomes\nSm=X\nat¯aq\na(iℏ∂t−EA\na)acl\na+X\nbt¯bq\nb(iℏ∂t−EB\nb)bcl\nb\n+X\nataq\na(iℏ∂t−EA\na)¯acl\na+X\nbtbq\nb(iℏ∂t−EB\nb)¯bcl\nb\n−2X\naa′t√sasa′Jaa′\u0002\n¯aq\naacl\na′+h.c.\u0003\n−2X\nbb′t√sbsb′Jbb′\u0002¯bq\nbbcl\nb′+h.c\u0003\n−2X\nabt√sasbJab\u0002\n¯aq\na¯bcl\nb+ ¯acl\na¯bq\nb+h.c.\u0003\n−γµ0ℏX\nat√sa\u0002\nHA\na−aq\na+HA\na+¯aq\na\u0003\n−γµ0ℏX\nbt√sb\u0002\nHB\nb−¯bq\nb+HB\nb+bq\nb\u0003\n, (41)where we wrote the time integral as a sum for compact\nnotation. In Eq. (41), h.c.denotes the hermitian conju-\ngate of the previous term. The fermion action becomes\nSe+S0=X\nst¯Csγcl(iℏ∂t−ϵ)Cs, (42)\nwhere we introduced vector notation for the 1 /2-fields,\n¯Cs= (¯c1\nsα,¯c2\nsα), and ϵis a diagonal matrix containing\nthe single-particle energies of the electrons. The Keldysh\nrotated first-order electron-magnon interaction is\nS1=−X\nat\nαβ1√sah\nacl\naWαβ\na↓↑¯C↓αγclC↑β+aq\naWαβ\na↓↑¯C↓αγqC↑β+h.c.i\n−X\nbt\nαβ1√sbh\n¯bcl\naWαβ\nb↓↑¯C↓αγclC↑β+¯bq\nbWαβ\nb↓↑¯C↓αγqC↑β+h.c.i\n, (43)\nand, finally, the second-order term reads\nS2=X\nat\nαβ1\n2sah\nWαβ\na↑↑\u0000¯AaγclAa¯C↑αγclC↑β+¯AaγqAa¯C↑αγqC↑β\u0001\n−Wαβ\na↓↓\u0000¯AaγclAa¯C↓αγclC↓β+¯AaγqAa¯C↓αγqC↓β\u0001i\n−X\nbt\nαβ1\n2sbh\nWαβ\nb↑↑\u0000¯BbγclBb¯C↑αγclC↑β+¯BbγqBb¯C↑αγqC↑β\u0001\n−Wαβ\nb↓↓\u0000¯BbγclBb¯C↓αγclC↓β+¯BbγqBb¯C↓αγqC↓β\u0001i\n,\n(44)\nwhere the magnon q/cloperators are consolidated in vec-\ntors ¯Aaand ¯Bb. The Keldysh rotated action proves to\nbe well-suited for the computation of an effective magnon\naction, a topic we delve into in the following section.B. Integrating out the fermionic degrees of\nfreedom\nFor the itinerant electrons in the normal metal and\nantiferromagnet, the total effective electron action is7\nSe,tot=Se+Sem, and can be expressed as\nSe,tot=X\nss′tt′¯Cs,tG−1\nss′,tt′Cs′,t′, (45)\nwhere the interacting Green function Gis given in terms\nof the noninteracting Green function G0and interaction\nterms as\nG−1=G−1\n0+˜W1+˜W2. (46)\nHere, ˜W1contains the first-order magnon operators on\nboth sublattices,\n˜W1=δ(t−t′)\u0002\nWA\n1+WB\n1\u0003\n, (47)\nwhere WA\n1andWB\n1are spin flip operators:\nWA\n1=−X\nxa1√saγx\u0002\nWa↑↓¯ax\naσ++Wa↓↑ax\naσ−\u0003\n(48a)\nWB\n1=−X\nxb1√sbγx\u0002\nWb↑↓bx\nbσ++Wb↓↑¯bx\nbσ−\u0003\n.(48b)\nIn Eq. (48), the variable x={cl, q}represents a Keldysh\nspace index, and\nσ+=\u0012\n0 1\n0 0\u0013\nσ−=\u0012\n0 0\n1 0\u0013\n(49)\nare the usual raising and lowering Pauli matrices. Sim-\nilarly, ˜W2contains the magnon operators to quadratic\norder for both sublattices,\n˜W2=δ(t−t′)\u0002\nWA\n2−WB\n2\u0003\n, (50)\nwith WA\n2andWB\n2given by\nWA\n2=X\naxy1\n2sa¯ax\naγxay\naγy\u0012\nWa↑↑ 0\n0−Wa↓↓\u0013\n(51a)\nWB\n2=X\nbxy1\n2sb¯bx\nbγxby\nbγy\u0012\nWb↑↑ 0\n0−Wb↓↓\u0013\n, (51b)\nwhere the spin structure is explicitly written out as a ma-\ntrix. The matrices WA(B)\n1 andWA(B)\n2 have a structure\nin the scattering states space from Wa(b), spin space from\nthe Pauli matrices, and Keldysh space from γx. The in-\nverse free electron Green function G−1\n0from Eq. (46) has\nthe conventional causality structure in Keldysh space,\nwith a retarded ( R), advanced ( A), and Keldysh ( K)\ncomponent:\nG−1\n0=\u0012\n[GR\n0]−1[GK\n0]−1\n0 [ GA\n0]−1\u0013\n, (52)\nand has equilibrium components that are diagonal in\nboth spin space and in the scattering states space,\n[G−1\n0]R(A)\nαβ,ss′=δαβδss′δ(t−t′)(iℏ∂t−ϵα±iδ),(53)where the upper sign corresponds to the retarded compo-\nnent, while the lower sign is applicable to the advanced\ncomponent. The Keldysh component includes informa-\ntion about the distribution function, and will be dis-\ncussed below, when we Fourier transform the Green func-\ntions. Inverting the matrix from Eq. (52), the fermionic\nGreen function acquires the following causality structure:\nG0=\u0012\nGR\n0GK\n0\n0GA\n0\u0013\n. (54)\nFrom the effective electron action in Eq. (45), it is\nevident that the partition function of Se,tottakes on a\nGaussian form with respect to the fermionic operators.\nHence, the fermionic integral in the partition function\ncan be evaluated exactly, with an inconsequential pro-\nportionality constant being disregarded:\nZ\nD[C]eiSe,tot/ℏ= eTr[ln[1+G0˜W1+G0˜W2]]. (55)\nIn Eq. (55), we have used the short-hand notation for\nthe functional integral measure of all fermionic states,\nD[C] =D[¯C↑C↑¯C↓C↓]. We have absorbed a normaliza-\ntion constant into the functional integral measure for sim-\nplicity. We note that as a consequence of the continuity\nof the time coordinate and scattering states energy that\nwe are employing, the unit matrix is a delta function in\ntime and energy, 1 ≡δ(t−t′)δ(ϵα−ϵβ), and thus quanti-\nties inside the logarithm carries dimension J−1s−1. The\ntrace, on the other hand, is an integral operator with unit\nJ s. As long as one interprets the logarithm in terms of\nits Taylor expansion, this does not lead to any problems,\nas the exponent of Eq. (55) becomes dimensionless for all\nterms in the expansion. The exponent is interpreted as\nan additional contribution to the magnon action,\ni\nℏSeff= Trh\nln[1 + G0˜W1+G0˜W2]i\n. (56)\nThe way forward is to treat this interaction as a pertur-\nbation, expanding the logarithm in first and second-order\ncontributions and disregarding higher-order terms,\nSeff≈ −iℏTrh\nG0˜W1i\n−iℏTrh\nG0˜W2i\n+iℏ\n2Trh\nG0˜W1G0˜W1i\n. (57)\nTo evaluate the trace in these terms, it is convenient to\nFourier transform all quantities from the time domain\nto the energy domain. This diagonalizes the noninter-\nacting Green functions, making calculations much more\nstraightforward.\nC. Fourier representation\nThe paper employs the Fourier transform convention\ndefined in Appendix D. In Fourier space, the fermion8\nequilibrium Green function components are particularly\nsimple:\n[G0]R(A)\nαβ,ss′(ω) =δαβδss′(ℏω−ϵ±iδ)−1, (58)\nwhere δ > 0 is an infinitesimal quantity ensuring con-\nvergence. The Keldysh component accounts for non-\nequilibrium phenomena though the spin-dependent dis-\ntribution nss′αdefined in Eq. (17),\n[G0]K\nαβ,ss′(ω) =−2πiδαβδ(ℏω−ϵα) [δss′−2nss′α].(59)\nThe Keldysh component has off-diagonal terms in spin\nspace if the distribution function nss′αhas off-diagonal\nelements, i.e. if there is a transverse spin accumulation\nin the normal metals.\nV. NON-EQUILIBRIUM SPIN DYNAMICS\nHaving derived the effective action as expressed in Eq.\n(57), we proceed by evaluating the traces and delving\ninto the resultant terms. The discussion unveils effective\nlongitudinal and transverse fields, which we ascribe to\nspin transfer torque and spin pumping originating from\nthe normal metal reservoirs.\nA. First-order contribution\nEvaluating the trace in the first order term in Eq.\n(57) corresponds to summing over the diagonal elements\nin spin space and Keldysh space, integrating over both\ntime variables, and summing over the space of scattering\nstates, we find\n−iℏTrh\nG0˜W1i\n=−X\naα2√saWαα\na↑↓n↓↑αZ\ndt¯aq\na(t)\n−X\naα2√saWαα\na↓↑n↑↓αZ\ndtaq\na(t)\n−X\nbα2√saWαα\nb↑↓n↓↑αZ\ndtbq\nb(t)\n−X\nbα2√saWαα\nb↓↑n↑↓αZ\ndt¯bq\nb(t).(60)\nHere, we have used the general Green function identity\nGR(t, t)+GA(t, t) = 0 [30], and written the time integra-\ntion explicitly. Comparing the first-order contribution in\nEq. (60) with the magnon action in Eq. (41), we observe\nthat the first-order effect of the spin accumulation in the\nnormal metal is equivalent to an effective deterministic\ntransverse magnetic field Hstt\ni, which act on a localized\nspin at site i={a, b}in the antiferromagnet. The ”stt”\nsuperscript indicates that this field will take the form of\na spin transfer torque, which will be elaborated on be-\nlow. The magnitudes of these effective transverse fieldsare given by\nγµ0Hstt\ni−=2\nsiℏX\nαWαα\ni↓↑n↑↓α (61a)\nγµ0Hstt\ni+=2\nsiℏX\nαWαα\ni↑↓n↓↑α, (61b)\nwhich implies that the Cartesian components read\nγµ0Hstt\nix=2\nsiℏX\nαRe\u0002\nWαα\ni↑↓n↓↑α\u0003\n(62a)\nγµ0Hstt\niy=2\nsiℏX\nαIm\u0002\nWαα\ni↑↓n↓↑α\u0003\n. (62b)\nRecalling that the spin accumulation is given by Eq. (24)\nand Eq. (21), we write the effective fields from Eq. (62)\nin the conventional spin transfer torque form:\nγµ0Hstt\ni=1\nℏX\nκ\u0002\nβR\niκz×µS\nκ+βI\niκz×(z×µS\nκ)\u0003\n,(63)\nwhere the appearance of zis a consequence of our the-\nory being restricted to small deviations for the equilib-\nrium magnetization ±z. This results in the spin transfer\ntorque given in Eq. (27). In Eq. (63), the superscripts\n”R” and ” I” denote the real and imaginary parts and the\nlead- and site-dependent constants βiκhave been intro-\nduced as sums over the transverse modes of the scattering\nmatrix elements,\nβiκ=−2i\nsiX\nnWκnκn\ni↑↓, (64)\nand where we have assumed that the transverse spin dis-\ntribution functions n↑↓andn↓↑are only significant close\nto the Fermi surface, such that the scattering states ma-\ntrix elements are well approximated by their value at\nthe Fermi surface. The expression for the spin transfer\nfield in Eq. (63) is valid in both the elastic and inelastic\nregime, and vanishes in equilibrium. We note that the\ncoefficient βiκ, for i={a, b}, depends not only on the\npotential at lattice site ibut also indirectly of all lattice\nsites on both sublattices through the scattering states.\nTo the lowest order, the sublattice magnetizations are\nparallel and antiparallel to the z-axis, mA≈zand\nmB≈ −z. Thus, to the lowest order in the magnon\noperators, the expressions for the transverse fields are\nambiguous, and we can write the transverse field in Eq.\n(63) in terms of mAormB. To the lowest order in the\nmagnon operators, the Keldysh technique cannot be used\nto identify which sublattice the transverse fields in Eq.\n(60) originate from.\nB. Second order contribution\nThe second order contribution in Eq. (57) has contri-\nbutions from ˜W2,\nS21=−iℏTrh\nG0˜W2i\n, (65)9\nas well as a contribution from ˜W1,\nS22=iℏ\n2Trh\nG0˜W1G0˜W1i\n. (66)\nProceeding in a manner analogous to the treatment of\nthe first-order term, the trace in S21is evaluated:\nS21=−X\naαπ\nsa\u0002\nWαα\na↑↑(1−2n↑↑α)−Wαα\na↓↓(1−2n↓↓α)\u0003Z\ndt¯Aa(t)γqAa(t)\n+X\nbαπ\nsb\u0002\nWαα\nb↑↑(1−2n↑↑α)−Wαα\nb↓↓(1−2n↓↓α)\u0003Z\ndt¯Bb(t)γqBb(t). (67)\nFrom Eq. (41), it is apparent that the second-order terms in S21are equivalent with a longitudinal magnetic field,\nwith magnitude\nγµ0HA21\niz=−π\nℏsiX\nα\u0002\nWαα\ni↑↑(1−2n↑↑α)−Wαα\ni↓↓(1−2n↓↓α)\u0003\n, (68)\nwhich, in this reference frame, renormalizes the energies of localized magnon excitations. However, such longitudinal\nmagnetic fields should not affect the spin dynamics since they, in the instantaneous reference field, correspond to\ncontributions to the total free energy proportional to S2\ni.\nThe final contribution S22to the effective action contains inter-lattice and intra-lattice terms and can be written\ncompactly by introducing a field di={aa,¯bb}and summing over the two field components, i.e.P\nidi=P\naaa+P\nb¯bb:\nS22=Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↑↓(t′, t)Wi↓↑γxG0,↑↓(t, t′)γx′Wj↓↑i\ndx\ni(t)dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↓↑(t′, t)Wi↑↓γxG0,↓↑(t, t′)γx′Wj↑↓i\n¯dx\ni(t)¯dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↓↓(t′, t)Wi↓↑γxG0,↑↑(t, t′)γx′Wj↑↓i\ndx\ni(t)¯dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↑↑(t′, t)Wi↑↓γxG0,↓↓(t, t′)γx′Wj↓↑i\n¯dx\ni(t)dx′\nj(t′), (69)\nwhere the trace is taken only over the 2 ×2 Keldysh space and the space of scattering states α. The interlattice terms,\ni.e.d=d′, are discussed in Ref. 35 for a macrospin ferromagnet. Here, we summarize this discussion and highlight\nthe addition of the inter-lattice terms not present in the macrospin ferromagnet.\nEvaluating the trace in the first and second line of Eq. (69), we note that only the Keldysh component has off-\ndiagonal elements in spin space, and find a contribution only from x=x′=q,\n˜Sqq\n22=ℏZ\ndtdt′X\nijh\ndq\ni(t)˜ΣK\n↑↓ij(t, t′)dq\nj(t′)i\n(70a)\n˜S¯q¯q\n22=ℏZ\ndtdt′X\nijh\n¯dq\ni(t)˜ΣK\n↓↑ij(t, t′)¯dq\nj(t′)i\n, (70b)\nwhere the self-energies are\n˜ΣK\n↑↓ij(t−t′) =−2i\nℏ2√sisjX\nαβn↑↓αn↑↓βWαβ\ni↓↑Wβα\nj↓↑ei(ϵα−ϵβ)(t−t′)/ℏ(71a)\n˜ΣK\n↓↑ij(t−t′) =−2i\nℏ2√sisjX\nαβn↓↑αn↓↑βWαβ\ni↑↓Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ. (71b)10\nThe reasoning behind identifying this self-energy as a Keldysh component is that it couples the quantum components\nof the fields, see Eq. (70). The terms in Eq. (70) do not have a direct analog in the magnon action in Eq. (41), and\ninterpreting these will be the subject of Sec. V C. The self-energies in Eq. (71) are invariant under a joint time and\nlattice site reversal, i.e. ˜Σij(t−t′) =˜Σji(t′−t). Moreover, due to the properties n↑↓=n∗\n↓↑andWαβ\ni↑↓= [Wβα\ni↓↑]∗, we\nsee that the self-energies are related by ˜ΣK\n↑↓ij(t−t′) =−[˜ΣK\n↓↑ij(t−t′)]∗, which will be important later.\nDisregarding terms of the order kBT/ϵFandµs/ϵF[35], we find that the Fourier-transformed self-energy becomes\n˜ΣK\n↑↓ij(ω) =−i\nℏX\nκλ˜σ↑↓ijκλ˜πκλ(ω), (72)\nwhere\n˜πκλ(ω) =−4Z\ndϵn↑↓κ(ϵ)n↑↓λ(ϵ+ω) ˜ σ↑↓ijκλ=−π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↓↑, (73)\nand where the matrix elements Ware evaluated at the Fermi surface. This is a straightforward generalization of\nthe macrospin ferromagnet case, with the addition of shot-noise contributions from inter-lattice and intra-lattice\ninteractions between different lattice sites. We can evaluate the quantity ˜ πκλ↑↓(ω) by using Eq. (17):\n˜π↑↓κλ(ω) =−uκ−uλ−Z\ndϵ[f↑κ(ϵ)−f↓κ(ϵ)] [f↑λ(ϵ+ℏω)−f↓λ(ϵ+ℏω)] (74)\nwhere we introduced the conventional ”lowering” vector u−=ux−iuy. This can be computed in equilibrium, elastic,\nand inelastic scattering cases[recall the definition of the chemical potentials in Eq. (20)]:\n˜πeq\n↑↓κλ(ω) = 0 (75a)\n˜πin\n↑↓κλ(ω) =uκ−uλ−\u0002\nF(µ↑λ−µ↑κ−ℏω) +F(µ↓λ−µ↓κ−ℏω)−F(µ↑λ−µ↓κ−ℏω)−F(µ↓λ−µ↑κ−ℏω)\u0003\n(75b)\n˜πel\nκλ↑↓(ω) =uκ−uλ−X\nll′[R↑κl−R↓κl] [R↑λl′−R↓λl′]F(µl′−µl−ℏω), (75c)\nwhere we defined the quantity F(µ2−µ1) =R\ndϵf(ϵ−µ1)[1−f(ϵ−µ2)] = [ µ2−µ1]n(µ2−µ1),nis the Bose-Einstein\ndistribution, and we used the identity from Eq. (23) in the last line.\nWe now turn our attention to the third and fourth lines of the second-order action in Eq. (69). The contributions\nfrom the two lines are equal, which is evident from interchanging summation indices and rearranging terms. Their\ntotal contribution to the action S22can be split into contributions S¯qq\n22,S¯qcl\n22, and S¯clq\n22. The contribution Sclclvanishes,\ndue to the quantity GR(t′−t)GR(t−t′) being nonzero only for t=t′, which has measure zero, and similarly for\nGA. This ensures that the action satisfies the general requirement S[ϕcl, ϕq= 0] = 0 [30]. Introducing, for notational\nconvenience, the vector ¯Di=\u0000¯dcl¯dq\u0001\n, we find\nS¯qq\n22+S¯qcl\n22+S¯clq\n22=ℏZ\ndtdt′X\nij¯Di(t)ˆΣij(t−t′)Dj(t′), (76)\nwhere the self-energy matrix has structure in Keldysh space and in the sublattice space,\nˆΣij(t−t′) =\u0012\n0 ΣA(t−t′)\nΣR(t−t′) ΣK(t−t′)\u0013\nij, (77)\nand its components are given by\nΣK\nij(t−t′) =2i√sisjℏ2X\nαβ(n↑↑α+n↓↓β−2n↑↑αn↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ(78a)\nΣR\nij(t−t′) =2i√sisjℏ2θ(t−t′)X\nαβ(n↑↑α−n↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ(78b)\nΣA\nij(t′−t) =−2i√sisjℏ2θ(t−t′)X\nαβ(n↑↑α−n↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ. (78c)11\nThe Keldysh component of this self-energy has the symmetry [ΣK\nij(t−t′)]∗=−ΣK\nji(t′−t). Imperatively, as a\nconsequence of this symmetry, the quantities\nΣK\nij(t−t′)−ΣK\nji(t′−t) = 2Re\u0002\nΣK\nij(t−t′)\u0003\n(79)\niΣK\nij(t−t′) + iΣK\nji(t′−t) =−2Im\u0002\nΣK\nij(t−t′)\u0003\n, (80)\nare real numbers, which will be important in the next section. We proceed by a similar analysis to what was done\nwith ˜Σ, writing it in terms of a shot-noise matrix. We assume that the matrices Wcan be approximated by their\nvalue on the Fermi surface, and write\nΣK\nij(ω) =i\nℏX\nκλσijκλπκλ(ω), (81)\nwhere we introduced the matrices\nπκλ(ω) =−2Z\ndϵ[2n↑↑κ(ϵ)n↓↓λ(ϵ+ℏω)−n↑↑κ(ϵ)−n↓↓λ(ϵ+ℏω)] σijκλ=2π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↑↓.(82)\nWe evaluate π(ω) in equilibrium, and for elastic and inelastic scattering :\nπeq\nκλ(ω) = 4 kBTξ\u0012ℏω\n2kBT\u0013\n(83)\nπin\nκλ(ω) = 4 kBTX\nss′p↑s′κp↓sλξ\u0012ℏω+µs′κ−µsλ\n2kBT\u0013\n(84)\nπel\nκλ(ω) = 4 kBTX\nss′ll′p↑s′κp↓sλRs′κlRsλl′ξ\u0012ℏω+µl−µl′\n2kBT\u0013\n, (85)\nwhere ξ(x) =xcothxis an asymptotically linear function\nfor high xandpss′κ= (1−uzκ)/2+uzκδssis a projection\nfactor introduced for notational convenience.\nComparing with the magnetic action in Eq. (41), we\nnotice that the terms with the retarded and advanced\nself-energies are equivalent with longitudinal fields, which\nwe in the following will show consists of dissipative\nGilbert-like terms and non-dissipative field-like terms.\nFourier transforming and applying the identity (D5), the\nretarded and advanced self-energies from Eq. (78b) and\nEq. (78c) become\nΣR,A\nij=−2√sisjℏX\nαβn↑↑α−n↓↓β\nℏω+ϵα−ϵβ±iδWαβ\ni↓↑Wβα\nj↑↓.(86)\nThis self-energy has equilibrium contributions as well\nas non-equilibrium contributions, however, the non-\nequilibrium contributions scale as µ↑↑/ϵFandµ↓↓/ϵFand\nare disregarded in the following. The equilibrium part of\nEq. (86) becomes particularly transparent when expand-\ning to first order in the frequency ω:\nΣR/A\n↑↓ij(ω)≈ΣR/A\n↑↓ij(ω= 0)±iωαij, (87)\nwhere we introduced the frequency-independent matrix\nelement\nαij=2π√sisjX\nαβ[−f′(ϵα)]δ(ϵα−ϵβ)Wαβ\ni↓↑Wβα\nj↑↓,(88)which can be approximated to\nαij=2π√sisjX\nκλnmWκnλm\ni↓↑Wλmκn\nj↑↓, (89)\nwhere the scattering states matrix elements are evalu-\nated at the Fermi surface. We note from the identity\n[Wαβ\ni↑↓]∗=Wβα\ni↓↑thatαis a Hermitian matrix in the space\nof lattice sites, i.e. [ αij]∗=αji. The zeroth order term\nin frequency is\n[S¯qcl\n22+S¯clq\n22]0=ℏX\nijZ\ndω¯dq\ni(ω)ΣR\n↑↓ij(0)dcl\nj(ω)\n+ℏX\nijZ\ndω¯dcl\ni(ω)ΣA\n↑↓ij(0)dq\nj(ω),(90)\nwhich is a constant longitudinal field that plays no role\nin the instantaneous reference frame, as discussed above.\nThe first-order term in frequency is finite even in equi-\nlibrium,\n[S¯qcl\n22+S¯clq\n22]1=ℏX\nijαijZ\ndt¯Diγq∂tDj, (91)\nand takes the form of a Gilbert damping term, includ-\ning both inter-lattice and intra-lattice contributions. The\nspin transfer torque coefficient αand the spin pumping12\ncoefficient βare related to each other as a consequence of\nthe Onsager reciprocal relations [36]. In Appendix B we\nderive this relation, which is given in Eq. (B11), and de-\nrive an optical theorem relating the scattering matrices,\ngiven in Eq. (B12).\nSummarizing this section, we have found that the cor-\nrections to the magnon action Smin the presence of spin\nand charge accumulations in surrounding normal metals\nisS1+S21+S¯qcl\n22+S¯clq\n22+˜Sqq\n22+S¯qq\n22, and found that the\nfirst three of these contributions appear like magnetic\nfields and (in the low-frequency limit) like Gilbert-like\ndamping terms in the effective magnon action. Impor-\ntantly, we find both longitudinal and transverse fields in\nthe general case. The last two contributions to the ac-\ntion consist of coupled quantum fields and are the result\nof purely quantum effects. These terms are the subjectof the next section.\nC. Fluctuating fields\nFrom the effective action in the last section, we were\nable to associate the ( q, cl) and ( cl, q) terms with longi-\ntudinal fields by comparing them with the magnon ac-\ntion in Eq. (41). Now, we must address the issue of\nhow to interpret the ( q, q) terms, which lack an ana-\nlog in the action described in Eq. (41). In this section,\nwe derive fluctuating forces from these terms by employ-\ning a Hubbard-Stratonovich (HS) transformation on the\nquadratic fields in the effective action, introducing aux-\niliary fields in the process. Commencing with the con-\ntribution from the term S¯qq\n22, we introduce the complex\nauxiliary field h¯qq\ni(in units of inverse second) via a con-\nventional Hubbard–Stratonovich transformation:\neiS¯qq\n22/ℏ= exp\u0014Z\ndtdt′X\nij¯dq\ni(t)iΣK\nij(t−t′)dq\nj(t′)\u0015\n=1\ndet [−iΣK]ZY\niD[h¯qq\ni] exp\u0014\niZ\ndtX\nih¯qq\ni(t)¯dq\ni(t) +h.c.−Z\ndtdt′X\nij¯h¯qq\ni(t)[−iΣK\nij(t−t′)]−1hqq\nj(t′)\u0015\n,\n(92)\nwhere a shorthand notation for the measure was in-\ntroduced as D[h¯qq\ni] = Π k\b\nd[Imh¯qq\ni(tk)]d[Reh¯qq\ni(tk)]/π\t\n,\nwhere kis the index used to order the discretization of\nthe time coordinate. From the Gaussian form of Eq. (92),\nthe correlators of the auxiliary field can be identified as\n⟨h¯qq\ni(t)⟩= 0 (93a)\n⟨h¯qq\ni(t)h¯qq\nj(t′)⟩= 0 (93b)\n⟨¯h¯qq\ni(t)h¯qq\nj(t′)⟩=−iΣK\nij(t−t′). (93c)The second term in the exponent is quadratic in the new\nfields, and gives no contribution to the magnon action,\nwhile the first term is linear in the magnon field diand is\ninterpreted as an effective transverse field in the magnon\naction.\nThe contribution from the terms ˜Sqq\n22+˜S¯q¯q\n22is HS trans-\nformed by performing an unconventional transformation\nin the two complex fields ˜hqq\niand˜hqq\niseparately:\nei˜S22/ℏ=1q\ndet [−2i˜ΣK\n↓↑]q\ndet [−2i˜ΣK\n↑↓]ZY\niD[˜hqq\ni] exp\"\niZ\ndtX\ni\u0010\n˜hqq(t)˜hqq(t)\u0011\ni\u0012\nd(t)\n¯d(t)\u0013\ni+h.c.\n−Z\ndtdt′X\nij\u0010\n˜hqq(t)˜hqq(t)\u0011\ni \n0 −i˜ΣK\n↓↑(t−t′)\n−i˜ΣK\n↑↓(t−t′) 0!−1\nij ˜hqq(t′)\n˜hqq(t′)!\nj#\n,(94)\nagain interpreting the exponent as an effective action in- cluding the field ˜hqq\ni, which has the correlators\n⟨˜hqq\ni(t)⟩= 0 (95a)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩=−i˜ΣK\n↑↓ij(t−t′) (95b)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩=−i˜ΣK\n↓↑ij(t−t′) (95c)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩= 0. (95d)13\nWe remark that the unconventional form of the Hubbard-\nStratonovich decoupling leads to non-zero correlators for\nequal fields, as opposed to the conventional approach\nwhere the non-zero correlators involve one field being\nthe complex conjugate of the other. The fields h¯qqand\n˜hqqare interpreted as fluctuating transverse fields with,\nin general, different amplitudes depending on the lattice\nsite, but with correlators between lattice sites. Compar-\ning the effective action in Eq. (92) and Eq. (94) with the\nmagnon action in Eq. (41), the components of the total\nfluctuating field Hfcan be identified as\nγµ0Hf\n+,i=−1√sih\n2˜hqq\ni+h¯qq\nii\n(96a)\nγµ0Hf\n−,i=−1√sih\n2˜hqq\ni+¯h¯qq\nii\n. (96b)In this expression, the factor of 2 arises from the uncon-\nventional nature of the Hubbard-Stratonovich transfor-\nmation in Eq. (94). The correlators between the Carte-\nsian components of the fluctuating field can be calculated\nusing Eq. (95) and Eq. (96),\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nxj⟩= ImΣK\nij+ 4Im ˜Σ↑↓ij (97a)\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nyj⟩=−ReΣK\nij−4Re˜Σ↑↓ij (97b)\n2√sisjγ2µ2\n0⟨Hf\nyiHf\nyj⟩= ImΣK\nij−4Im˜Σ↑↓ij, (97c)\nfrom which we conclude that the correlators in the fluc-\ntuating field Hfare real numbers. In Eq. (97), we omit-\nted the time arguments for notational simplicity. Fur-\nthermore, it is evident that for i=jandt=t′, the\ncorrelators in Eq. (97a) and (97c) are positive, aligning\nwith the conditions expected for representing the vari-\nance of a real field.\nD. Equations of motion\nAfter HS decoupling the qqcomponents, the effective action reads\nSeff=−γℏµ0Z\ndt\"X\na√sa\u0000\nHstt\na++Hf\na+\u0001\n¯aq\na(t) +X\nb√sb\u0000\nHstt\nb−+Hf\nb−\u0001¯bq\nb(t) +h.c.#\n+ℏZ\ndt\"X\naa′βaa′¯aq\na∂tacl\na+X\nabβabaq\na∂tbcl\nb+X\nbaβba¯bq\nb∂t¯acl\na+X\nbb′βbb′¯bq\nb∂tbcl\nb′+h.c.#\n. (98)\nHaving cast the total action Sm+Seffin a form that is linear in the quantum fields ¯ aqand¯bqand their complex\nconjugates, we can integrate over these fields in the partition function, producing the functional delta function imposing\nthe semiclassical equations of motion for the fields aclandbcl[30]. Using acl\na=S+a/(ℏ√sa) and ¯bcl\nb=S+b/(ℏ√sb) in\nthe semiclassical limit, we find the coupled equations of motion:\ni∂tSi+=ℏ−1EiSi++siℏµ0γ\u0000\nHi++Hf\ni++Hstt\ni+\u0001\n−X\njβij∂tSj+, (99)\nas well as its complex conjugated counterpart. Both in the definition of this field and in Eq. (99), the upper sign holds\nfor sublattice A, and the lower sign holds for sublattice B. We find the Cartesian components by taking the real and\nimaginary parts and divide with ℏsito find an equation for the vector mi=Si/(ℏsi),\n∂tmi=τb\ni+τf\ni+τsp\ni+τstt\ni, (100)\nwhere\nτb\ni=−z×\u0000\nℏ−1Eimi+γµ0Hi\u0001\n(101a)\nτf\ni=−γµ0z×Hf\ni (101b)\nτstt\ni=−γµ0z×Hstt\ni (101c)\nτsp\ni=X\njReβijz×∂tmj+X\njImβijz×(z×∂tmj) (101d)\nis microscopic expressions for the bulk torque τb, the fluctuating torque τf, the spin pumping torque τsp, and the\nspin transfer torque τstt.\nVI. CONCLUSION\nIn this paper, we have presented a general quantum\ntheory of spin dynamics in magnet-normal metal systems,generalizing earlier results to a general antiferromagnetic14\nor ferrimagnetic bipartite lattice. Spin and charge ac-\ncumulations in the normal metals influence the magne-\ntization dynamics in the magnet through spin transfer\ntorque, and the damping is enhanced due to spin pump-\ning, including both inter- and intra-lattice contributions.\nWe derived expressions for transverse fluctuating fields\narising due to the electron magnon interactions. These\nfields have contributions from equilibrium terms as well\nas charge and spin accumulation in the normal metals.\nWe found site-dependent shot noise contributions that\nare non-negligible at low temperatures.\nACKNOWLEDGMENTS\nThis work was supported by the Research Council\nof Norway through its Centers of Excellence funding\nscheme, Project No. 262633, ”QuSpin”.\nAppendix A: Holstein-Primakoff transformation\nIn this Appendix, we discuss the transformations used\nto diagonalize the magnon Hamiltonian of Eq. (9). To\ngo from the SU(2) spin operators to bosonic annihila-\ntion and creation operators, we employ the Holstein-\nPrimakoff transformation [37, 38] at sublattices AandB\nand expand to the lowest order in the bosonic operators,\nassuming the antiferromagnet is close to the N´ eel state,\ni.e. that all spins on sublattice A(B) is close to being\nparallel (antiparallel) to the z-direction. At sublattice A,\nwe expand\nˆSa+=ℏ√\n2sa\u0012\n1−ˆa†\naˆaa\n2sa\u00131/2\nˆaa≈ℏ√\n2saˆaa (A1)\nˆSa−=ℏ√\n2saˆa†\na\u0012\n1−ˆa†\naˆaa\n2sa\u00131/2\n≈ℏ√\n2saˆa†(A2)\nˆSaz=ℏ(sa−ˆa†\naˆaa), (A3)\nwhere aaannihilates a localized magnon and sais the\ntotal spin at lattice site a. In the expansion of the square\nroots in Eq. (A1) and Eq. (A2), we assumed sa≫1 and\nexpanded the square root to lowest order in 1 /sA. We\nhave employed the standard raising and lowering spin\noperators, defined as S±=Sx±iSy.\nSimilarly, at sublattice B, we expand\nˆSb+=ℏ√\n2sbˆb†\nb \n1−ˆb†\nbˆbb\n2sb!\n≈√\n2sbˆb†\nb, (A4)\nˆSb−=ℏ√\n2sb \n1−ˆb†\nbˆbb\n2sb!\nˆbb≈√\n2sbˆbb, (A5)\nˆSbz=ℏ\u0010\n−sb+ˆb†\nbˆbb\u0011\n, (A6)\nwhere ˆbannihilates a localized spin-up magnon.Appendix B: Relating spin transfer torque and spin\npumping coefficients\nWe relate the spin transfer pumping coefficients de-\nfined in Eq. (88) to the spin transfer coefficients found\nin Eq. (64) in the case of one normal metal reservoir us-\ning the Onsager reciprocal relations [36]. We start by\ndefining the pumped spin current (in units of electrical\ncurrent, i.e. Ampere) into normal metal as the change in\ntotal spin inside the antiferromagnetic due to spin pump-\ning, i.e.\nIS=−e\nℏX\njSjτsp\nj. (B1)\nThe appearance of Sj=ℏp\nsj(sj+ 1) is due to the way\nwe have defined the torques in the main text, causing\nthem to have the dimension of inverse time. The dynam-\nics of the localized magnetic moment µj=−γSjmjand\nthe spin current are driven by the external effective field\nHeffand the spin accumulation µS, which are the ther-\nmodynamic forces in our system. In linear response, we\ncan then write the equations for the spin dynamics and\nthe spin current in matrix form:\n\u0012−γSi∂tmi\nIS\u0013\n=\u0012Lmm\nijLms\ni\nLsm\njLss\u0013\u0012µ0Heff\nj\nµS/e\u0013\n, (B2)\nwhere the matrix elements 3 ×3 tensors that effectively\napply the relevant cross products to make Eq. (B2) con-\nsistent with the Landau-Lifshitz equation, and where\nwe use the Einstein summation convention for repeated\nLatin indices.\n1. Identifying Lsm\nInserting the spin pumping torque from Eq. (29), the\nspin current becomes\nIS=−Xj∂tmj, (B3)\nwhere we defined the 3 ×3 matrix Xjas\nXj=e\nℏSjX\nih\nαR\nij˜O+αI\nij˜O2i\n, (B4)\nand the 3 ×3 matrix ˜Oimplements the cross product\nz×v=˜Ovand can be defined in terms of the Levi-\nCivita tensor. The LLG equation in the absence of spin\naccumulation (causing the spin transfer torque to vanish)\nreads\n(1−αb˜O)∂tmi=˜O(−γµ0Heff\ni), (B5)\nwhere αbis the (bulk) Gilbert damping constant. Hence,\nwe identify\nLsm\nj=γXj˜O(1−αb˜O)−1. (B6)15\n2. Identifying Lms\nInserting the spin transfer torque from Eq. (27) into\nthe LLGS equation in the absence of an effective field,\nwe find\n∂tmi=ℏ−1(1−αb˜O)−1h\nβI\ni˜O−βR\ni˜O2i\nµS,(B7)\nmeaning that we can identify the linear response coeffi-\ncient Lmsas (no Einstein summation)\nLms\ni=−Siγe\nℏ(1−αb˜O)−1h\nβI\ni˜O−βR\ni˜O2i\n. (B8)\n3. Deriving relations from the Onsager reciprocal\nrelations\nWe are now looking to employ Onsager’s reciprocal\nrelation:\n[Lsm\ni({−mj})]T=Lms\ni({mj}), (B9)\nwhere the superscript Tindicates a matrix transpose in\nthe 3 ×3 Cartesian space. Using the matrix identity\n˜O3=−˜O, we find that Eq. (B9) implies that\nβI\nj˜O−βR\nj˜O2=X\nih\nαI\nij˜O−αR\nij˜O2i\n(B10)\nThis equality is satisfied if\nβj=X\niαij, (B11)\nwhich generalizes the result from Ref. 35. Inserting the\ndefinitions of these coefficients in the low-temperature\nlimit, we find that\nX\nnWnn\nj↑↓= iπX\ninmWnm\ni↓↑Wmn\nj↑↓, (B12)\nwhich we classify as a generalized optical theorem, since\nin the diagonal case i=j, we can rewrite the imaginary\npart of this to\nIm\"X\nnWnn\ni↑↓#\n=πX\ninm|Wnm\ni↓↑|2, (B13)\nwhich is reminiscent of the optical theorem in wave scat-\ntering theory.\nAppendix C: Contour fields and Keldysh rotations\nIn this Appendix, we show how the action can be writ-\nten in the ±basis, and introduce the Keldysh rotated\nfields, which differ in the case of fermionic and bosonic\nfields. In the ±field basis, the action of the scattering(electron) states, corresponding to the Hamiltonian in\nEq. (7), reads\nSe+S0=X\nsZ∞\n−∞dt¯c+\ns(iℏ∂t−ϵ)c+\ns\n−X\nsZ∞\n−∞dt¯c−\ns(iℏ∂t−ϵ)cs−\n=X\nsξt¯cξ\ns(iℏ∂t−ϵ)cξ\ns, (C1)\nwhere now csis a vector containing the scattering fields,\n¯csdenotes its complex conjugate, and ϵis a diagonal\nmatrix containing all energy eigenvalues of the scatter-\ning states. In the final line, we have written the time\nintegration as a sum for concise notation. Additionally,\nwe introduced the sum over ” ±” fields as a sum over\nξ={+,−}, with an implicit negative sign before the ”-”\nfield, i.e.P\nξ. . .ξ=. . .+−. . .−. A similar notation will\nalso be used for the magnon fields below. The negative\nsign ( ξ=−) in the integral in Eq. (C1) and in the other\nactions below originates from reversing the integration\nlimits on the backward contour. The magnon action is\nSm=X\nξabt[¯aξ\na(iℏ∂t−EA\nab)aξ\na+¯bξ\nb(iℏ∂t−EB\nab)bξ\nb]\n−2X\naa′Jaa′√sasa′¯aξ\naaξ\na′\n−2X\nbb′Jbb′√sbsb′¯bξ\nbbξ\nb′\n−2X\nξabtJab√sasb[aξ\nabξ\nb+ ¯aξ\na¯bξ\nb]\n−γµ0ℏX\nξatrsa\n2[HA\na−aξ\na+HA\na+¯aξ\na]\n−γµ0ℏX\nξbtrsb\n2[HB\nb−¯bξ\nb+HB\nb+bξ\nb]. (C2)\nThe first-order electron-magnon interaction is\nS1=−X\nξat\nαβr\n2\nsah\naξ\na¯cξ\n↓αWαβ\na↓↑cξ\n↑β+ ¯aξ\na¯cξ\n↑αWαβ\na↑↓cξ\n↓βi\n−X\nξbt\nαβr\n2\nsbh\n¯bξ\nb¯cξ\n↓αWαβ\nb↓↑cξ\n↑β+bξ\nb¯cξ\n↑αWαβ\nb↑↓cξ\n↓βi\n,\n(C3)\nand the second-order term is\nS2=X\nξat\nαβ1\nsa¯aξ\naaξ\nah\n¯cξ\n↑αWαβ\na↑↑cξ\n↑β−¯cξ\n↓αWαβ\na↓↓cξ\n↓βi\n−X\nξbt\nαβ1\nsb¯bξ\nbbξ\nbh\n¯cξ\n↑αWαβ\nb↑↑cξ\n↑β−¯cξ\n↓αWαβ\nb↓↓cξ\n↓βi\n.(C4)16\nFor a general bosonic field ϕ, the classical (cl) and\nquantum (q) fields are defined as [30]:\nϕcl/q=1√\n2(ϕ+±ϕ−)¯ϕcl/q=1√\n2(¯ϕ+±¯ϕ−).(C5)\nIn our case, we have ϕ={a, b}. The upper (lower) sign\nholds for the classical (quantum) fields. For a fermionic\nfield c, the rotated fields are denoted by 1 and 2, and\ndefined as\nc1/2=1√\n2(c+±c−) ¯c1/2=1√\n2(¯c+∓¯c−).(C6)\nFor fermions, ¯ candcare independent variables, not re-\nlated by complex conjugation.\nAppendix D: Fourier transform\nFor a general function of relative time t−t′, we define\nthe Fourier transform between the relative time domain\nand the energy domain as\nf(ω) =Z∞\n−∞d(t−t′)eiω(t−t′)f(t−t′), (D1)\nf(t−t′) =Z∞\n−∞dω\n2πe−iω(t−t′)f(ω). (D2)The delta function can be represented as\nδ(t−t′) =Z∞\n−∞dω\n2πe−iω(t−t′), (D3)\nδ(ω) =1\n2πZ∞\n−∞d(t−t′)eiω(t−t′). (D4)\nFinally, we note a frequently employed identity,\n−iZ∞\n−∞d(t−t′)eiω(t−t′)θ(t−t′) = (ω+ iδ)−1,(D5)\niZ∞\n−∞d(t−t′)eiω(t−t′)θ(t′−t) = (ω−iδ)−1,(D6)\nwhere δis an infinitesimal positive quantity.\n[1] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu,\nB. Di´ eny, P. Pirro, and B. Hillebrands, Journal of Mag-\nnetism and Magnetic Materials 509, 166711 (2020).\n[2] A. Brataas, B. van Wees, O. 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White, Physical Review B 14,\n2939 (1976)." }, { "title": "1502.00742v1.Generation_of_Spin_Currents_in_the_Skyrmion_Phase_of_a_Helimagnetic_Insulator___mathrm_Cu_2OSeO_3__.pdf", "content": "arXiv:1502.00742v1 [cond-mat.mes-hall] 3 Feb 2015Generation of Spin Currents in the Skyrmion Phase of a\nHelimagnetic Insulator Cu2OSeO3\nDaichi Hirobe∗and Yuki Shiomi\nInstitute for Materials Research, Tohoku University, Send ai 980-8577, Japan\nYuhki Shimada and Jun-ichiro Ohe\nDepartment of Physics, Toho University, 2-2-1,\nMiyama, Funabashi, Chiba 274-8510, Japan\nEiji Saitoh\nInstitute for Materials Research, Tohoku University, Send ai 980-8577, Japan\nWPI Advanced Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan\nCREST, Japan Science and Technology Agency,\nChiyoda, Tokyo 102-0075, Japan and\nThe Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\n(Dated: June 22, 2021)\nAbstract\nWe report spin-current generation related with skyrmion dy namics resonantly excited by a mi-\ncrowave in a helimagnetic insulator Cu 2OSeO3. A Pt layer was fabricated on Cu 2OSeO3and\nvoltage in the Pt layer was measured upon magnetic resonance of Cu2OSeO3to electrically detect\ninjected spin currents via the inverse spin Hall effect (ISHE) in Pt. We found that ISHE-induced\nelectromotive forces appear in the skyrmion phase of Cu 2OSeO3as well as in the ferrimagnetic\nphase, which shows that magnetic skyrmions can contribute t o the spin pumping effect.\n∗Electronic address: daichi.kinken@imr.tohoku.ac.jp\n1I. INTRODUCTION\nA magnetic skyrmion, a nano-scale vortex-like spin texture in the re al space, has at-\ntracted much attention in condensed matter physics.[1, 2] Figure 1 (a) illustrates a magnetic\nskyrmion. Magnetic moments point downward at the center of the s kyrmion and upward\nalong the periphery; intermediate magnetic moments vary their dire ctions continuously such\nthat the moments wrap the whole solid angle at the center. The skyr mion’s diameter is typ-\nically 10 to 100 nm and the magnetic skyrmion has been usually observe d in a crystalline\nform(theskyrmioncrystal).[3–6]Sincetheskyrmion’s spinconfigur ationpossesses thescalar\nspin chirality,[1] topological phenomena appear in the Hall effect,[7–12 ] which is formu-\nlated by introducing an emergent electromagnetic field.[1, 7–12] Bes ides its rich physics, the\nskyrmion has potential for applications especially in spintronics. Uniq ue properties of the\nskyrmion, i.e., particle-like nanostructure and topological stability, are highly promising for\nnovel spintronic devices such as information storage or logic device s.[1, 2] The skyrmion is\nalso appealing as an efficient information carrier; in fact, recent exp eriments showed that\nmagnetic skyrmions can be moved by electric current at low energy c osts.[12–14]\nIn spite of the skyrmion’s potential for spintronic applications, few studies have inves-\ntigated skyrmions in terms of the pure spin current physics. A pure spin current refers to\na flow of spin angular momentum with no charge currents; its unders tanding and control\nare central issues of spintronics.[15] If the pure spin current can be generated from mag-\nnetic skyrmions, the generation should be interesting for basic res earch and would be a step\nforward in skyrmionic devices as well. Here, we experimentally demons trate spin-current\ngeneration froma resonantly driven skyrmion motion ina helimagnetic insulator Cu 2OSeO3.\nUsing a Cu 2OSeO3/Pt two-layer structure, we show that spin angular momentum tra nsfers\nfrom the skyrmion crystal in Cu 2OSeO3to conduction electrons in Pt via the spin pumping\neffect.[16–18] The injected spin current is electrically detected in Pt via the inverse spin Hall\neffect (ISHE),[19–24] which converts a spin current into a transve rse electromotive force\nESHEvia large spin-orbit interaction of Pt [Fig. 1(b)]. ESHEis given by[20]\nESHE∝Js×σ, (1)\nwhen a spin current carries the spin polarization σin the spatial direction Js.\n2We used Cu 2OSeO3for this work because the high-insulating property of Cu 2OSeO3\nenables electric detection of spin currents free from galvanomagn etic contamination.[25]\nThe skyrmion crystal in Cu 2OSeO3has been observed by Lorentz transmission elec-\ntron microscopy[6] and neutron diffraction measurements.[26, 27] The helimagnetic order-\ning temperature is approximately 60 K and a helical spin structure is s tabilized by the\nDzyaloshinskii-Moriya interaction at the zero magnetic field. The sky rmion-crystal phase\nappears in the vicinity of the ordering temperature in a restricted w indow of magnetic fields\nfor bulk Cu 2OSeO3crystals.[6] Skyrmions form a two-dimensional triangular lattice with in\nthe plane perpendicular to an applied magnetic field.[6, 26, 27] The mag netic phase diagram\nof Cu2OSeO3is almost the same for any direction of an applied magnetic field.[26]\nII. METHOD\nWe grew single crystals of Cu 2OSeO3by a chemical vapor transport method.[30] The\nvolume of the crystals was typically 1 mm3. The magnetic moment Mfor the prepared\ncrystals was investigated with a vibrating sample magnetometer (VS M) in a Physical Prop-\nerties Measurement System (Quantum Design, Inc.). To make a Cu 2OSeO3/Pt sample, we\npolished the surface of Cu 2OSeO3and sputtered a 5-nm-thick Pt film on the polished sur-\nface in an argon atmosphere. Figure 1(c) illustrates an experiment al set-up for detecting the\nspin pumping effect. To conduct microwave experiments, we used a r eflection-type copla-\nnar waveguide and a network analyzer. Since the skyrmion phase is r estricted in a narrow\nwindow of magnetic fields, we fixed a magnetic field and swept a microwa ve frequency to\nmeasure magnetic resonance. A static in-plane magnetic field was ap plied perpendicularly\nto a microwave magnetic field. When a microwave frequency fulfills res onance conditions of\nCu2OSeO3, magnetic moments are forced to precess (magnetic resonance) . If this precession\ntransfersspinangularmomentum toconduction electronsinPt, th espincurrent isconverted\nintoESHEin Pt via the ISHE. To measure ESHE, we attached two electrodes to the ends of\nthe Pt film [Fig. 1(c)]. Applied microwaves were set at 10 mW. The used C u2OSeO3sample\nwas of column shape and 1.5 mm across and 0.5 mm high. The spin-pumpin g measurement\nwas performed in a cryogenic probe station, where the sample was c ooled by attaching the\nbottom of a sample holder to a cold head. A system temperature was measured with a\nthermometer near the cold head; the sample temperature appear ed higher than the system\n3temperature by about 5 K in the present study.\nIII. RESULTS AND DISCUSSION\nThe prepared Cu 2OSeO3single crystals were characterized by measuring M. Figure 2\nshows temperature dependence of Mat 300 mT. With decreasing temperature, Msharply\nincreases at approximately 60 K and saturates at 0 .52µB/Cu2+at 2 K. The ordering tem-\nperature and the saturation moment are consistent with those of preceding studies.[31] The\ninset to Fig. 2 shows magnetic-field dependence of Mat 5 K.Msaturates above 100 mT,\nwhich suggests transition into a ferrimagnetic phase. In the magne tization curve below 100\nmT, a step-like change appears near 30 mT. This change correspon ds to a phase transition\nbetween multi-domain and single-domain helimagnetic states.[6, 31] He nce, the magnetic\nproperties of the prepared crystals are well consistent with the r eported ones.\nFigure 3 shows microwave responses of a Cu 2OSeO3/Pt sample at 45 K and 50 K. Here-\nafter, temperatures refer to a system temperature Tof the probe station. Figure 3(a) shows\nfrequency dependence of microwave absorption ∆S11at 45 K, which is well lower than\nthe helimagnetic ordering temperature. At 114 mT, ∆S11shows a dip around 3.5 GHz.\nSince the resonance frequency decreases with decreasing a magn etic field, the microwave\nabsorption is attributed to a ferrimagnetic resonance of Cu 2OSeO3. Below 54 mT, two dips\nappear and their resonance frequencies increase with decreasing a magnetic field. The field\ndependence of the resonance frequencies is attributed to a helima gnetic phase induced by\nthe Dzyaloshinskii-Moriya interaction.[32, 33] Note that the two dips in the helimagnetic\nphase correspond to two spin-wave modes.[32–34] Figure 3(c) sum marizes field dependence\nof resonance frequencies at 45 K.\nWe observed magnetic resonances of the skyrmion crystal betwe en 47 K and 51 K. Figure\n3(b) shows frequency dependence of ∆S11at 50 K. The resonance frequency decreases with\ndecreasing a magnetic field from 84 mT, and then increases below 39 m T. This behavior\nis similar to that of 45 K. From 30 mT through 6 mT, however, an additio nal excitation\nappears around 1.4 GHz while the excitation around 2 GHz is suppress ed. Below 3 mT,\nthe additional dip disappears and the helimagnetic one revives aroun d 2 GHz. Figure 3(d)\nsummarizes field dependence of resonance frequencies at 50 K. Wh ile helimagnetic and fer-\nrimagnetic modes appear both at 45 K and at 50 K, the new resonanc e mode appears at 50\n4K only, as highlighted by the shading of Fig. 3(d). Its resonance fre quency is lower than\nthose of the other modes and monotonically increases with increasin g a magnetic field. This\nmagnetic-field dependence obviously differs from that of helimagnet ic modes. Moreover, the\nnew magnetic excitation occurs in a restricted window of magnetic fie lds. These results\nstrongly indicate that magnetic excitations near 1.5 GHz originate fr om a collective motion\nof the skyrmion crystal. According to preceding studies on magnet ic resonance of skyrmion\ncrystals,[33, 35, 36] the newly observed mode is attributed to a cou nterclockwise rotation\nof magnetic skyrmions, in which skyrmion’s core rotates counterclo ckwise. Note that the\nexcitation observed around 2 GHz in the skyrmion-crystal phase is due to either a clock-\nwise rotation of skyrmions[38] or magnetic resonance of remnant h elimagnetic states[3, 33];\nthe origin of this excitation was unidentified in this experiment. Accor dingly, we confine\nourselves to the counterclockwise skyrmion motion to investigate t he spin pumping effect.\nFigure 4(a) shows a skyrmion phase diagram determined by magnetic resonance in the\nTrange slightly below the ordering temperature. Here, critical magn etic fields Hc1,Hc2,\nandHc3are defined as shown in Fig. 3(d); from Hc1throughHc2the skyrmion excitation\ncontinues while across Hc3a phase transition occurs from the helimagnetic state to the\nferrimagnetic state. Tdependence of the critical magnetic fields reveals that magnetic\nexcitations of the skyrmion crystal are restricted in a narrow wind ow ofT(approximately 4\nK). This observation is consistent with a phase diagram determined b y measurements of ac\nmagnetic susceptibility,[6, 33] althougherror bars for Hc1andHc2arerather largebecause of\nsmall skyrmion excitations around Hc1andHc2. Note again that the skyrmion-crystal phase\nwas observed at Tlower than those reported elsewhere[6, 26, 27] because in the pre sent set-\nup we could not attach the thermometer directly to the sample as me ntioned in the method\npart.\nWedemonstrate spin-current generationby measuring ISHE-indu ced electromotive forces\nin Pt at 50 K. We first investigated the ferrimagnetic spin pumping at 6 6 mT [point A\nin Fig. 4(a)]. Figures 4(b) and 4(c) represent a microwave-absorp tion spectrum and an\nelectromotive-forcespectrumatpointA,respectively. Inthemic rowave-absorptionspectrum\n[Fig. 4(b)], a single dip of ferrimagnetic resonance appears around 2 GHz. Correlating with\nthe resonance, a single dip of electromotive forces appear around the resonance frequency,\nas shown in Fig. 4(c). Moreover, the electromotive forces show op posite polarity with an\nopposite magnetic field. This reversal is consistent with the ISHE [Eq . (1)] and provides\n5evidence for spin-current generation from the ferrimagnetic res onance.\nNext, we conducted the spin-pumping measurement in the skyrmion -crystal phase [point\nB in Fig. 4(a)]. A magnetic field was set at 15 mT to maximize microwave-a bsorption\nintensity within the skyrmion-crystal phase [see also Fig. 3(b)]. Figu res 4(d) and 4(e)\nare a microwave-absorption spectrum and an electromotive-forc e spectrum at point B, re-\nspectively. In the microwave-absorption spectrum [Fig. 4(d)], two absorption dips appear;\nthe dip at the lower frequency ( ≈1.3 GHz) corresponds to the counterclockwise rotation of\nskyrmions [see also Figs. 3(b) and 3(d)]. In the electromotive-forc e spectrum [Fig. 4(e)],\nelectromotive forces appear around 1.3 GHz in correlation with the s kyrmion excitation.\nAgain, the electromotive forces show opposite polarity with an oppo site magnetic field.\nMoreover, the electromotive force is of the same sign for skyrmion and ferrimagnetic phases\nfor a given field direction. The results are consistent with Eq. (1) be cause the direction of\nspinpolarizationshouldbedetermined byastaticmagneticfieldalsoint heskyrmion-crystal\nphase. These observations show that the resonantly excited sky rmion motion can give rise\nto the spin pumping effect and that the skyrmion dynamics can be det ected electrically via\nthe ISHE. Dividing the voltage by the intensity of microwave absorpt ion[39] yields 16 µV/W\nfor the ferrimagnetic phase and 8 µV/W for the skyrmion-crystal phase, which implies the\nlower efficiency of the spin pumping in the skyrmion-crystal phase.\nWe numerically calculated spin currents related with skyrmion dynamic s to demonstrate\ntheoretically that the counterclockwise skyrmion motion can contr ibute to the spin pumping\neffect. The dynamics of local magnetizations M(r,t) were simulated with the Landau-\nLifshitz-Gilbert (LLG) equation[36, 40] for a single skyrmion on a two -dimensional square\nlattice in the x-yplane. The used Hamiltonian is\nH=−J/summationdisplay\ni,jMi·Mj+D/summationdisplay\ni(ˆ x·(Mi×Mi+ˆ x)+ˆ y·(Mi×Mi+ˆ y))−/summationdisplay\niMi·H,(2)\nwhereJ= 3.9 meV and D= 0.5Jare coupling constants of ferromagnetic interaction and\nDzyaloshinskii-Moriya interaction, respectively, and H=Hˆ zis a static magnetic field in\nthezdirection, i.e., perpendicular to the square lattice. The system size w as 41×41 sites.\nThe parameters used for the LLG equation were: the gyromagnet ic ratioγ= 1.76×1011\nHz/T; the Gilbert damping constant α= 0.01; a static magnetic field µ0H= 800 mT in\nthezdirection; a microwave magnetic field µ0h= 8 mT in the ydirection; the microwave\n6frequency f= 0.78 GHz. With these parameters, the skyrmion’s core resonantly ro tates\ncounterclockwise as shown in Figs. 5(a)-(d). Note that values of t he static magnetic field\nand the resonance frequency are not equal to those of our meas urement because the simplest\nmodel of a single skyrmion was adopted to capture the essence of t he spin pumping from\nthe skyrmion dynamics.\nGenerated spin currents were calculated using the time average of Dz(t) =\n(M×∂M/∂t)z[29] at the bottom edge of the system r= (x,0), namely ∝angbracketleftJsz∝angbracketright ∝\n/integraltext\ndx∝angbracketleftDz(t)∝angbracketrighttime, where∝angbracketleftDz(t)∝angbracketrighttimerepresents the time average of Dz(t). As shown in Figs.\n5(e)-(h), both the magnitude and the sign of Dz(t) depend heavily on space and time. The\nspatial dependence reflects a swirling spin structure due to the Dz yaloshinskii-Moriya inter-\naction; such behavior has not been observed for centrosymmetr ic systems such as Y 3Fe5O12,\nwhere the Dzyaloshinskii-Moriya interaction is absent. The spatial a nd temporal depen-\ndence of Dz(t) at the edge gives rise to the complicated time evolution of generate d spin\ncurrents Jz\ns(t) as indicated by a blue dashed line in Fig. 5(i). The temporal dependen ce of\nJz\ns(t) cannot be described by a single sine wave, which implies that nonlinear magnetiza-\ntion dynamics were also driven for µ0h= 8 mT. Nevertheless, ∝angbracketleftJsz∝angbracketrightis found to be finite as\nhighlighted by the shading in Fig. 5(i), which theoretically substantiat es our experimental\nresults. As illustrated in Figs. 5(a)-(d), magnetic moments even at the edge form a non-\ncollinear configuration owing to the Dzyaloshinskii-Moriya interaction . Although a similar\nnoncollinear configuration at the edge was found also for the ferrim agnetic phase in the\ncalculation with µ0H= 2.4 T, the temporal dependence of Jz\ns(t) forµ0h= 8 mT was simply\nsinusoidal. We note that ∝angbracketleftJsz∝angbracketrightfor the skyrmion phase was approximately 30 % smaller than\nfor the ferrimagnetic phase in the present calculation. This result in dicates the lower effi-\nciency of the spin pumping in the skyrmion phase, which is consistent w ith our experimental\nresult. The numerical calculation presented here demonstrates t hat non-uniform spin con-\nfigurations such as skyrmions can give rise to the spin pumping effect while the observation\nof the spin pumping has been confined to uniform spin configurations such as ferromagnetic\nand ferrimagnetic states so far.\n7IV. SUMMARY\nTo summarize, we have observed spin-current generation from sk yrmion dynamics in a\nCu2OSeO3/Pt two-layer structure. We resonantly excited magnetic skyrmio ns in Cu 2OSeO3\nby a microwave and electrically detected spin currents via the ISHE in Pt. A numerical cal-\nculationshowed that a rotationalmotionofskyrmions cangive rise t o afinite dcspin current\nvia the spin pumping effect. Spin-current generation via skyrmion dy namics demonstrated\nhere can be a new functionality operating at microwave frequencies in spintronic applica-\ntions.\nWe thank S. Seki for fruitful discussions and R. Iguchi for techn ical assistance. 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Arrows represent magnetic moments.\n(b) A schematic illustration of the inverse spin Hall effect in duced by the skyrmion spin pumping\nin a Cu 2OSeO3/Pt two-layer structure. A microwave magnetic field his applied perpendicularly\nto a static magnetic field H.σandJsdenote the spin polarization and the spatial direction of a\nspin current, respectively. ESHEdenotes an electromotive force induced by the inverse spin H all\neffect. (c) Experimental set-up for the spin-pumping measure ment.\n12FIG. 2: Temperature dependence of the magnetization of Cu 2OSeO3. An applied magnetic field\nwas set at 300 mT. The inset shows magnetic-field dependence o f the magnetization of Cu 2OSeO3\nat 5 K. The shading at ±30 mT highlights transitions between multi-domain and sing le-domain\nhelimagnetic phases.\nFIG. 3: (a),(b) Frequency dependence of microwave absorpti on (a) at 45 K and (b) at 50 K at\nvarious magnetic fields. Black and blue curves represent dat a in ferrimangetic and helimagnetic\nphases, respectively. Red triangles in (b) indicate the mag netic resonance of the skyrmion crystal.\nMicrowave-absorption spectra are shifted vertically for c larity. (c),(d) Magnetic-field dependence\nof resonance frequencies (c) at 45 K and (d) at 50 K. The skyrmi on resonance persists from the\ncritical field Hc1through the critical field Hc2. Across the critical field Hc3, a transition occurs\nbetween helimagnetic and ferrimagnetic phases.\n1380 \n60 \n40 \n20 \n0Critical Field (mT) \n52 50 48 46 44 \nTemperature (K) Hc3 \nHc2 \nHc1 A\nB -10010 Voltage (nV) \n3 2 1\nFrequency (GHz) -0.2-0.10B (skyrmion-crystal phase)\n50 K\n15 mT\n-15 mT \n+15 mT(a) (b) (d)\n(c) (e)\nS11 (dB) \n3 2 1\nFrequency (GHz) -0.4\n-0.80\n50 K\n66 mT \n050 100 \n-50 \n-100 Voltage (nV) S11 (dB) A (Ferrimagnetic phase)\n-66 mT \n+66 mT\nFIG.4: (a) Temperaturedependenceofcritical magneticfiel dsHc1,Hc2, andHc3. Fordefinitionsof\nHc1,Hc2, andHc3, seethemaintextandFig. 3(d). Thelight pinkbackgroundsh owstheskyrmion-\ncrystal phase. Spin-pumping measurements were conducted a t points A (ferrimagnetic phase) and\nB (skyrmion-crystal phase). (b)-(e) Frequency dependence of (b),(d) microwave absorption and\n(c),(e) electromotive forces at 50 K. (b) and (c) are for the f errimagnetic phase at 66 mT; (d) and\n(e) are for the skyrmion-crystal phase at 15 mT. Symbols are e xperimental data and solid curves\nare (multi-)Lorentzian fits to the data. A red dashed curve in (d) represents skyrmion-crystal\nresonance.\n14 0 10 20y (nm)A B C D t=176.18 ns\n-1 0 1 t=176.36 ns t=176.62 ns t=177.04 ns \n010 20 \n0 10 20 y (nm )\nx (nm )-10 0 10 \n0 10 20 \nx (nm )0 10 20 \nx (nm )0 10 20 \nx (nm )\n-1 -0.500.51\n175 176 177 178 179 -0.01 00.01 Spin current (arb. unit) \nTime (ns)A B C D\nac field(a) (b) (c) (d) \n(e) ( f ) (g) (h) \n( i )\nh / H \nDz (arb. unit) mz\nJsz<>\nJsz\nFIG. 5: (a)-(d) Representative snapshots of the coutercloc kwise skymion rotation. Black arrows\nrepresent xandycomponents of local magnetizataions while the contour plot represents the zcom-\nponent. Magnified views are shown in the panels below (a)-(d) . (e)-(h) Representative snapshots\nof the spatial distribution of the damping torque, Dz(t) = (M×∂M/∂t)zfor the skyrmion phase.\nSpin currents generated at the edge, y= 0, were calculated. (i) Time evolution of generated spin\ncurrents Jszfor the skyrmion phase (blue dashed curve) and a microwave ma gnetic field divided\nby a static magnetic field h/H(black solid curve). The red line represents the magnitude o f a\ntime-averaged spin current ∝angbracketleftJsz∝angbracketright.\n15" }, { "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 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. The depth\nsensitivity depends on the photon energy, absorption cross-section, and on the stoichiometry of the\nfilm. As such, the FY provides magnetic information for the whole film thickness, which we denote\nas ”bulk” sensitive. The spectra are recorded as function of the x-ray energy and normalized by\nthe same magnetically shielded x-ray monitor.\nReferences\n[1] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak. Antiferromag-\nnetic spintronics. Rev. Mod. Phys. , 90:015005, Feb 2018.\n[2] Naoto Nagaosa, Jairo Sinova, Shigeki Onoda, A. H. MacDonald, and N. P. Ong. Anomalous\nhall effect. Rev. Mod. Phys. , 82:1539–1592, May 2010.\n[3] Jairo Sinova, Sergio O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth. Spin hall\neffects. Rev. Mod. Phys. , 87:1213–1260, Oct 2015.\n9[4] Florin Radu and Jaime S´ anchez-Barriga. 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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": "2008.06628v2.Magnon_laser_based_on_Brillouin_light_scattering.pdf", "content": "arXiv:2008.06628v2 [physics.optics] 18 Aug 2020Magnon laser based on Brillouin light scattering\nZeng-Xing Liu∗\nSchool of Electronic Engineering &Intelligentization, Dongguan University of Technology, D ongguan, Guangdong 523808, China\nHao Xiong†\nSchool of Physics, Huazhong University of Science and Techn ology, Wuhan 430074, People’s Republic of China\n(Dated: August 19, 2020)\nAn analogous laser action of magnons would be a subject of int erest and is crucial for the study of nonlinear\nmagnons spintronics. Here, we demonstrate the magnon laser behavior based on Brillouin light scattering in a\nferrimagnetic insulator sphere which supports optical whi spering gallery modes and magnon resonances. We\nshow that the excited magnon plays what has traditionally be en the role of the Stokes wave and is coherently\namplified during the Brillouin scattering process, making m agnon laser possible. Furthermore, the stimulating\nexcited magnon number increasing exponentially with the in put light power can be manipulated by adjusting the\nexternal magnetic field. In addition to providing insight in to magneto-optical interaction, the study of magnon\nlaser action will help to develop novel technologies for han dling spin-wave excitations and could a ffect scientific\nfields beyond magnonics. Potential applications range from preparing coherent magnon sources to operating on-\nchip functional magnetic devices.\nPACS numbers: 72.10.Di, 75.30.Ds,75.50.Gg\nMagnons, the quasiparticle of spin-wave excitation, are\nwell known for its favorable compatibility with a wide range\nof carriers such as microwaves [ 1–3], phonons [ 4–6], and su-\nperconducting qubits [ 7,8]. Recently, experimental works\nhave reported that the coupling between magnons and mi-\ncrowave photons can reach a strong and even ultrastrong\nregime [ 9–11]. In addition, the yttrium iron garnet (YIG)\nferrimagnetic material has a high magnetic quality factor a nd\nsupports magnons with long coherence times, which envision s\nthe promising candidates for quantum information processi ng\n[12]. In particular, the YIG sphere supports optical whisper-\ning gallery modes (WGMs) and magnon resonances [ 13–15].\nSimultaneously, the magnon-photon interaction in the YIG\nsphere gives rise to Brillouin light scattering [ 16], which is\na well-established tool for the characterization of magnon ic\nfeatures and magnetic dynamics.\nBrillouin light scattering, in a quantum mechanical lan-\nguage, is essentially an inelastic scattering of light exci ted\nby various quasiparticles [ 17], such as phonons, polarons, or\nmagnons within a medium. In the YIG sphere, akin to the me-\nchanical effect of light [ 18], the coherent conversion between\nthe magnons and photons can be achieved via the Brillouin\nscattering process even though the large frequency mismatc h\nbetween the optical and magnon modes. Previous work has\nshown that the Brillouin light scattering in the YIG sphere\nwill be greatly enhanced when the triple-resonance conditi on\nis satisfied [ 19], i.e., the input and output optical modes are\nresonance with the frequency di fference given by the magnon\nmode. Some coherent e ffects, ranging from optomagnoni-\ncally induced transparency [ 20] and magnon-induced high-\norder sideband generation [ 21,22] to magnon blockade e ffect\n[23] have been revealed.\nAn analogous laser action of magnons would be a subject of\ninteresting for a broad range of physics from the areas of spi n-\ntronics, magnonics, and photonics. The purpose of this workis to establish a theoretical framework for magnon laser ac-\ntion. Some important characteristics of magnon laser, for i n-\nstance, the gain factor, threshold power, laser control, an d en-\nvironment temperature have been analyzed. We found that the\nexcited magnon in the YIG sphere plays what has traditionall y\nbeen the role of the Stokes wave and is coherently amplified\nduring the Brillouin scattering process, making magnon las er\naction possible [ 18]. The effective magnon gain is propor-\ntional to the input light power, while the threshold power of\nthe magnon laser is inversely proportional to the square of\nthe magnon-photon coupling strength. Moreover, we shown\nthat one can achieve the magnon laser control by adjusting th e\nexternal magnetic field. In addition, we theoretically eval u-\nated the possibility of observing the magnon laser action at\nroom temperature under the current experimental condition s\n[14,15]. Beyond their fundamental scientific significance,\nour results offer attractive prospects for preparing coherent\nmagnon sources [ 24,25], designing magnon-laser amplifiers,\nand manipulating on-chip magnetic devices [ 26,27].\nThe physical model is schematically shown in Figure 1, in\nwhich a micrometer-scale YIG sphere supports optical WGMs\nand magnon resonances. A bias magnetic field B perpendic-\nular to the plane of the WGMs (x-y plane) is applied to sat-\nurate the magnetization. The frequency of the uniform Kitte l\n(magnon) mode in the YIG sphere can be flexibly tuned by ad-\njusting the external magnetic field [ 28], i.e.,Ωm=̺Hm, where\n̺=2π×28 GHz/T is the gyromagnetic ratio and H mis the\nmagnetic field strength. The input light is introduced throu gh\na polarization controller and then evanescently coupled to\nthe optical WGMs via a tapered nanofiber. The transverse-\nelectric (TE) mode and the transverse-magnetic (TM) mode\nin the optical WGMs with the same mode index have dis-\ntinct frequency differences due to the geometrical birefrin-\ngence, which is demonstrated to be valid regardless of the\ncirculation direction of the photon [ 29]. Assuming that the2\nInelastic Scattering \nPolarization Controller Tapered Nanofibre TM wmw\nTE w\nxyInput \nScattered \nMagnon \nYIG sphere B\n) (+sTM ) (pTE \nFIG. 1: Schematic diagram of the physical model. Similar to t he\noptical WGMs, the YIG ferrimagnetic sphere also supports op tical\nWGMs and magnon resonances. The inelastic scattering is a pr ocess\nof three-particle interaction, i.e., ωM→ωE+ωm, where, respec-\ntively, the input photon with σ+polarization in the TM mode and the\nscattered photon with πpolarization in the TE mode.\ninput light is adjusted to couple to the TM mode (TM input),\nas shown in Fig. 1, the light in the resonator is σ+polar-\nized [ 14,15]. The inelastic scattering process occurs in the\nYIG sphere, i.e., the annihilation of a σ+-polarized photon\nin the TM mode accompanied by the creation of a magnon\nand a down-converted red-sideband photon with πpolariza-\ntion in the TE mode. The interaction between the two light\nphotons and one magnon can be well described by the Hamil-\ntonian H int=/planckover2pi1g(aMa†\nEm†+a†\nMaEm), where aM(E)(a†\nM(E)) and\nm(m†) are the annihilation (creation) operators of the pho-\nton in the TM (TE) mode and the magnon mode, respectively.\ng=Vc\nnr/radicalBig\n2\nnspinVis the magnon-photon coupling strength [ 15],\nwith the Verdet constant V, the vacuum speed of light c, the\nrefractive index nr, the spin density nspin, and the YIG sphere\nvolume V. In this circumstance, the magnon mode plays what\nhas traditionally been the role of the Stokes wave and is co-\nherently amplified during the Brillouin scattering process .\nIn order to model the magnon laser action more rigor-\nously, we proceed from the Hamiltonian including the TE and\nTM modes and the magnon mode, as well as the interaction\nHamiltonian\nH=/planckover2pi1ΩMa†\nMaM+/planckover2pi1ΩEa†\nEaE+/planckover2pi1Ωmm†m\n+/planckover2pi1g(aMa†\nEm†+a†\nMaEm), (1)\nwhereΩM(E) andΩmare, respectively, the frequency of the\nTM (TE) mode and the magnon mode. We assume that the\nTM mode is driven by a input light with the frequency ωM,\nwhich allows us to replace the operator aMwith a classical\nfield√nMe−iωMt, where\nnM=ΥMPin\n/planckover2pi1ωM[∆2\nM+(γM\n2)2], (2)\nindicates the average number of photons in the TM mode with\nthe detuning∆M=Ω M−ωM, the input light power P inand\nthe decay rate γM=κM+Υ M(κMthe intrinsic decay rateandΥMthe external coupling). Next we would like to turn\nto a rotation framework subject to an unitary transformatio n\nU(t)=exp(iωMa†\nMaMt+iωEa†\nEaEt+iωmm†mt) withωM(E)and\nωmthe frequency of the input (output) field for the TM (TE)\nmode and the magnon mode. The Hamiltonian describes the\ncoupling between the TE and the magnon modes driven by the\nTM mode, therefore, can be obtained as\nH=/planckover2pi1∆Ea†\nEaE+/planckover2pi1∆mm†m\n+/planckover2pi1g√nM(a†\nEm†e−iδt+aEmeiδt), (3)\nwhere∆E(m)=Ω E(m)−ωE(m)denotes the detuning from the TE\nmode and magnon mode resonances, respectively. Under the\ncondition of triple resonance, i.e., ωM−ωE=ωm, the index\nfactor e±iδt≡1 forδ≡ωM−ωE−ωm=0. The evolution of the\nphotons and magnons can be well described by the following\ncoupling equations\n˙aE=(−i∆E−γE\n2)aE−ig√nMm†+√κEψ(in)\nE(t),\n˙m=(−i∆m−γm\n2)m−ig√nMa†\nE+√κmϕ(in)(t),(4)\nwhereγE(κE) andγm(κm) are, respectively, the total (intrinsic)\ndecay rates of the TE and magnon modes. ψ(in)\nE(t) andϕ(in)(t)\nrepresent the thermal noise terms of the photon and magnon\nmodes, and characterized by the following temperature-\ndependent correlation functions [ 30]/angbracketleftψ(in)\nE(t)ψ(in)†\nE(t′)/angbracketright=\n[nth(ωE)+1]δ(t−t′),/angbracketleftψ(in)†\nE(t)ψ(in)\nE(t′)/angbracketright=[nth(ωE)]δ(t−t′),\nand/angbracketleftϕ(in)(t)ϕ(in)†(t′)/angbracketright=[mth(ωm)+1]δ(t−t′),/angbracketleftϕ(in)†(t)ϕ(in)(t′)/angbracketright\n=[mth(ωm)]δ(t−t′), where nth(ωE)=[exp(/planckover2pi1ωE\nKBT)−1]−1, and\nmth(ωm)=[exp(/planckover2pi1ωm\nKBT)−1]−1with the Boltzmann constant K B\nand the ambient temperature T, are, respectively, the equi-\nlibrium means thermal photon and magnon numbers. For\nan experiment temperature of T ∼100 mK [ 11], the thermal\nmagnon numbers are mth≈0.0083, which is far less than the\nemitted magnon numbers, and thus the environment thermal\nnoises can be safely ignored.\nTransferring the variables to the rotating frame by setting\n˜aE=aEei∆Etand ˜m=mei∆mt, we shall have\n˙˜aE=−γE\n2˜aE−ig√nMei(∆E+∆m)t˜m†,\n˙˜m=−γm\n2˜m−ig√nMei(∆E+∆m)t˜a†\nE. (5)\nWe can adiabatically eliminate the optical mode degrees of\nfreedom since γm≪γE[31], and then by solving for the an-\ntiderivative, we can obtain that\n˜aE=e−/integraltextγE\n2dt/integraldisplay\n−ig√nM˜m†ei(∆E+∆m)te/integraltextγE\n2dtdt\n=−ig√nM\ni(∆E+∆ m)+γE\n2ei(∆E+∆m)t˜m†. (6)\nSubstituting this solution into Eq. ( 5) yields\n˙˜m=/braceleftBigg\n−γm\n2+g2nM\n−i(∆E+∆ m)+γE\n2/bracerightBigg\n˜m. (7)3\nFIG. 2: The threshold power of the magnon laser P th(mW) as a func-\ntion of the magnon-photon coupling strength g/(2π×39.2 Hz) and\nthe detuning from the magnon mode ∆m/Ωm. The red, yellow, green\nsolid lines correspond to the threshold power P th=22, 10, 4 mW,\nrespectively. The simulation parameters we use are [ 14,15]ΩM/2π\n≈ΩE/2π=300 THz,Ωm/2π=10 GHz,γM/2π=γE/2π=15 MHz,\nγm/2π=1 MHz,ΥM=γM/2, and∆M=∆ E=0.\nFrom Eq. ( 7) we observe that the Brillouin scattering process\nbetween the optical photon and magnon in YIG sphere con-\ntributes an effective magnon gain as\nG=Re/braceleftBiggg2nM\n−i(∆E+∆ m)+γE\n2/bracerightBigg\n=g2nMγE\n2\n(∆E+∆ m)2+(γE\n2)2.\n(8)\nWe define M[ γ−1\nm]=exp[(2G−γm)/γm] as the steady state\nnumber of magnons. We can see that the magnon gain fac-\ntor is proportional to the input light power enlightening us of\nthe possibility of implementing magnon laser by using light .\nBy settingγm/2=G, we obtain the threshold power of the\nmagnon laser\nPth=/planckover2pi1ωMγmγ2\nM[(∆E+∆ M)2+(γE\n2)2]\n4g2ΥMγE, (9)\nreferring to the incident power required when the magnon gai n\ninduced by Brillouin scattering overcomes the dissipation of\nthe magnon, which is the necessary condition for the genera-\ntion of magnon laser.\nFigure 2shows the threshold power of the magnon laser\nPthvaries with the magnon-photon coupling strength gand\nthe detuning∆m/Ωm. We can clearly see that the threshold\npower is the lowest when the magnon mode satisfies the reso-\nnance condition, i.e., ∆m=Ω m−ωm=0, but will exponen-\ntially increase when it deviates from the resonance positio n.\nFor a YIG sphere with a diameter of 200- µm[9], without as-\nsuming further optimization process of the YIG sphere, the\nmagnon-photon coupling constant is theoretically evaluat ed\nto be g=2π×39.2 Hz [ 15]. In this realistic assumption, the\nthreshold power of the magnon laser is about 22 mW (the red\nline shown). In addition, increasing the coupling strength to\ng=1.5×2π×39.2 Hz and g=2.4×2π×39.2 Hz, as the0 5 10 15 20 25 30\nInput power Pin (mW)015101520Magnon number M [ γm-1]g=2πx39.2Hz\ng/1.3=2πx39.2Hz\ng/3=2πx39.2Hz\ng/5=2πx39.2Hz\n0.99 10.0 10.1012M [γm-1]\n(a)\n0.99 10.0 10.1\nFrequency fm (GHz)0.20.3M [γm-1](b)x104 Pin = 15 mW\nPin = 1 mW\nFIG. 3: The stimulated emitted magnon number M[ γ−1\nm] as a func-\ntion of the input power in the context of di fferent magnon-photon\ncoupling strength gunder the resonance condition ∆m/Ωm=0. The\nblack dotted line represents the threshold conditions for t he gener-\nation of magnon laser, i.e., M[ γ−1\nm]=1. The threshold condition\ndenoted by the thick points is obtained for γm/2=G. The insets (a)\nand (b) plot the magnon line shape function in the case of P in=15\nand 1 mW, respectively. The other parameters are the same as t hose\nin Fig. 2.\nyellow and green lines shown, the threshold power is corre-\nspondingly reduced from P th=10 mW to 4 mW. The phys-\nical mechanism as Eq. ( 9) reveals that the threshold power\nof the magnon laser is inversely proportional to the square o f\nthe magnon-photon coupling strength g. In experiment, the\nimprovement of the magnon-photon coupling strength might\nbe realized in several aspects, for instance, scaling down t he\nYIG sphere size to reduce the mode volume [ 15], manufactur-\ning the nanostructured magnets [ 26,27] to increase the spatial\noverlap between the optical and magnon fields, and further pu -\nrifying and chemically processing of the YIG sphere are also\nrecommended [ 32]. With these improvements, the magnon-\nphoton coupling strength is expected to increase two orders of\nmagnitude [ 20], and the threshold power is decreased to P th∼\n2.3µW. In this context, even if the input power is quite weak,\nthe magnon laser action can be observed experimentally in th e\nnear future.\nFrom Eqs. ( 7) and ( 8), the stimulated emitted magnon num-\nber can be derived as M[ γ−1\nm]=exp(η−1) [18], where\nη=8g2Pin\n/planckover2pi1ωMγMγEγm, (10)\nrepresents the pure gain factor of the emitted magnon number\nunder the triple-resonance condition. In the case of η>1 im-\nplies that the magnon gain caused by the Brillouin light scat -\ntering overcomes the mode dissipation and yields the magnon\ncoherent amplification. As the green solid line shown in Fig.\n3, for the magnon-photon coupling strength g=2π×39.2 Hz,\nthe magnon laser action would be observed when the incident\nlight power reaches about 22 mW, which is highly accessi-\nble under the current experimental technique [ 15]. More im-\nportantly, the emitted magnon number M[ γ−1\nm] increases ex-4\nFIG. 4: Three-dimensional graph of the stimulated emitted m agnon\nnumber M[ γ−1\nm] varies with the magnetic field strength H /Hm, and\n(a) the optical power P in(mW), (b) the magnon-photon coupling\nstrength g. The red solid line represents the threshold condition of th e\nmagnon laser. We use g=2π×39.2 Hz in Fig. 3(a) and P in=1mW\nin Fig. 3(b), and the other parameters are the same as those in Fig. 2.\nponentially with the incident power. In particular, when th e\nmagnon-photon coupling strength increase to g=3×2π×39.2\nHz, as the blue solid line shown, the threshold power of the\nmagnon laser is reduced to 2.5 mW. And the steady state num-\nber of stimulated emitted magnons is calculated to be M[ γ−1\nm]\n≈1.82×104when the input light power P in=15 mW, which is\nmuch larger than the number of thermally populated magnons\nmth≈624.8 at room temperature (T =300 K), consequently,\nthe magnon laser e ffect can be measured at room tempera-\nture. Besides, the magnon line shape function are plotted in\nthe illustration (a) and (b) of Fig. 3in the context of the pump-\ning points below (P in=1 mW) and above (P in=15 mW) the\nthreshold, respectively. We can see that the stimulated emi t-\nted magnons are mainly populated in the vicinity of the fre-\nquencyωm=2π×10 GHz≃62.83 GHz, and the linewidth\nof the magnon laser above the threshold is significantly nar-\nrower [ 33]. These results, therefore, are enlightening for the\nrealization of magnon operation by using light, and may of-\nfer theoretical support for the manufacture of high-intens ity\nmagnon-laser amplifiers.\nFinally, achieving the e ffective control of the magnon laser\naction is of fundamental importance and is also a key link\nfor the practical application of the magnon laser in the fu-\nture. Obviously, we can see from Figure 4that the stimulated\nmagnon number reaches the peak only when the magnetic\nfield strength was adjusted to H =Hm=Ω m/̺≈357.14 mT.\nIn this circumstances, the frequency of the magnon mode in\nYIG sphere is resonant with the frequency di fference between\nthe incident photons and the scattered photons. Otherwise,\nthe emitted magnon number decreases exponentially when the\nmagnetic field strength deviates from H m, i.e., H>Hmor\nH 1, the magnon\nlaser action can be achieved, in which the external magnetic\nfield plays a paramount role. Our proposal thus provides a\npathway for implementing magnetic-field-modulated magnon\nlaser that can be applied to both fundamental problems in cav -\nity optomagnonics and influences a broad range of scientific\nfields beyond magnonics.\nIn conclusion, the magnon laser action based on Brillouin\nlight scattering in a YIG ferrimagnetic sphere has been theo -\nretically demonstrated. We manifest that the stimulated em it-\nted magnon number increases exponentially with the input\nlight power, similar to the photon amplification by stimulat ed\nemission of radiation, providing a theoretical foundation for\nthe realization of magnon laser. 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Lee* \nDepartment of Physics, Yonsei University, Seoul 03722, Korea \n \nExploring new magnetic materials is essential for finding advan tageous functional properties such as \nmagnetoresistance, magnetocaloric effect, spintronic functional ity, and multiferroicity. Versatile \nclasses of double perovskite compounds have been recently inves tigated because of intriguing physical \nproperties arising from the proper combination of several magne tic ions. In this study, it is observed \nthat the dominant ferrimagnetic phase is coexisted with a minor multiferroic phase in single-crystalline \ndouble-perovskite Er 2CoMnO 6. The majority portion of the ferrimagnetic order is activated by the long-\nrange order of Er3+ moments below TEr = 10 K in addition to the ferromagnetic order of Co2+ and Mn4+ \nmoments arising at TC = 67 K, characterized by compensated magnetization at TComp = 3.15 K. The \ninverted magnetic hysteresis loop observed below TComp can be described by an extended Stoner–\nWohlfarth model. The additional multiferroic phase is identifie d by the ferroelectric polarization of ~0.9 \nμC/m2 at 2 K. The coexisting ferri magnetic and multiferroic phases a ppear to be strongly correlated in \nthat metamagnetic and ferroelectric transitions occur simultane ously. The results based on intricate \nmagnetic correlations and phases in Er 2CoMnO 6 enrich fundamental and a pplied research on magnetic \nmaterials through the scope of di stinct magnetic characteristic s in double perovskites. \n \n \nCorrespondence and requests for materials should be addressed t o Y. J. C. \n(phylove@yonsei.ac.kr) or N . L. (eland@yonsei.ac.kr). \n Introduction \n \nOne of the ideas behind examining magnetic materials aims to de velop desired functional \nproperties utilized in a wide range of technologies, for exampl e, energy storage, 1 memory \ndevices 2, medical appliances 3, and environmental monitoring sensors 4. In particular, magnetic \noxides comprising metal cations and oxygen anions have been ext ensively investigated owing \nto the abundance of the constituents and stability of the compo unds. A prominent example can \nbe found in perovskite rare-earth m anganites that have been the focus of research on magnetic \nmaterials over the last few decades. In mixed-valence manganite s, the subtle balance between \nhopping and localization of charge carriers leads to the phase coexistence of ferromagnetic \n(FM) metallic and antiferromagnetic (AFM) insulating states via the kinetic arrest of the phase \ntransition. The formation of mixed magnetic glass, which is sus ceptible to an external magnetic \nfield ( H), is essential to the origin of colossal magnetoresistance 5,6. The variation of rare-earth \nions in perovskite manganites also generates several types of m ultiferroic (MF) phases 7-10. In \nmedium-sized rare-earth ions, the spiral spin modulation can be stabilized, inducing \nferroelectricity via antisymmetric exchange strictions with str ong controllability of \nferroelectric properties by external magnetic fields 7. With a smaller radius, the crystallographic \nstructure changes into a hexagonal structure, which represents a unique improper \nferroelectricity due to structural trimerization 9. However, the perovskite structure remains \nintact under high pressure and accompanies the E-type AFM phase that results in another type \nof the MF phase driven by sy mmetric exchange strictions 10. \n As an extension of studies on perovskite manganites, double per ovskites of R\n2CoMnO 6 (R = \nLa, …, Lu, and Y) have recently been explored owing to their fa scinating magnetic and \nfunctional properties, such as metamagnetism 11-13, spin-glass state 14-16, exchange bias effect \n17-19, magnetocaloric effect 20-22, and multiferrocity 23-27. By replacing half the Mn ions with Co \nions in perovskite manganites, a double perovskite structure is formed with Co2+ (S = 3/2) and \nMn4+ (S = 3/2) ions, alternatingly locate d in corner-shared octahedral environments. As the size \nof rare-earth ions decreases, the magnetic transition temperatu re (T) arising from the dominant \nCo2+ and Mn4+ superexchange interactions decreases from 204 K for La 2CoMnO 6 to 48 K for \nLu2CoMnO 6 28. In these compounds, the difficulty in attaining the impeccabl e alteration of \nCo2+ and Mn4+ ions naturally entails additional AFM clusters which involve an other valence \nstate of Co3+- Mn3+, and anti-sites of ionic disord ers and/or antiphase boundaries leading to Co2+- Co2+ or Mn4+- Mn4+ pairs. The formation of anti-sites in addition to the dominant FM \norder 27,29,30 of Co2+ and Mn4+ moments is known as the mechan ism for the observed magnetic \nexchange bias in polycrystalline Y 2CoMnO 6 19. In Tm 2CoMnO 6 and Er 2CoMnO 6 (ECMO), the \nneutron diffraction studies c onfirm that the order of Co2+ and Mn4+ moments is FM and the \norder of Er3+/Tm3+ moments at lower temperature activates the additional ferrimag netic (FIM) \norder between Er3+/Tm3+ and ferromagnetic Co2+/Mn4+ sublattices 31-33. The FIM order exhibits \nan inversion of the magnetic hysteresis loop in polycrystalline ECMO 34. In Yb 2CoMnO 6 and \nLu2CoMnO 6, the Co2+ and Mn4+ ions display the up-up-down-down (↑↑↓↓) spin configuration \nin which the ferroelectricity emerges perpendicular to the c-axis from the cooperative O2- \ndisplacements through the sym metric exchange striction 23-25. Evidently, a scientific \nunderstanding of diverse magnetic phases and interactions is cr ucial for finding novel \nfunctional properties in double perovskites. \n In this work, the magnetic and magnetoelectric properties of si ngle crystals of double-\nperovskite ECMO were studied to reveal the characteristics corr esponding to the mixed FIM \nand MF phases. The dominant FIM order between Er\n3+ and FM Co2+/Mn4+ sublattices was \nidentified by compensated magnetization ( M) occurring at TComp = 3.15 K. From our precise \nmeasurement of isothermal M in the low T regime, the inversion of the magnetic hysteresis \nloop was observed below TComp, which can be explained by the delicate balance between \ndifferent magnetic moments, and qualitatively by an extended St oner–Wohlfarth model 35-38. \nThe ferroelectric polarization ( P) and dielectric constant ( ') measurements demonstrated an \nadditional inclusion of the MF phase as found in Yb 2CoMnO 6 and Lu 2CoMnO 6 23,24. Associated \nwith the coexistence of FIM and MF phases, the disappearance of MF phase by an external H \noccurs simultaneously with the m etamagnetic transition, reveali ng exclusive characteristics of \nthe double perovskite. \nResults and Discussion \n \nFigure 1(a) shows the X-ray powder diffraction pattern for the ground single crystals of double \nperovskite ECMO at room T. The crystallographic structur e was refined as a monoclinic \nstructure with the P2\n1/n space group. The lattice constants were found to be a = 5.228 Å, b = \n5.594 Å, and c = 7.477 Å with = 90.244º with good agreement factors, χ2 = 1.74, Rp = 7.97, \nRwp = 6.23, and Rexp = 4.72. The crystal structures viewed from the a- and c-axes are depicted in Figs. 1(b) and (c), respectively. Co2+ and Mn4+ ions are alternatingly located in corner-shared \noctahedral environments. The oxygen octahedral cages are strong ly distorted due to the small \nradius of the Er3+ ion 28. \n To investigate intricate magnetic properties as anticipated in the double perovskite \nincorporating three different magnetic ions, the T-dependence of magnetic susceptibility ( χ = \nM/H) was obtained. The anisotropic χ in H = 0.05 kOe along ( H//c) and perpendicular ( Hc) \nto the c-axis was measured upon warming in H after zero-field cooling (ZFC) and upon cooling \nat the same H (FC), as shown in Fig. 2(a). The overall T-dependence of χ’s for two different \norientations exhibits strong magnetic anisotropy, which indicat es that the spins are mainly \naligned along the c-axis. The FM order relevant to the dominant Co\n2+ and Mn4+ superexchange \ninteractions sets in at TC = 67 K, which can be determined by the sharp anomaly in the T \nderivative of χ in H//c. The T-dependence of heat capacity divided by the temperature ( C/T) \nmeasured upon warming in zero H also exhibits the anomaly starting from TC, shown in Fig. \n2(b). Upon further cooling, C/T shows an abrupt increase below TEr ≈ 10 K, which corresponds \nto the ordering of Er3+ moments. Below TEr, the reversal of χ was observed in both ZFC and \nFC measurements (Fig. 2(a)) as a c haracteristic signature of a ferrimagnet 39-46. \n A ferrimagnet is a substance that involves a portion of opposin g magnetic moments as in \nantiferromagnetism, but generates a net M from unequal magnetic moments in the opposite \ndirections, thus exhibiting dis tinct characteristics of magneti sm. The FIM interaction between \nEr\n3+ and FM Co2+/Mn4+ sublattices generates the intriguing T-dependence of χ following the \ndifferent sequence of measurement. In the FC measurement in H//c, χ increases smoothly below \nTC with the parallel alignment of Co2+ and Mn4+ moments. Upon cooling further below TEr, the \nEr3+ moments begin to align oppositely to the Co2+/Mn4+ moments, which leads to a gradual \ndecrease in χ. At lower T, χ intersects the zero point owing to the large moment of Er3+ spin. \nOn the other hand, ZFC χ shows a positive value at 2 K since the Er3+ moments tend to orient \nalong the H direction. The decrease in the effective Er3+ moments upon increasing T results in \nthe sign change of χ. Above TEr, the negatively magnetized Co2+/Mn4+ spins begin to flip along \nthe applied H due to thermal fluctuation, which causes another sign change o f χ at 48 K. To \nfind the compensation T precisely, the thermoremanent magnetization ( Mrem) 47 was measured \nin H//c (Fig. 2(c)). At 2 K, H = 50 kOe was applied in H//c and then turned off, and Mrem was recorded in the absence of H upon warming from 2 K. The sign reversal of Mrem occurs at TComp \n= 3.15 K, which manifests the FIM feature of this double perovs kite compound. \n \nThe anisotropic M in H//c and Hc was measured up to ±90 kOe at T = 2 K, shown in Fig. 3(a). \nFor the hysteresis loop in H//c, solid and dashed lines denote sweeping H from +90 to −90 kOe \nand from −90 to +90 kOe, respectively. M in H//c is not saturated at +90 kOe with the magnetic \nmoment of 17.2 B/f.u., but it is much larger than the moment in Hc (8.71 B/f.u.), indicating \nthe magnetic easy c-axis. Upon decreasing H from +90 kOe, M decreases smoothly until it \ndrops precipitously be low 15 kOe. At low H, M intersects the zero poin t at 1 kOe and exhibits \nthe negative remanent M (Mr) of −1 B/f.u. (inset of Fig. 3(a)). Further decrease in H in the \nnegative direction induces a sharp drop in M at HC = −26.5 kOe. The measurement of M in H//c \nin the opposite direction completes the inverted magnetic hyste resis loop. The inversion of the \nhysteresis loop in H//c can be analysed by an extended Stoner–Wohlfarth model within t he \nf r a m e o f t h e F I M o r d e r b e t w e e n E r3+ and Co2+/Mn4+ sublattices with a different magnetic \nanisotropy 35-38 (see Experimental section for detail). The experimental observ ation of inversed \nmagnetic hysteresis loop in ECMO s uggests the considerable diff erence of magnetic anisotropy \nenergies between Er3+ and Co2+/Mn4+ moments. In our calculation, we assumed that the \nmagnetocrystalline anisotropy energy of Co2+/Mn4+ moments is three times larger than that of \nEr3+ moments. With qualitative similarity, the magnetic hysteresis loop was attained from the \nmodel, as illustrated in Fig. 3(b). Based on the result, the ev olution of the spin configuration \nfor Er3+ and Co2+/Mn4+ ions during the sweeping of H from +90 to −90 kOe in H//c i s \nschematically depicted in Fig. 3(b). The red and blue arrows in dicate the effective moments of \nEr3+ and Co2+/Mn4+ ions, respectively. At high H, the Er3+ and Co2+/Mn4+ moments tend to be \naligned in the same direction due to the dominant Zeeman energy . Upon decreasing H, the \nnegative exchange c oupling between Er3+ a n d C o2+/Mn4+ spins accompanied by a smaller \nmagnetocrystalline anisotropy en ergy and larger moment of Er3+ ions leads to the progressive \ndecrease in the net Er3+ moments, followed by zero net M even at a positive H and negative Mr. \nDecreasing H further in the negative direction induces an abrupt drop in M, where the \nCo2+/Mn4+ spins are fully reversed because the Zeeman energy of Co2+/Mn4+ sublattices \novercomes the anisotropy energy. Since the change in magnitude of M caused by the reversal \nof Co2+/Mn4+ moment at the metamagnetic transition is found to be ~9 B/f.u. (Fig. 3(a)), the \nnet magnetic moment of Co2+/Mn4+ spins should be ~ 4.5 μB/f.u., which is smaller than the \nsummation of Co2+ and Mn4+ moments (6 B/f.u.). The smaller net magnetic moment of Co2+/Mn4+ spins is acceptable because a small portion of Co2+/Mn4+ spins is naturally reversed \nduring the magnetization process from +90 kOe to −26.5 kOe and antiferromagnetic exchange \ncouplings of Co2+- Co2+ or Mn4+- Mn4+ pairs are originally include d from the presence of anti-\nsites of ionic disorders an d/or antiphase boundaries. \n The close relevance of M\nr to TComp was cautiously examined by the T dependent evolution of \nMr. The full hysteresis curves up to ±90 kOe were recorded in H//c at various T’s. The \nhysteresis loops below and above TComp are shown within the range of ±5 kOe in Fig. 3(c) and \nd, respectively. Below TComp, all the curves present the i nverted magnetic hysteresis. Upon \nincreasing T, the inverted loop becomes narrow and the magnitude of negativ e Mr decreases \nlinearly, resulting from the reduced net Er3+ moments by thermal fluctuation. By crossing TComp, \nthe sign of Mr changes and it increases gr adually with an increasing T. \n Recently, new magnetism-driven ferroelectrics, i.e. type-II mul tiferroics, were found in \ndouble-perovskite Yb\n2CoMnO 6 and Lu 2CoMnO 6 23,24. The initial polycrystalline analysis of \nneutron diffraction and bulk electric properties for Lu 2CoMnO 6 suggested that the \nferroelectricity arises from the symmetric exchange striction o f the ↑↑↓↓ spin chains with \nalternating Co2+ and Mn4+ charge valences 48, consistent with the Ising spin chain magnet of \nCa3CoMnO 6 49. However, studies on the single crystals of Yb 2CoMnO 6 and Lu 2CoMnO 6 \nrevealed that the ferroelectricity emerges perpendicular to the c-axis below TC = 52 and 48 K, \nrespectively. Several theoretical works provided a plausible ex planation for the ferroelecticity, \nin which the symmetric exchange strictions along the ↑↑↓↓ spin chain with alternatingly shifted \nO2- ions generate cooperative O2- displacements perpendicular to the c-axis 50-52. \n The possible formation of an additional MF phase in ECMO was ex amined by the H-\ndependence of P obtained by integrating magnetoelectric current density ( J), measured \nperpendicular to the c-axis ( Ec) at 2 K, shown in Figs. 4(a) a nd (b). After poling from 100 K \nto 2 K in H = 0 kOe and E = 5.7 kV/cm, the J in H//c exhibits a very sharp peak with peak \nheight of ~0.76 μA/m\n2 at the metamagnetic transition, HC = 26.5 kOe. The corresponding P \nvalue at H = 0 kOe and 2 K was estimated as ~0.9 μC/m2, which is only two orders of magnitude \nsmaller than the P observed in Lu 2CoMnO 6 and signifies the presence of a small amount of the \nMF phase. The tiny magnitude of P at 2 K implies that the exact magnetic configuration of MF \nphase could hardly be identified by the neutron diffraction exp eriment. Upon increasing H, the P shows the sharp step at HC and disappears above HC. The simultaneous transitions at HC for \nthe suppression of the ferroel ectricity and the reversal of Co2+/Mn4+ spins in the FIM state \nsuggest that the small amount of the additional MF phase is str ongly influenced by the \ndominant FIM phase. In analogy w ith the ferroelectricity in Lu 2CoMnO 6, the P e m e r g e d \nperpendicular to the c axis at H=0 kOe in ECMO suggests that the most plausible spin \nconfiguration of the minor MF phase would be ↑↑↓↓. The disappea rance of the P by applying \nH along the c axis can be explained by the cha nge of spin configuration from the ↑↑↓↓ to ↑↑↑↑. \n \nIn Fig. 4(c), the H-dependence of ' in Ec is plotted, measured in H//c up to ±90 kOe at f = \n100 kHz and T = 2 K. By sweeping H between +90 to −90 kOe, the whole variation of ' is \nonly about 1 % with strong hysteretic behaviour. The maximum va lues occur at H = ±6 kOe, \nfollowed by the sharp transitions at HC. The ' shows the rather complicated H dependence in \ncomparison with the H dependence of P. In addition to the small portion of MF phase, the \nadditional AFM clusters formed by anti-site disorders and antip hase boundaries in the \nferromagnetic Co2+/Mn4+ sublattices would also affect the isothermal '. The complicated but \ntiny magnitude variation of isothermal ' may result from the intricate contributions from the \nsmall portions of MF phase and AFM clusters. For a comparison w ith '(H), the H derivative \nof isothermal M, dM/dH at 2 K is also plotted in Fig. 4(d). The d M/dH reveals the similar \nhysteretic variation of '. The isothermal M mainly reflects the response of the FIM order \nbetween the Er3+ and Co2+/Mn4+ moments to the external H, as illustrated in Fig. 3(b), but also \naffects strongly on the hysteretic behaviour of '(H) . \n \nThe T-dependence of the d ielectric constant ( ') and tangential loss (tan ) is displayed in Figs. \n5(a) and (b), respectively, m easured perpendicular to the c-axis ( Ec) at f = 100 kHz in H//c \nwith H = 0, 10, 20, and 30 kOe. At zero H, a small and broad peak of ' at TC = 67 K was \nobserved in Fig. 5(a), which signifies the emergence of a small amount of MF phase. Compared \nto the peak height of ~15 %, normalized by the value at TC = 48 K in Lu 2CoMnO 6,23 it can be \nestimated as only about 1 % in ECMO. Despite a small portion of the MF phase in ECMO, TC \nis fairly enhanced. The broad peak of ' is gradually suppressed by applying H along the c-axis, \nascribed to the change in the spin configuration from ↑↑↓↓ to ↑ ↑↑↑, similar to that in \nLu2CoMnO 6.23 Upon decreasing T, ' decreases linearly until it declines faster below 20 K. The overall T-dependence of ' and tan (Fig. 5(b)) below TC appears similar to those of \nLu2CoMnO 6. \n While the intrinsic coupling phenomena between magnetic and fer roelectric states in single-\nphase type-II multiferroics were extensively explored, detailed properties of an MF phase \nmixed with another magnetic phase have scarcely been revealed. The T evolution of \nmagnetoelectric effect in the mixed FIM and MF phases was exami ned by comparison between \nisothermal P and M at T’s below T\ncomp. Figures 6(a) and (b) show the H-dependence of P’s and \nM’s, respectively, in Ec and H//c at T = 2, 2.25, 2.5, 2.75, and 3 K, indicating that both of P \nand M vary delicately to the change of T. The estimated P’s at 2.25 and 2.5 K were 0.79 and \n0.47 μC/m2, respectively. As H is increased, the P’s are suppressed with steep steps at H = 28.0 \nand 28.7 kOe. The i nitial curve of M at 2, 2.25, and 2.5 K also show s the step at the same H as \nP, suggesting the strong intercorr elation between FIM and MF pha ses. At 2.75 and 3 K, P \nmagnitudes at 0 kOe are reduced as 0.43 and 0.32 μC/m2, respectively. Upon increasing H, the \nP’s are gradually reduced and vanish above ~37 kOe, correspondin g to the overall broad feature \nof M’s. Note that P above Tcomp could not hardly be obtained because of the almost suppressed \nmagnitude of J with a broadened feature. Figur es 6(c) and (d) display isother mal ' in Ec at f \n= 100 kHz and J in H//c, respectively, at T = 2, 2.25, 2.5, 2.75, and 3 K. The initial curve of ' \nat 2 and 2.25 K indicates both a sharp peak and step-like featu re at the metamagnetic transition \nbut the ' at 2.5 K shows only a step. The sharp peak of the J at 2 K shifts to higher H and the \npeak height is reduced upon slightly increasing the T. The weak anomaly was observed in the \n' at 2.75 and 3 K, corresponding to the disappearance of P. As shown in the inset of Fig. 6(d), \nJ’s at 2.75 and 3 K exhibit wide and small peaks around 35 kOe. \n The T evolution of the magnetodielect ric effect in a wide range of T’s in the mixed FIM and \nMF phases was also investigated by comparison between isotherma l \n' and M. Figure 7 displays \nthe isothermal ' in Ec at f = 100 kHz and M in H//c and Hc, at T = 5, 10, 20, 35, 50, and 65 \nK. At 5 K, a butterfly-like shape of ' was observed with a strong magnetic hysteresis, with the \nabsence of the step-like metama gnetic transition (Fig. 7(a)). T he broadened feature of ' is \ncompatible with the modulation of M in H//c with the narrow magnetic hysteresis described as \nsmall values of Mr = 1.22 B/f.u. and the coercive field of Hc = 2.10 kOe (Fig. 7(g)). At 10 K, \nthe butterfly-like shape of ' is maintained (Fig. 7(b)), but the magnetic hysteresis is considerably reduced. The central part of the hysteresis loop i n H//c is extended as Mr = 2.73 \nB/f.u. and Hc = 7.24 kOe (Fig. 7(h)), indicativ e of the reduced strength of t he Er3+ spin order. \nIn addition, the slight and elonga ted hysteretic behaviour of M in Hc emerges. As T increases \nfurther, the magnetic hysteresis in both ' and M is progressively reduc ed. At 65 K, just below \nTC, the sharp cusp of ' occurs at zero H with the hysteresis loop in M vanishing. \n \nIn summary, we explor ed the magnetic and magnetoelectric proper ties of mixed ferrimagnetic \nand multiferroic phases of single-crystalline double-perovskite E r 2CoMnO 6. The dominant \nCo2+ and Mn4+ superexchange interactions lead t o the ferromagnetic order bel ow TC = 67 K, \naligned mainly along the c-axis. The long-range order of Er3+ m o m e n t s b e l o w TEr = 10 K \ninduces the ferrimagnetic order and magnetization compensation at TComp = 3.15 K, delicately \nbalanced with the ferromagnetic Co2+/Mn4+ sublattice. The extende d Stoner–Wohlfarth model \ndepicts qualitatively the invert ed magnetic hysteresis loop obs erved below TComp. The \nobservation of electric polarizat ion at low temperature is indi cative of the presence of a small \nportion of a multiferroic phase simultaneously with the ferrima gnetic phase. The strong \nmagnetoelectric correlation at the metamagnetic transition in t he phase coexistence reveals the \nunique characteristic of the double perovskite compound, which o f f e r s c r u c i a l c l u e s f o r \nexploring suitable materials for magnetoelectric functional app lications. \n \nMethods \n \nRod-shaped single crystals of ECMO with a typical size of 2 2 5 mm3 were grown by the \nconventional flux method with Bi 2O3 flux in air. Er 2O3, Co 3O4, and MnO 2 powders were mixed \nin the stoichiometric ratio for ECMO and ground in a mortar, fo llowed by pelletizing and \ncalcining at 1000 °C for 12 h in a box furnace. The calcined pe llet was delicately reground and \nsintered at 1100 °C for 24 h. Th e same sintering procedure afte r regrinding was carried out at \n1200 °C for 48 h. A mixture of pre -sintered polycrystalline pow der and Bi 2O3 flux with a ratio \nof 1:12 ratio was heated to 1300 °C in a Pt crucible. It was me lted at the soaking T for 5 h, \nslowly cooled to 985 °C at a rate of 2 °C/h, and cooled to room T at a rate of 250 °C/h. The \ncrystallographic structure and absence of a second phase were c hecked by the Rietveld \nrefinement 53 using the FullProf program 54 for the power X-ray diffraction data. The data were \nobtained with a Rigaku D/Ma x 2500 powder X-ray diffractometer u sing Cu-K α radiation. \n The T and H dependences of DC M were examined by using a VSM magnetometer in a \nQuantum Design PPMS (Physical Properties Measurement System). T he specific heat ( C) was \nmeasured with the standard relaxation method in PPMS. The T and H dependences of ' were \nobserved at f = 100 kHz using an LCR meter (E4980, Agilent). The H dependence of electric \npolarization ( P) was obtained by the integrati on of magnetoelectric current me asured with the \nH variation of 0.1 kOe/s after polin g in a static electric field of E = 5.7 kV/cm. \n In our extended Stoner–Wohlfarth model \n35-38, the magnetic energy density can be expressed \nas \nܧൌെ ܯ ாܪcosߠ ாെܯ/ெܪcosߠ /ெെܬcos൫ߠ ாെߠ/ெ൯ \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t \tܭ ாsinଶሺߠாെ߮ாሻܭ/ெsinଶ൫ߠ/ெെ߮/ெ൯ ( 1 ) \nThe first two terms represent the Zeeman energy density, where ܯா and ܯ/ெ are the \neffective magnetic moments of Er3+ and Co2+/Mn4+ ions, respectively, and ߠ is the angle \nbetween the corresponding M and applied H. The third term denotes the exchange energy \ndensity, where the moments in the Er3+ and Co2+/Mn4+ sublattices tend to be ordered oppositely, \nand thus, the exchange coupling constant ( ܬ )is negative. The last two terms signify the densities \nof magnetocrystalline anisotropy energy ( ܭா and ܭ/ெ ) for Er3+ and Co2+/Mn4+ ions, \nrespectively, where ߮ is the angle between H and the magnetic easy axis . The energy density \ncan be minimized to determine the direction of net M at an applied H by solving both பா\nபఏಶൌ0 \nand பா\nபఏ/ಾൌ0. T h e e s t i m a t e d n e t M can be written as follows: ܯൌܯ ாcosߠா\nܯ/ெcosߠ/ெ . The calculated hysteresis loop in Figure 3(b) was obtained wi th ܬൌ\nെ4.241ൈ10ହ\tJ / mଷ, ܭாൌ 8.482ൈ10ସ\tJ / mଷ, and ܭ/ெൌ 2.545ൈ10ହ\tJ / mଷ. \n \nAcknowledgements \n \nThis work was supported by the NRF Grant ( NRF-2016R1C1B2013709, NRF-\n2017K2A9A2A08000278, 2017R1A5A1014862 (SRC program: vdWMRC center), and NRF-\n2018R1C1B6006859 ). \n \nAuthor contributions \n Y.J.C. and N.L. designed the experiments. M.K.K. calculated Sto ner–Wohlfarth model, and \nS.H.O. and J.Y.M. synthesised th e single crystals. M.K.K., J.Y. M. and D.G.O. performed \nmagnetization, heat capacity, diel ectric constant, and magnetoe lectric current measurements. \nM.K.K., Y.J.C., and N.L. analysed the data and prepared the man uscript. All the authors have \nread and approved the final ve rsion of the manuscript. \n \nAdditional information \n \nThe authors declare no competing interests . \n \n \n \n \n \n \n \n \n \nFigure 1. Crystallographic structure of Er 2CoMnO 6. (a) Observed (open circles) and \ncalculated (solid line) powder X-ray diffraction patterns for g round Er 2CoMnO 6 (ECMO) \nsingle crystals. Y obs, Y cal, and Y obs−Y cal r e p r e s e n t t h e i n t e n s i t i e s of the observed patterns, \ncalculated patterns, and their difference, respectively. The gr een short lines denote the Bragg \npositions. (b) and (c) Views of the crystal structure of double perovskite ECMO from the a- \nand c-axes, respectively. The purple, pink, blue, and yellow spheres represent Er3+, Co2+, Mn4+, \nand O2 ions, respectively. \n \n \n \nFigure 2. Temperature-dependent magnetic properties of Er 2CoMnO 6. (a) Temperature \n(T) dependence of the magnetic susceptibility ( χ = M/H , 1 emu = 4π × 10-6 m3) of a double-\nperovskite ECMO single crystal along ( H//c) and perpendicular ( Hc) to the c-axis, measured \nupon warming in H = 0.05 kOe after zero-magnetic-fie ld cooling (ZFC) and upon coo ling in \nthe same field (FC), shown up to 80 K. (b) T-dependence of specific heat divided by \ntemperature ( C/T) measured without magnetic field ( H). (c) T-dependence of the \nthermoremanent magnetization ( Mrem) of the ECMO crystal, measured in H//c warming from \n2 K in the absence of H after cooling in 50 kOe. The v ertical dashed lines indicate th e \nferromagnetic transition temperature ( TC), the Er3+ spin ordering temperature ( TEr), and the \ncompensation temperature ( TComp), respectively. \n \n \n \nFigure 3. Observed and calculated isothermal magnetization at 2 K and temperature \nevolution of inverted magnetic hysteresis. (a) Isothermal magnetization ( M) of the ECMO \ncrystal in H//c and Hc, measured at T = 2 K up to 90 kOe after ZFC. The inset shows the \nmagnified view in the range of H = ±10 kOe of the hysteresis loop in H//c. For the hysteresis \nloop in H//c, the solid and dashed curves indicate the data obtained by swe eping H from +90 \nkOe to −90 kOe, and by sweeping H from −90 kOe to +90 kOe, respectively. (b) Calculated \nhysteresis loop in H//c by adopting the extended Stoner-W ohlfarth model. The schematic spin \nconfigurations depicted as net moments of Er3+ (light red arrows) and Co2+/Mn4+ (light blue \narrows) spins are illustrated for the curve of sweeping H from +90 kOe to −90 kOe. (c) and (d) \nMagnified views in the range of H = ±5 kOe of isothermal M’s in H//c, measured at various \nT’s below TComp (T = 2, 2.25, 2.5, 2.75, and 3 K) and above TComp (T = 3.25, 3.5, 4, 4.5, and 5 \nK), respectively. \n \n \n \n Figure 4. Isothermal ferroelectric polarization and dielectric constant a t 2 K. ( a ) H-\ndependence of ferroelectric polarization ( P) at 2 K, obtained by integrating the magnetoelectric \ncurrent in b). (b) H-dependence of current density ( J), measured with the H variation of 0.1 \nkOe/s in H//c after poling from 100 K to 2 K in E = 5.7 kV/cm perpendicular to the c axis. (c) \nH-dependence of \n' i n Ec, measured up to ±90 kOe in H//c at 2 K. (d) H-derivative of \nisothermal M (dM/dH) at 2 K, taken from the data in Figure 3a). \n \n \n \n \n \n \n \nFigure 5. Temperature dependence of the dielectric properties o f Er 2CoMnO 6. (a) and (b) \nT-dependences of dielectric constant ( ') and dielectric tangential loss (tan ), respectively, \nmeasured upon warming from 2 K to 100 K in an applied AC voltag e of V = 1 V at f = 100 kHz \nperpendicular to the c-axis (Ec), and H = 0, 10, 20, and 30 kOe along the c-axis (H//c). \n \n \n \n \n \nFigure 6. Temperature evolution of ferroelectric polarization below Tcomp in comparison \nwith that of magnetization and dielectric constant. (a) H-dependence of P at T = 2, 2.25, \n2.5, 2.75 and 3 K s hown in the range of 25-40 kOe. (b) Initial curves of isothermal M at T = 2, \n2.25, 2.5, 2.75 and 3 K. (c) H-dependence of ' a t T = 2, 2.25, 2.5, 2.75 and 3 K. (d) H-\ndependence of J, measured with the H variation of 0.1 kOe/s in H//c at T = 2, 2.25, 2.5, 2.75 \nand 3 K after poling in Ec. The inset shows the magnified view of J. \n \n \n \n \nFigure 7. Temperature evolution of the isothermal dielectric co nstant. 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Recent advances in magnetic structur e determination by neutron \npowder diffraction. Phys. B (Amsterdam, Neth.) 192, 55-69, doi:10.1016/0921-\n4526(93)90108-I (1993). \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 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\nJ’ =1.5, M =16\n00.20.4J’ =0.87, M =12\n20 40 60 80 \ni–0.1 00.10.2\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. DMRG calculations\nwere carried out using the ALPS DMRG application.31)\nSome of the calculations were carried out at the Super-\ncomputer Center, Institute for Solid State Physics, Uni-\nversity of Tokyo.\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) K. Takano, K. Kubo, and H. Sakamoto: J. Phys.: Condens.\nMatter8(1996) 6405.\n5) K. Okamoto, T. Tonegawa, Y. Takahashi, and M. Kaburagi:\nJ. Phys.: Condens. Matter 11(1999) 10485.\n6) T.Tonegawa, K.Okamoto, T.Hikihara, Y.Takahashi, and M.\nKaburagi: J. Phys. Soc. Jpn. 69(2000) Suppl. A, 332.\n7) M.Ishii, H.Tanaka, M.Hori, H.Uekusa, Y.Ohashi, K.Tatan i,\nY. Narumi, and K. Kindo: J. Phys. Soc. Jpn. 69(2000) 340.\n8) As a candidate compound of the diamond chain system, nat-\nural mineral azurite, Cu 3(CO3)2(OH)2, is proposed.9,10)\n9) H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, and T. Idehara:\nPhysica B 329-333 (2003) 967.\n10) H. Ohta, S. Okubo, T. Kamikawa, T. Kunimoto, Y. Inagaki,\nH.Kikuchi, T.Saito, M.Azuma, and M.Takano: J.Phys.Soc.\nJpn.72(2003) 2464-3467.\n11) H. Nakano and M. Takahashi: J. Phys. Soc. Jpn. 66(1997)\n228.\n12) C. K. Majumdar and D. K. Ghosh: J. Math. Phys. 10(1969)\n1399.\n13) S. R. White: Phys. Rev. Lett. 69(1992) 2863.\n14) S. R. White: Phys. Rev. B. 48(1993) 10345.\n15) S. Sachdev and T. Senthil: Ann. Phys. 251(1996) 76.\n16) L.Bartosch, M.Kollar,and P.Kopietz: Phys.Rev.B 67(2003)\n092403.\n17) N. B. Ivanov and J. Richter: Phys. Rev. B 69(2004) 214420.\n18) S. Yoshikawa and S. Miyashita: J. Phys. Soc. Jpn. 74(2005)\nSuppl. 71.\n19) K. Hida: J. Phys.: Condens. Matter 19(2007) 145225.\n20) K. Hida and K. Takano: Phys. Rev. B 78(2008) 064407.\n21) R.R.Montenegro-Filho and M.D.Coutinho-Filho: Phys.R ev.\nB78(2008) 014418.\n22) H.Nakano, T.Shimokawa, and T.Sakai: submitted to J.Phy s.\nSoc. Jpn.\n23) J. Schulenburg and J. Richter: Phys. Rev. B 65(2002) 054420\n24) T. Tonegawa, I. Harada, and J. Igarashi: Prog. Theor. Phy s.\nSuppl.101(1990) 513.\n25) I. Harada and T. Tonegawa: J. Magn. Magn. Mater. 90&91\n(1990) 234.\n26) T. Tonegawa, H. Matsumoto, T. Hikihara, and M. Kaburagi:\nCan. J. Phys. 79(2001) 1581.\n27) T. Tonegawa and M. Kaburagi: J. Magn. Magn. Mater. 272-\n276(2004) 898.\n28) M. Kaburagi, T. Tonegawa, and M. Kang: J. Appl. Phys. 97\n(2005) 10B306\n29) T. Hamada, J. Kane, S. Nakagawa, and Y. Natsume: J. Phys.\nSoc. Jpn. 57(1988) 1891\n30) T.Tonegawa and I.Harada: J. Phys.Soc. Jpn. 58(1989) 2902\n31) A. F. Albuquerque, et al.: J. Magn. Magn. Mater. 310(2007)\n1187 (see also http://alps.comp-phys.org)." }, { "title": "0811.2455v1.Half_metallic_ferrimagnet_formed_by_substituting_Fe_for_Mn_in_semiconductor_MnTe.pdf", "content": "arXiv:0811.2455v1 [cond-mat.mtrl-sci] 15 Nov 2008physica statussolidi, 4 November2018\nHalf-metallic ferrimagnet formed\nby substituting Fe for Mn in\nsemiconductorMnTe\nLi-Fang Zhu1,2*, Bang-Gui Liu1,2.\n1Institute of Physics,Chinese Academy of Science, Beijing1 00190, China\n2BeijingNational Laboratory forCondensed Matter Physics, Beijing100190, China\nReceivedXXXX, revisedXXXX,accepted XXXX\nPublishedonline XXXX\nPACS75.90.+w, 75.50.Pp, 75.47.-m, 75.30.-m\n∗Corresponding author: e-mail lfzhu@aphy.iphy.ac.cn , Phone +86-10-82649438, Fax+86-10-62553698\ne-mailbgliu@aphy.iphy.ac.cn , Phone +86-10-82649437, Fax+86-10-62553698\nA ternary ferrimagnetic half-metal, constructed through s ubstituting 25% Fe for Mn in zincblende semiconductor\nMnTe, is predicted in terms of accurate first-principles cal culations. It has a large half-metallic (HM) gap of 0.54eV\nand its ferrimagnetic order is very stable against other mag netic fluctuations. The HM ferrimagnetism is formed\nbecause the complete moment compensation in the antiferrom agnetic MnTe is replaced by an uncomplete one in\ntheFe-substitutedMnTe.Thisshouldmakeanovelapproacht onewHMmaterials.Thehalf-metalcouldbefabricated\nbecauseFe hasgoodaffinitywithMn,andusefulforspintroni cs.\nCopyrightlinewillbe provided by the publisher\n1 Introduction Half-metallic(HM)ferromagnetshave\nattracted much attention because they have band gaps at\nthe Fermi energy for one electronic spin channel and are\nmetallicfortheotherchannel[1,2].AlotofHMferromag-\nnetic(FM)materialshavebeenfound[3,4,5,6,7,8,9].Ac-\ncurate first-principles calculations have revealed HM fer-\nromagnetism in binary transition metal chalcogenidesand\npnictides in the zincblende and wurtzite structures[10,11 ,\n12,13,14,15]. It is excitingthat a Singaporegroup,stimu-\nlated by the theoretical prediction of zincblende CrTe (z-\nCrTe)[14,15], has fabricated z-CrTe samples of 100 nm\nthickness[16]. It has also been reported that half-metalli c\nferrimagnets can be formed by introducing Cr antisites in\nCrAs or CrSb[17]. It is still highly desirable to search for\nnovelsemiconductor-compatiblehalf-metalswithhighCur ie\ntemperatureforpotentialspintronicapplications[18].\nMagneticmaterialswithandbasedonzincblendestruc-\ntureareveryinterestingtospintronicapplications.Zinc ble-\nndeMnTe(z-MnTe)isoneofafewantiferromagnetic(AF-\nM)semiconductors.AlthoughMnTecrystallizesintoaNiAs\nphase, the metastable z-MnTe has been grown by molecu-\nlarbeamepitaxy(MBE)growthtechnique[19]andsemibulk\n(about 1 micrometer thick) film samples of z-MnTe have\nbeen fabricated[20] because z-MnTe is only 0.02eV performulaunithigherintotalenergythantheNiAs-typeMnTe.\nTernaryCr-dopedNiAs-typemanganesetellurides,Mn 1−x-\nCrxTe, withxbeing up to 14%, have been fabricated, in\nwhichthesubstitutionofCrforMnleadstoachangefrom\nan AFM semiconductor of MnTe to a FM (or ferrimag-\nnetic) semiconductor of Mn 1−xCrxTe[21]. Therefore, z-\nMnTe should be an interesting novel approach to explore\npromisingmagneticsemiconductorsandHM compounds.\nInthispaper,weperformfirst-principlesstudyonstruc-\ntural, electronic, and magnetic properties of the 25%-Fe-\ndoped z-MnTe. The substitution of Fe for Mn results in a\ntransition from the AFM semiconductor of z-MnTe to the\nferrimagnetichalf-metalof Mn 3FeTe4. We understandthe\nmechanism of the magnetism and the magnetic transition\nthrough investigating the atomic and electronic structure s\nofMn3FeTe4incomparisonwiththoseofz-MnTe.\nThe remaining part of this paper is organized as fol-\nlows. In next section we present our computational detail.\nIn the third section we shall present our optimized results\nof crystal structuresand investigatethe stability of the f er-\nrimagnetism against magnetic fluctuations. In the fourth\nsection we shall present the electronic structures and dis-\ncuss the mechanism for the half-metallic ferrimagnetism.\nCopyrightlinewillbe provided by the publisher2 Li-Fang Zhu and Bang-Gui Liu: Half-metallicferrimagnet f ormed bysubstituting Fefor Mninsemiconductor MnTe\nFinally we shall make some discussions and give our con-\nclusion.\n2 Computationaldetail Toperformthecalculations,\nwe use the package WIEN2K[22], which is based on full-\npotential linearized augmented plane wave method within\nthedensity-functionaltheory(DFT)[23].ThePerdew-Bur-\nke-Ernzerhof1996version[24] of the generalizedgradient\napproximation(GGA)isusedfortheexchange-correlation\npotential. Full relativistic effects are calculated for co re\nstates, and the scalar relativistic approximationis used f or\nvalence states. We investigate the effect of the spin-orbit\ncoupling,butstillpresenttheresultswithoutspin-orbit cou-\npling in the following because it does not affect our main\nconclusions. For different magnetic structures we use dif-\nferent but appropriate k points in the first Brillouin zones\nandmaketheexpansionupto l=10inmuffintins. Rmt×Kmax\nis set to 8.5 for z-MnTe and to 7.0 for Mn 3FeTe4with-\nout affecting our conclusions. The self-consistent calcul a-\ntions are considered to be converged when the integrated\ncharge differenceper formula unit between input and out-\nputchargedensityislessthan0.0001.\n3 Optimizedcrystalstructures Recentinelasticne-\nutron-scatteringexperimenthasrevealedthatthestablem a-\ngnetic structure of z-MnTe is collinear type-III AFM or-\nder of Mn spins in a double conventionalunit cell[25,26],\nrather than early type-I AFM order in single conventional\nunitcell[27]ornoncollineartype-IIIAFMordersuggested\nin terms of previous neutron-diffractionresult[20]. Ther e-\nfore, we consider only collinear spin configurationsin the\nfollowing.\n(d1)\n (d0) (e0)\nFigure 1 (color online). AFM-I ( d0) and AFM-III ( e0)\nstructures of zincblende MnTe and the most stable struc-\nture (d1) of Mn 3FeTe4. The black (red) ball denotes Fe\n(with arrow), the grey (blue) one Mn (with arrow) or Te.\nThearrowsrepresentthespinsonthesites.\nFive spin configurations a0,b0,c0,d0ande0can be\nconstructedforz-MnTe. a0isaFMstructurewithfourMn\nmomentsbeinginparallel. b0andc0,obtainedbyreversing\none Mn moment respectivelyat the face-centerand on theTable1Thespacegroups(SG),themagneticorders(MO),\nthe lattice constants ( aora/c), the relative energy Er(de-\nfined with respect to the lowest structure for the same for-\nmula), the absolute value of total magnetic moment ( M),\nand the Kohn-Sham gaps ( Eg) or the HM gaps ( Eh).Er\nandMare normalizedin termsof those of d1for compar-\nison.\nz-MnTe\n(a0) (b0) (c0) (d0) ( e0)\nSG 216 215 215 111 122\nMO FM FM FM AFM AFM\na(/c˚A) 6.393 6.314 6.314 6.290 6.290/12.580\nEr(eV) 0.712 0.196 0.196 0.020 0\nM(µB) 20.000 10.00 10.00 0.000 0.000\nEg(eV) −0.90 0.90 1.30 1.35\nMn3FeTe4\n(a1) (b1) (c1) (d1) ( e1)\nSG 215 215 111 111 35\nMO FM FM FM FM AFM\na(/c˚A) 6.305 6.244 6.251 6.223 8.822/12.476\nEr(eV) 0.764 0.114 0.222 0 0.125\nM(µB) 18.226 11.000 9.000 1.000 0.000\nEh(eV) −0.21 0.16 0.54 −\nvertex of a0, are ferrimagnetic structures and equivalent\nwith each other. d0ande0are the type-I AFM (AFM-I)\nstructurewithsingleconventionalcell[27]andthetype-I II\nAFM (AFM-III) structure with double unit cells[25], re-\nspectively, as shown in Fig. 1. By substituting Fe for Mn\non the vertex in the single MnTe cell of a0∼d0, we get\ncorrespondinglyfour FM (or ferrimagnetic) structures a1,\nb1,c1, andd1of Mn3FeTe4.d1is the most stable among\nthem andis shown in Fig. 1. Generallyspeaking,to get an\nAFM structure we construct a supercell of two unit cells\nandmakethemomentsinoneunitcelloppositetothosein\nthe other. a1∼d1are all possible FM (or ferrimagnetic)\nstructures one can construct without enlarging the mag-\nnetic unit cell. We construct all possible AFM structures\nbased on them, and the results for the most stable AFM\nstructure e1(astherepresentative)areshownin Table1.\nAll the above structures, both FM and AFM, are opti-\nmizedfully.Themomentandelectronicstructuresare cal-\nculated with the lattice constants of the optimized struc-\ntures. Our calculated results are summarized in Table 1. It\nis clear that the most stable structuretendsto havea small\nequilibrium lattice constant. As is shown in Table 1, the\ntwo FM structures ( a0anda1) and the four ferrimagnetic\nstructures ( b0,c0,b1andc1), having large magnetic mo-\nments, areunfavorablein total energy.For z-MnTe,AFM-\nIIIe0and AFM-I d0, with the total moments being 0, are\nfavorable in total energy,and e0is 5meV per formula unit\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3\nTable 2The partial magnetic moments ( µB) projected in\nthemuffin-tinspheresofMn1,Mn2,Fe,andTeatomsand\nintheinterstitialregion(Inter)andthetotalmoment(Tot al)\ninthemoststablestructuresz-MnTe e0andMn 3FeTe4d1.\nz-MnTee0Mn3FeTe4d1\nMn1 4.180 4.130\nMn2 -4.180 -4.114\nFe − 3.157\nTe 0.000 0.036\nInter 0.000 -0.095\nTotal 0.000 -1.000\nlower than d0, being in agreement with experimental fact\nthate0istheground-statephaseofz-MnTewithasemicon-\nducting gap of about 3.2eV[19]. The most stable structure\nforMn 3FeTe4, however,is notanyAFM structure,but the\nferrimagneticstructure d1withanabsolutetotalmomentof\n1.000µB. It is lower by 0.125eV per formula unit in total\nenergythanthelowestAFM structure e1.\nWesummarizethepartialmagneticmoments( µB)pro-\njected in the muffin-tin spheres of Mn1, Mn2, Fe, and Te\natomsandintheinterstitialregioninthemoststablestruc -\ntures z-MnTe e0and Mn 3FeTe4d1in Table 2. The cor-\nrespondingtotal magnetic moments also are presented for\ncomparison.ItisworthnotingthattherearetwoMn1atoms\nand two Mn2 ones in z-MnTe e0, but we have one Mn1\natom, one Fe atom, and two Mn2 atoms in Mn 3FeTe4d1,\nasshowninFig.1.Itisobviousthatthepartialsubstitutio n\nof Fe for Mn leads to the transferring of a little magnetic\nmomentsfromtheMnatomstotheTeatomsandtheinter-\nstitial region. Mn 3FeTe4d1has a total moment of -1.000\nµBbecause Fe has one more delectron, or one µBless\nmagneticmoment,thanMn.\n4 Electronicstructuresandmagneticmechanism\nThe spin-dependent density of states (DOS) of the AFM-\nIII MnTe are presented in Fig. 2( a). The primitive cell of\nAFM-III MnTe consists of 2 Mn1 (with spin up), 2 Mn2\n(with spin down), and 4 Te atoms. The valence bands are\nformedby 10 dand 12pstates. The 10 lowest conduction\nbands originate from Mn dstates. The Mn moments are\ncoupledwithasuperexchangeinteractionthroughthenear-\nest Te atoms, which yields the antiferromagnetism. The\nspin exchange splitting is about 4.7eV, as shown in Fig.\n2(a).\nThespin-dependentdensityofstates(DOS)andenergy\nbandsof the Mn 3FeTe4are presentedin Fig. 2( b) and Fig.\n3,respectively.TheFermilevel EFissettozero.Thefilled\nbands between -5eV and -0.6eV, consisting of 10 dstates\nand12pstatesforeachspinchannel,aresimilartothoseof\ntheMnTe.Themaindifferenceisthattherearepartly-filled\nmajority-spin bands across the Fermi level in the case of\ntheMn 3FeTe4.Theminority-spinbandsstill haveagapof/s49/s50/s54/s48/s54/s49/s50/s49/s50/s54/s48/s54/s49/s50\n/s45/s52 /s45/s50 /s48 /s50/s49/s50/s54/s48/s54/s49/s50/s32/s116/s111/s116/s97/s108\n/s32/s70/s101\n/s32/s77 /s110/s49\n/s32/s77 /s110/s50\n/s32/s84/s101\n/s32/s105/s110/s116/s101/s114/s32\n/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s32/s116/s111/s116/s97/s108\n/s32/s77 /s110/s49\n/s32/s77 /s110/s50\n/s32/s84/s101\n/s32/s105/s110/s116/s101/s114/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115\n/s40 /s99 /s41/s40/s98 /s41 /s40/s97 /s41 \n/s32/s32/s116/s111/s116/s97/s108\n/s32/s67/s114\n/s32/s77 /s110/s49\n/s32/s77 /s110/s50\n/s32/s84/s101\n/s32/s105/s110/s116/s101/s114\nFigure 2 (coloronline).Spin-dependenttotal(thicksolid)\nand partial DOS (state/eV per formula unit) for the AFM-\nIIIMnTe( a),themoststable d1structureofMn 3FeTe4(b),\nand the most stable structure of Mn 3CrTe4(c). The upper\nhalf of each panel is DOS for majority spin and the lower\none for minority spin. The partial DOS are those in Fe/Cr\n(dot),Mn1(dash),Mn2(longdash),andTe(dotdash)muf-\nfin tinsandininterstitial region(thinsolid).\n1.12eV,alittlesmallerthantheKohn-Shamgap,1.35eV,of\ntheMnTe.TheMn 3FeTe4has0.54eVasitsHMgapwhich\nisdefinedasthesmallerof Ec\nmin-EFandEF-Ev\nmax,where\nEc\nminis the bottom of the minority-spinconductionbands\nandEv\nmaxthetopoftheminority-spinvalenceones[14,15].\nThe 25% Fe substitution for Mn results in a cell con-\nsistingof2Fe,6Mn,and8Teatoms.Thespinorientations\ncannotremainthesameasthoseofAFM-IIIMnTebecause\nFe has one more delectron than Mn. Instead, the mag-\nneticorderisreorganizedsothatthe16-atomcellisdivide d\nintotwoequivalentsmaller8-atomoneswhichwouldhave\nAFM-I structure if we neglect the difference between Fe\nand Mn. As a result, we obtain a ferrimagnetic order be-\ncause the Fe moment cannot completely compensate the\nopposite Mn moment. The substitution does not substan-\ntiallychangethevalencebands,butmovessomemajority-\nspindstates downwards with respect to those of the z-\nMnTebecausetheFe dstatesarealittlelowerthanthoseof\nCopyrightlinewillbe provided by the publisher4 Li-Fang Zhu and Bang-Gui Liu: Half-metallicferrimagnet f ormed bysubstituting Fefor Mninsemiconductor MnTe\nΓX M ΓZ R A Z E F Energy (eV) 0.0 1.0 2.0 3.0\n -1.0\n -2.0\n -3.0\n -4.0\n -5.0\nΓX M ΓZ R A Z 0.0 1.0 2.0 3.0\n -1.0\n -2.0\n -3.0\n -4.0\n -5.0\nFigure 3 Spin-dependent energy bands (plotted with cir-\ncles) of the d1structure of Mn 3FeTe4. A larger circle im-\nplies more Fe dcharacter. The left panel is for majority-\nspinandthe rightoneforminority-spin.\nMndonesinenergy.Themajority-spinbandsattheFermi\nlevel, belonging to a doublet, are half-filled because there\nis only one electron for them. The HM ferrimagnetism is\nachievedbecausewestillhaveagapacrosstheFermilevel\nforminority-spinchannel.\nWehavestudiedsimilar25%-Cr-dopedMnTe,Mn 3Cr-\nTe4. Its stable structure also exhibits HM ferrimagnetism.\nTheresultsforMn 3CrTe4areconsistentwithNakamura et\nal’s through doping 75% Mn into z-CrTe [28]. The spin-\ndependentdensityof states forthe moststable structureof\nMn3CrTe4are also given in Fig. 2( c). Cr has four delec-\ntrons, one less than Mn. The Cr dstates are a little higher\nthan those of Mn in energy, which results in the partially\noccupiedCrimpuritybandsinthemajority-spinbandsand\ntheopengapin theminority-spinbands.\nBycomparingtheDOSsoftheMn 3FeTe4andMn 3Cr-\nTe4in Fig. 2, we can explain the origin of their ferrimag-\nnetismuniformlyaccordingtothenumberof delectronsin\nthetransitionmetalatomsandtheenergylevelsof dstates.\nThe substitution of Fe for Mn or Cr for Mn changes the\ndistribution of dstates at the fermi level and results in the\nferrimagnetism.\n5 Discussion and conclusion All of our presented\nresults are calculated with GGA, although local density\napproximation (LDA) yields almost the same results. It\nis worth noting that a developed single-ion implantation\ntechnique recently was used to implant dopant ions one-\nby-oneintoa semiconductor[29].Thatis, boththenumber\nand the position of the dopantatoms in the semiconductor\nare precisely controlled. As a result, the promising half-\nmetals predicted in this paper could be realized by using\nsuchtechniques.\nIn summary, we have predicted a ternary half-metal\nMn3FeTe4, constructedbysubstitutingFe forMnin semi-\nconductorz-MnTe,intermsofouraccuratefirst-principlescalculations. The substitution results in a transition fro m\nthe AFM semiconductor MnTe to the HM ferrimagnet of\nthe Mn 3FeTe4. The HM ferrimagnetism is stable against\nantiferromagneticfluctuations.ThelargeHMgapimpliesa\npossiblehighCurietemperature[30].TheMn 3FeTe4could\nbefabricatedexperimentallysoonbecauseofthegoodaffin-\nity ofFeto Mn,andit couldbeusedin spintronics.\nAcknowledgements ThisworkissupportedbyNatureSci-\nenceFoundationofChina(GrantNos.10874232,10774180,90 406010,\nand60621091),bytheChineseAcademyofSciences(GrantNo. KJC-\nX2.YW.W09-5),andbyChineseDepartmentofScienceandTech -\nnology (GrantNo. 2005CB623602).\nReferences\n[1] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M.\nDaughton, S. von Molnar, M.L. 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B 68,\n134407 (2003).\n[14] W.-H. Xie, Y.-Q. Xu, B.-G. Liu, and D.G. Pettifor, Phys.\nRev. Lett. 91, 037204 (2003).\n[15] B.-G. Liu, in: Half-metallic Alloys-Fundamentals and Ap-\nplications, edited by I Galanakis and P. H. Dederichs, Lec-\nture Notes in Physics Vol. 676, (Springer, Berlin, 2005),\npp.267-291.\n[16] M.G. Sreenivasan, K.L. Teo, M.B. A. Jalil, T. Liew, T.C.\nChong, and A.Y. Du, IEEE Transactions on Magnetics 42,\n2691 (2006).\n[17] I. Galanakis, K. Ozdogan, E. Sasloglu, and B. Aktas, Phy s.\nRev. B74, 140408(R) (2006).\n[18] C.M.Fang,G.A.deWijs,andR.A.deGroot,J.Appl.Phys.\n91, 8340 (2002).\n[19] S.M. Durbin, J. Han, O. Sungki, M. Kobayashi, D.R.\nMenke, and R.L. Gunshor, Appl. Phys. Lett. 55, 2087\n(1989).\n[20] T.M.Giebultowicz,P.Klosowski,N.Samarth,H.Luo,J. K.\nFurdyna, and J.J.Rhyne, Phys.Rev. B 48, 12817 (1993).\n[21] Y.B.Li,Y.Q.Zhang, N.K. Sun,Q. Zhang, D.Li,J.Li,and\nZ.D.Zhang, Phys. Rev. B 72, 193308 (2005).\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 5\n[22] P. Blaha, K. Schwarz, P.Sorantin, and S.B. Trickey, Com p.\nPhys. Comm. 59, 399 (1990).\n[23] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);\nW.Kohn and L.J.Sham, Phys.Rev. 140, A1133 (1965).\n[24] J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett .\n77, 3865 (1996).\n[25] B. Hennion, W. Szuszkiewicz, E. Dynowska, E. Janik, and\nT. Wojtowicz,Phys.Rev. B 66, 224426 (2002).\n[26] S.-H.Weiand A.Zunger, Phys. Rev. B 48, 6111 (1993).\n[27] S.-H.Weiand A.Zunger, Phys. Rev. Lett. 56, 2391 (1986).\n[28] K. Nakamura, T. Ito, and A.J. Freeman, Phys. Rev. B 72,\n064449 (2005).\n[29] S.Takahiro, O.Shintaro, K.Takahiro, O. Iwao, Nature 437,\n1128 (2005).\n[30] J. Kubler,Phys. Rev. B 67, 220403 (2003).\nCopyrightlinewillbe provided by the publisher" }, { "title": "1301.4754v3.Ferrimagnetism_of_dilute_Ising_antiferromagnets.pdf", "content": "arXiv:1301.4754v3 [cond-mat.dis-nn] 18 Apr 2013Ferrimagnetism of dilute Ising antiferromagnets\nP. N. Timonin∗\nSouthern Federal University, 344090, Rostov-on-Don, Russ ia\n(Dated: September 3, 2018)\nIt is shown that nearest-neighbor antiferromagnetic inter actions of identical Ising spins on im-\nbalanced bipartite lattice and imbalanced bipartite hiera rchical fractal result in ferrimagnetic order\ninstead of antiferromagnetic one. On some crystal lattices dilute Ising antiferromagnets may also be-\ncome ferrimagnets due to the imbalanced nature of the magnet ic percolation cluster when it coexists\nwith the percolation cluster of vacancies. As evidenced by t he existing experiments on FepZn1−pF2,\nsuch ferrimagnetism is inherent property of bcc lattice so t hermodynamics of these compounds at\nlowpcan be similar to that of antiferromagnet on imbalanced hier archical fractal.\nThe system of the identical Ising spins on the sites\nof some crystalline lattices with the nearest-neighbor\nantiferromagnetic (AF) exchange may have magnetized\nground states. In such states there would be antipar-\nallel neighboring spins, as interaction dictates, but the\nwhole numbers of up-spins and down-spins would dif-\nfer. One such 2 dlattice is shown in Fig. 1(a). Here\ntwo sublattices with parallel up and down spins in the\nground state are shown by filled and empty circles cor-\nrespondingly. We see that in the unit cell there are\none filled circle and two empty ones so we get ±1/3\nmagnetizations in two globally-reversed ground states\nfor nearest-neighbor AF on this lattice. Thus this AF\nmodel has a couple of ferrimagnetic ground states with\nboth staggered L= (/angbracketleftSA/angbracketright−/angbracketleftSB/angbracketright)/2 and homogeneous\nM= (2/angbracketleftSA/angbracketright+/angbracketleftSB/angbracketright)/3 magnetizations.\nOne can easily show that this ordering persists up to\nfiniteTc. Summing the Gibbs function over spins on sub-\nlatticeA(empty circles)wegetthe Gibbs distributionfor\nthe spins on the sublattice B having effective ferromag-\nnetic Hamiltonian. Indeed, for each link with SAspin we\nhave (Jis AF exchange)\n/summationdisplay\nSA=±1exp[−SA(SB+S′\nBJ/T)] = 2cosh[( SB+S′\nBJ/T)]\n= 2exp[KB(SBS′B+1)],2KB≡lncosh(2 J/T)\nHence, the ordering of SBspins is described by the ferro-\nmagnetic Ising model on the square lattice, so /angbracketleftSB/angbracketright /negationslash= 0\nforKB>1\n2ln/parenleftbig√\n2+1/parenrightbig\n[1] or\nT < T c= 2J/ln/parenleftBigg\n√\n2+1+/radicalbigg\n2/parenleftBig√\n2+1/parenrightBig/parenrightBigg\n.\nAs/angbracketleftSB/angbracketrightis a linear combination of LandM, the ordered\nphase is a ferrimagnetic one. This implies that homoge-\nneous magnetic field Hhas a part conjugated with the\norder parameter so magnetic susceptibility diverges at Tc\nwhenH= 0and the transition becomessmearedat finite\nH.\nThus we have a simple example showing that nearest-\nneighbor AF interaction of identical Ising spins may re-\nsult in the macroscopic ferrimagnetic order. This is inab\ncd\nA\nAB\nFIG. 1. Examples of imbalanced bipartite graphs with dif-\nferent numbers of sites in sublattices A (open circles) and B\n(filled circles), NA> NB. (a) - fragment of regular 2 dlattice,\ndotted line shows the unit cell; (b, c, d) - clusters of dilute\nsquare lattice. In the ground state short-range Ising AF on\nthem would have parallel spins on A and B sublattices and,\nhence, a nonzero magnetization.\napparent distinction with conventional ferrimagnets hav-\ning several different magnetic moments in a cell. It may\nlook rather exotic in the realm of real crystals yet such\nsituation can be frequent in disordered Ising AF, first of\nall, in dilute Ising AF on bipartite lattices. These lattices\ncan be divided in two subsets of sites, A and B, such that\nall bonds are of the A-B type, i. e. there are no bonds\ninside A and B subsets [2]. Apparently, the Ising AF on\nsuch lattice is non-frustrated having all spins up on sub-\nlattice A and down on sublattice B or vice versa in its\ntwo degenerate ground states. Their magnetizations are\nm=±NA−NB\nNA+NB=±1−η\n1+η, η≡NB\nNA<1\nHereNAandNBare the numbers of sites in A and B\nsublattices and we choose η <1 for definiteness.\nSeemingly, all known non-frustrated AF crystals with\njust one sort of magnetic ions have bipartite lattices that\nare the balanced ones, that is with η= 1 and purely AF\nground states, while Fig. 1(a) shows the imbalanced bi-\npartite lattice with η <1 (η= 0.5). Yet the dilution\nof balanced bipartite lattices results in appearance of a\nnumberofisolatedclusters,mostlywith η <1,thosewith\nη= 1being the rareexceptions. Figs. 1(b, c, d) show the\nimbalanced clusters on the square lattice. So at T= 0\nand arbitrarily small magnetic field dilute bipartite AF2\nmust show nonzero magnetization due to the presence of\nsuch imbalanced finite clusters. This circumstance was\nfirst noticed by Neel [3]. Still it stays unnoticed that for\nsome concentrations of magnetic ions pthe giant perco-\nlation cluster may also have the average imbalance ratio\nηp<1.\nIndeed, in finite sample the role of percolation cluster\nbelong to that with the largest number of sites and very\nprobably it is imbalanced, as most of them. However,\nin the thermodynamic limit ( N→ ∞)ηpwill tend to\nunity if there are only finite clusters of vacancies. Ap-\nparently, such finite clusters cannot make infinite lattice\nimbalanced as for every cluster deleting unequal number\nof sites from A and B sublattices there exists (with the\nsame probability) the shifted cluster of the same form\nwhich restores the balance. Thus at 1 −pc< p <1 the\nimbalanced percolation cluster can only exist as a finite-\nsize effect. Meanwhile, at pc< p <1−pcthere is infinite\npercolation cluster of vacancies to which this argument\ndoes not apply. Hence, here ηp<1 may also hold in the\nN→ ∞limit in some crystal lattices. Then the ground\nstate magnetization of dilute AF in this interval will be\nmp(H= +0) =1/integraldisplay\n01−η\n1+ηWp(η)dη+1−ηp\n1+ηp\nHereWp(η) is the imbalance distribution function of fi-\nnite clusters. Nowit seems that neither Wp(η) norηpare\nknown for the crystal lattices. So to find them is quite\nrelevant task for the physics of dilute short-range AF.\nThe magnetization of finite clusters vanishes at finite\ntemperatures, but that of the imbalanced percolation\ncluster would persist up to a finite TcandM∼Lat allT\ndue to its geometrical origin. Then macroscopic features\nof dilute AF would be those of ordinary ferromagnet in\nspite of the presence of antiparallel neighboring spins in\nthe ordered phase. In such a case, the mapping of this\nmodel onto random-field Ising magnet (RFIM) [4] would\nbe no longer valid as it suggests purely AF transition in\nDAFF. This possibility of AF order breaking is missed in\nRef. [4] which is a consequence of the mean-field treat-\nment of homogeneous magnetization.\nThe evidences in favorof DAFF ferrimagnetismcan be\nfound in experiments on several dilute Ising AF with p <\n1−pcshowingtheremanentmagnetizationwiththeusual\norder-parameter behavior [5]-[9] and prominent peak in\ntemperaturedependence ofmagneticsusceptibilitywhich\nappears in low fields as a result of dilution and becomes\ngradually smeared in higher fields [8],[9].\nWe should note that on the lattices having perfectly\nbalanced percolation cluster with ηp= 1 DAFF also have\na ferrimagnetic phase in its ground state. The differ-\nence with the imbalanced case is that it appears above\nsome finite critical field HAF(p) while this field is zero if\nηp<1. The schematic ground state phase diagrams are\nshown in Fig. 2. The validity of these pictures followsfromquitesimpleconsiderations. Letusconsidertheper-\nfectly balanced AF percolation cluster. As it necessary\nhas some imbalanced (magnetized) parts, the field will\ninduce the energy-reducing global flipping of their spins\nif the magnetic moment Mof the part points opposite to\nthe field and His greater thanB\nMJ. HereBis the num-\nber of AF bonds connecting the given part with the rest\nof percolation cluster, Jis AF exchange. First the large\nclusters with smallB\nMratio will be flipped in low fields\nwhile the field growth will induce the flipping of smaller\nand smaller ones. At last the remaining single spins flip\nalong the field at H=zJ(zis the lattice coordination\nnumber). The corresponding jumps of sublattice magne-\ntizations are seen in the numerical study of the ground\nstate of 3 d(simple cubic) DAFF with pas large as 0.9\nandH >2J[10].\n0 1pH\nFerromagnet\nImproper\nFerrimagnet\nAF\npc(a)\n0 1pH\nFerromagnet\nFerrimagnetAF\npc(b)zJzJ\n1-p cImproper\nFerrimagnet\nFIG.2. Schematicgroundstate H−pphasediagrams ofdilute\nAF. (a)ηp= 1 for all p, (b)ηp<1 forp <1−pc. The lines\nbetween phases are defined by HAF(p) andHF(p) discussed\nin text. In the improper ferrimagnetic regions sharp AF tran -\nsition is preserved at finite Tcwhile it becomes smeared fer-\nrimagnetic one at H >0 in genuine ferrimagnetic region in\n(b).\nApparently, this process results in appearance of a\nnonzero magnetization of percolation cluster in fields\nabove some HAF(p) and vanishing of its staggered mag-\nnetizationabovesomegreaterfield HF(p). Sothe ground\nstate at HAF(p)< H < H F(p) is ferrimagnetic. Yet in\nthis case Mappears at Tcas a secondary order param-\neterM∼L2and here sharp AF transition is preserved3\nas well as DAFF-RFIM mapping. So we may call this\nphase ’improper ferrimagnetic’ to distinguish it from the\ngenuine ferrimagnetic one in Fig 2b.\nStill the improper ferrimagnetic ground state would\ncause a drastic change in the dynamics of AF phase.\nThis is the consequence of huge degeneracy of the fer-\nrimagnetic ground state as at rational H/Jthere can be\na huge amount of parts of the percolation cluster with\nH/J=B/Mso their flipping does not change the en-\nergy. This degeneracy is explicitly demonstrated in nu-\nmerical studies of realistic DAFF systems [10], [11]. At\nfiniteTthis results in many (nearly) degenerate min-\nima of thermodynamic potential so the system can be\ntrapped in each of them, depending on the previous his-\ntory ofTandHvariations. The particular manifestation\nof these phenomena is the difference between field-cooled\nand zero-field-cooled thermodynamic parameters. Ap-\nparently, it would be also present in the ferrimagnetic\nphase of imbalanced DAFF right down to H= 0.\nConcerning the behavior of HAF(p) andHF(p) in Fig\n2 we can note that it is quite apparent that HAF(1) =\nHF(1) =zJwhile their diminishing to zero at p=pc\nin Fig 2a is the consequence of sparse structure of per-\ncolation cluster near pc. Here it is divided into loosely\nconnected parts with B/M→0 so their flipping fields\nalso go to zero resulting in HAF(pc) =HF(pc) = 0. In\nFig 2bHAF(p) seized to exist at p= 1−pcwhen, ac-\ncording to our surmise, ηpbecomes less than 1 in some\nlattices.\nThe notion of HAF(p) andHF(p) behavior one can get\nfrom the results of extensive numerical studies of DAFF\nground state on simple cubic and bcc lattices [11]. Here\nthe boundaries of the so called ”domain state” are de-\ntermined. In this state the percolation cluster of the\nflipped spins coexists with that of unflipped ones. Its\nupper boundary coincides with HF(p) while the lower\none can be somewhat higher than HAF(p), yet its behav-\nior for the simple cubic lattice [11] resembles that in Fig.\n2a. So, most probably, this lattice has ηp= 1 for all p.\nThe results for bcc lattice are less conclusive, here the\npercolation cluster of flipped spins can appear at rather\nlow fields, depending on the boundary conditions and\ndisorder realization [11]. This makes bcc lattice a valid\ncandidate for having ηp<1 (andHAF(p) = 0 ) at some\np > pc.\nTo get some notion of the DAFF thermodynamics in\nthe ferrimagnetic phase which may result from ηp<1\natp <1−pcwe consider here the nearest-neighbor AF\non the simplest hierarchical lattice, imitating the perco-\nlation cluster with fractal dimension d= 2 and η= 1/3.\nAs well, it may describe qualitatively large planar aggre-\ngates of AF particles or disordered AF thin films which\nmay have the imbalanced structure of a set of magnetic\nions. The model also exhibits a number of field-induced\nground state transitions marked by the magnetization\njumps which are discussed above.LOW-FIELD THERMODYNAMICS OF\nHIERARCHICAL ANTIFERROMAGNET\nWe consider the short-range Ising AF on the sim-\nplest ”diamond” hierarchical lattice [12]. Its building\nprocess is shown in Fig.3. On the n-th level of hier-\narchy the lattice has Nnsites,Nn=2\n3(4n+2), see\nRef. [13]. The coordination numbers of the sites are\nthe powers of 2: z= 2,4,8,.... At all levels of the hi-\nerarchy the lattice is bipartite and for n >0 the sites\nwith coordination number z= 2 constitutes the sub-\nlattice A (open circles in Fig.3) while the others be-\nlong to the sublattice B (filled circles), NA,n≥NB,n.\nAt then-th level NA,n= 2·4n−1forn >0 [13], so\nηn= (Nn−NA,n)/NA,n=/parenleftbig\n1+2·41−n/parenrightbig\n/3. We are in-\nn=0 n=1 n=2AA\nB B\nFIG. 3. Construction of hierarchical lattice. It is biparti te\nat all levels. Different circles designate its partitioning , open\ncircles sublattice A, filled circles sublattice B, NA≥NB.\nterested in the thermodynamic limit of infinite levels of\nhierarchy. In this limit η= 1/3 and fractal dimension\nd= 2 [13].\nFor the Ising spins Si=±1 placed on the sites of this\nlattice we consider the AF Hamiltonian\nH=J/summationdisplay\nSiSj−HA/summationdisplay\ni∈ASi−HB/summationdisplay\nj∈BSj(1)\nwhere< i∈A,j∈B >means the summation over near-\nestneighborsanddifferentfieldsforthesublatticesarein-\ntroduced. This allows to find the averagemagnetizations\nof each sublattice and the order parameter for the transi-\ntion. Homogeneous field corresponds to HA=HB=H.\nThe usual way to get the partition function of the\nmodel is through the recursion relations for partial par-\ntition functions at different levels of hierarchy Zn(S,S′)\nhaving fixed values of the outmost left and right spins S\nandS′[12]. These relations read\nZn+1(S,S′) =/bracketleftBigg/summationdisplay\nS1=±1Zn(S,S1)ehnS1Zn(S1,S′)/bracketrightBigg2\n,(2)\nhn=Hn/T, H 0=HAandHn=HB, n≥1. The\ninitial condition for them is\nZ0(S,S′) =e−KSS′, K=J/T. (3)\nUsing the representation\nZn(S,S′) = exp1\n2[Cn+unSS′+(vn−hn)(S+S′)]4\nwe get from Eqs.(2,3)\nu0=−2K, v 0=hA, C 0= 0 (4)\nun+1= 2lncosh un+ln/parenleftbig\n1−tanh2untanh2vn/parenrightbig\n,\nvn+1= 2vn+2tanh−1(tanhuntanhvn)−2hn+hn+1,\nCn+1= 4Cn+un+1+4ln(2cosh vn).\nThe last of Eqs.(4) gives for n >0\nCn=un−4nu0+n−1/summationdisplay\nl=04n−l[ul+ln(2cosh vl)]\nso then-th level free energy per spin is\nFn=−T\nNnln/summationdisplay\nS,S′Zn(S,S′)ehB(S+S′)=\n−3\n4Tn−1/summationdisplay\nl=04−l[ul+ln(2cosh vl)]−3\n2J+O(1/Nn) (5)\nAtHn= 0 the model has phase transition at K=\nKc≈0.609 being the solution to the equation Kc=\nlncosh2Kc.uc= 2Kcis the stationary point of the\nzero-field equations, un+1= 2lncosh un,vn= 0. In the\nparamagneticphaseat K < K cun→0 forn→ ∞, while\nin the ordered phase at K > K cun→ ∞. According\nto above considerations the ground states of the model\nhave magnetizations ±(1/2), so we may expect that the\nordered phase is ferrimagnetic. To show this we consider\nEqs.(4) at\n0<(Tc−T)/Tc≡τ≪1,|hn| ≪τ.(6)\nIn this case unandvncan be found approximately in the\nthree regions of n:\n1) 1≤n≤λ,un−uc≤uc,|vn| ≪1\nun≈uc+κn−1(u1−uc),(7)\nvn≈hB+(2+κ)n−1κ˜h\n1+κ,˜h=hB−(κ+1)hA(8)\nu1= 2lncosh2 K, κ = 2tanh2 Kc≈1.68,\nThe value of λis defined by\nuλ= 2uc, κ−λ=u1−uc\nκuc≈τ (9)\nwhile|vn| ≪1 requires\n|˜h|(2+κ)λ=|˜h|τ−ln(2+κ)/lnκ≪1.(10)2)λ < n≤µ,1≪un≪ |vn|\nun≈2n−λ+1uc−n−1/summationdisplay\nk=λ2n−kln2coshvk,(11)\nvn≈hB\n3+4n−λ/parenleftbigg\nvλ−hB\n3/parenrightbigg\n,(12)\nvλ≈(2+κ)λ−1κ˜h\n(1+κ)=κ˜h\n(1+κ)(2+κ)τln(2+κ)/lnκ(13)\nThe value of µis defined by the equation uµ=\n|vµ|. Asuµ≈2µ−λ+1uc−µ−1/summationtext\nk=λ2µ−kln2cosh4k−λvλ≈\n2µ−λ2(uc−ln2)+4µ−λvλ, vµ≈4µ−λvλ, we get\n2µ=(uc−ln2)2λ\n|vλ|, uµ=|vµ|=(uc−ln2)2\n|vλ|.(14)\n3)µ > n,|vn| ≫1,un≈0,vn≈2n−µvµ.(15)\nNote that in the sums we consider the large numbers\nλandµas integers neglecting its fractional parts.\nUsing the above approximations for unandvnwe can\nfind from Eq.(5) free energy in the thermodynamic limit\nnear the transition point in a small field (cf. Eqs.(6),\n(10)). Thus, dividing the sum in (5) in three parts and\nn= 0 term,\n−4\n3F\nT= ln2cosh hA+Σλ+Σλµ+Σµ,\nwe get\nΣλ=λ/summationdisplay\nn=1/bracketleftBig\n4−n(uc+ln2)+ ucτ/parenleftBigκ\n4/parenrightBign/bracketrightBig\n+\n+λ/summationdisplay\nn=1˜h2\n2(1+κ)2(2+κ)2/parenleftBig\n1+κ\n2/parenrightBig2n\n≈1\n3(ln2+uc)+κ\n4−κucτ−4−λ\n3/parenleftbigg2+κ\n4−κ2uc+ln2/parenrightbigg\n+κ˜h2\n2(4+κ)(1+κ)2/parenleftBig\n1+κ\n2/parenrightBig2λ\nΣλµ=µ/summationdisplay\nn=λ+1/parenleftbig\n2−n−λ+1uc+4−nln2coshvn/parenrightbig\n−\n−µ/summationdisplay\nn=λ+14−nn−1/summationdisplay\nk=λ2n−kln2coshvk\n=µ/summationdisplay\nn=λ+1/parenleftbig\n2−n−λ+1uc+4−nln2coshvn/parenrightbig\n−\n−µ/summationdisplay\nk=λ/parenleftbig\n4−k−2−k−µ/parenrightbig\nln2coshvk\n= 4−λ2uc−4−µuµ+4−µln2coshvµ−4−λln2coshvλ\n≈4−λ(2uc−ln2)5\nΣµ≈∞/summationtext\nn=µ+14−n|vn| ≈4−µ|vµ|= 4−λ|vλ|.\nHere we used |vλ| ≪1,|vµ| ≫1, Eqs.(7, 8, 11, 12,\n15) and the relation following from Eq.(11),\nµ−1/summationdisplay\nk=λ2−k−µln2coshvk= 2−λ−µ+1uc−4−µuµ.\nFinally we have from Eqs.(5, 9, 13, 14)\nF/Tc≈ −Kc/2−ln2+τsc−aτ2−α−bτβ/vextendsingle/vextendsingle/vextendsingle˜h/vextendsingle/vextendsingle/vextendsingle−cτ−γ˜h2,\n(16)\nsc= ln2−2Kcκ−1\n4−κ≈0.34,a= 2Kc5−2κ\n4−κ−ln2≈0.17,\nb=3κ\n4(2+κ)(1+κ)≈0.13,c=3κ\n8(4+κ)(1+κ)2≈0.015.\nα= 2−ln4\nlnκ≈ −0.67, (17)\nβ=ln4−ln(2+κ)\nlnκ≈0.16,\nγ=2ln(2+ κ)−ln4\nlnκ≈2.35\nIn homogeneous field ˜h=−κh(cf. Eq.(8)) so Fin\nEq.(16) has the standard scaling form of a ferromag-\nnet with spontaneous magnetization m∼τβand diver-\ngent susceptibility χ∼τ−γ. This expression is valid at\n0< τ≪1,|h| ≪τβ+γ, cf. Eq.(10). Scaling indices\n(17) obey the usual relation α+2β+γ= 2. Negative α\nmeans that specific heat is finite at the transition point\nand has a cusp at Tc. Note also that scis the entropy at\nthe transition point. So this AF system looks like gen-\nuine ferromagnet, even featuring the absence (smearing)\nof transition in a finite field. The last is evident as the\nnontrivial stationary point of finite-field recursion rela-\ntions (4) cannot be reached from any initial conditions.\nYet the dependence of Fon˜hfrom Eq.(8) shows that\ntrue order parameter for the transition is a linear com-\nbination of MA=/summationtext\ni∈ASiandMB=/summationtext\ni∈BSiconjugate\nwith˜H=HB−(κ+1)HA. To distinguish the order\nparameter in the Hamiltonian (1) we perform a coordi-\nnate rotation in 2 dspace of vectors H= (HA,HB) and\nM= (MA,MB) to bring the term −MHin (1) to the\nform−MH=−˜M˜H−M′H′where\n˜M=MB−(κ+1)MA\n1+(κ+1)2,M′=(κ+1)MB+MA\n1+(κ+1)2,\nH′= (κ+1)HB+HA\nThus˜Mis the order parameter while M′andH′are\nnon-critical variables. Hence, /angbracketleftM′/angbracketright= 0 atH→0 so the\nspontaneous magnetic moments of the sublattices obey\nthe relation /angbracketleftMA/angbracketright=−(κ+1)/angbracketleftMB/angbracketright. Then for the spon-\ntaneous magnetizations mν=/angbracketleftMν/angbracketright/Nν(ν=A, B) we\nhave\nmA=−η(κ+1)mB≈ −0.9mBThis differs from the ground state relation mA=−mB.\nWe may suggestthat this is a consequenceof criticalfluc-\ntuations diminishing mAmore strongly than mBas all\nsites of sublattice A have the lowest coordination num-\nberzA= 2. To some extent this effect would be present\nin all dilute AF on imbalanced bipartite graphs since the\nsublattice A with larger amount of spins would necessary\nhave lower average coordination number ¯ zA=C/NA<\n¯zB=C/NB. HereCis the number of bonds and we used\nthe fact that all bonds are of A-B type. Thus ¯ zA=η¯zB\nsothe lower ηthestrongercanbe thefluctuation-induced\ndisbalance between mAandmBnearTc. Atη= 1 this\neffect vanishes so its observation in neutron-diffraction\nexperiments can certify the onset of imbalance in the\nmagnetic percolation cluster.\nGROUND STATE TRANSITIONS\nHere we assume HA=HB=H. AtT= 0 we define\n˜un= lim\nT→0Tun,˜vn= lim\nT→0Tvn, E= lim\nT→0Fto obtain\nfrom Eqs.(4, 5)\n˜u0=−2J˜v0=H\n˜un+1= 2(|˜un|−|˜vn|)ϑ(|˜un|−|˜vn|),(18)\n˜vn+1= 2˜vn+2min( |˜un|,|˜vn|)sgn(˜un˜vn)−H,(19)\n−(4/3)E=H+∞/summationdisplay\nn=14−n(˜un+|˜vn|)\nϑinEq.(18)isHeaviside’sstepfunction. SolvingEqs.(18,\n19), we get\n−(4/3)E=H+∞/summationdisplay\nn=14−n/vextendsingle/vextendsingleH−2n+1J/vextendsingle/vextendsingle,\nm=−∂E\n∂H=3\n4/bracketleftBigg\n1+∞/summationdisplay\nn=14−nsgn/parenleftbig\nH−2n+1J/parenrightbig/bracketrightBigg\n.\nSo atH/negationslash= 2kJ\nm= 2−1ϑ(2J−H)+/parenleftbig\n1−2·4−r/parenrightbig\nϑ(H−2J),\nwherer= [log2(H/J)] is an integer part of log2(H/J).\nAtHr= 2rJ,r/greaterorequalslant2, we have mr= 1−6·4−r. Field\ndependence of the ground state magnetization is shown\nin Fig.4. Due to the imbalance ratio η= 1/3 the system\nhas spontaneous magnetization m= 1/2 atH→+0.\nUnfortunately, the data on the percolation cluster mag-\nnetization on cubic and bcc lattices are totally absent in\nRef. [11] which deprives us of the opportunity to decide\nif there is the imbalance in real 3 dpercolation clusters at\npc< p <1−pc.\nThe jumps at Hr= 2rJresult from the flipping along\nthe field of single spins in sublattice B having the coordi-\nnation number 2r. As we discussed above in dilute crys-\ntalline lattices there are many more jumps appearing at6\n0.40.60.81m\nH/J 4 8 12 16 0\nFIG. 4. Field dependence of the ground state magnetization.\nrational values of H/Jwhere flipping of the magnetized\nparts of percolation cluster takes place [10]. Such jumps\nwere observed in low- Texperiments in FepZn1−pF2[14].\nDISCUSSION AND CONCLUSIONS\nThe decades of experimental investigations of dilute\nIsing AF have shown that DAFF - RFIM correspondence\nworks reasonably well at low dilution and low fields [15].\nMeanwhile the field-induced rounding of the transition\nappears at lower pwhich is impossible in the case of the\nAF ordered phase. One explanation assumes that this is\nnonequilibrium effect due to the pinning of AF domain\nwalls by the vacancies which results in very slow relax-\nation to the equilibrium AF structure [16]. Also one may\nsuggest that AF transition transforms at lower pinto a\nspin-glass one [15], [17].\nHere we argue that one more reason for the vanish-\ning of AF transition could be the imbalance of perco-\nlation cluster which makes transition ferrimagnetic at\nH= 0 and smeared at finite fields. Just these phe-\nnomena were found in FepZn1−pF2[5] - [7] and several\nother dilute AFs [8], [9]. Also the inspection of neutron-\ndiffraction data on metastability and domain formation\ninFepZn1−pF2family [18] makes authors to conclude\nthat AF order vanishes right at p= 1−pc. AF region\nin theH−pphase diagram of these compounds in Ref.\n[18] is quite similar to that in Fig.2b. Thus our surmise\nof possible imbalance of percolation cluster at p <1−pc\nseems to be true for bcc lattice.\nThis implies that qualitative features of the consid-\nered here model could apply to the thermodynamics\nofFepZn1−pF2compounds with vacancies’ percolation.\nThey are a small scaling index of remanent magnetiza-\ntion, ratherhigh index γ, largenegative αand disbalance\nin the sublattice magnetizations near Tc. But to observe\nthese features of ferrimagnetic transition the measure-ments in ultra-low fields (same as in Refs. [5]-[9]) are\nneeded to avoid its smearing. Also the irreversibility\nshould be taken into account as the theoretical results\nrefer only to the most stable state, which seemingly is a\nfield-cooled one in the ferrimagnetic phase.\nNow we do not know on which lattices DAFF would\nalso have the phase diagram of Fig.2b. Moreover, the\nexact form of HAF(p) andHF(p) is not known for the\nvariety of crystalline lattices of the known easy-axes an-\ntiferromagnets. Yet investigation of DAFF ground state\nin Ref. [11] shows that their determination is feasible\nwith modern numerical methods. Such studies and fur-\nther experiments revealing the details of low- Tand low-\nHbehavior of magnetization may help to elucidate the\nnature of transition in nearest-neighbor dilute AF.\nAuthorgratefullyacknowledgesuseful discussionswith\nV.P. Sakhnenko and M.P. Ivliev.\n∗pntim@live.ru\n[1] R.J. Baxter, Exactlysolvedmodels instatistical mechan-\nics, Academic Press, 1982.\n[2] G. Chatrand, Introductory Graph Theory , Dover, 1984.\n[3] L. Neel, C. R. Acad. Sci. Paris 252, 4075 (1961).\n[4] J.L. Cardy, Phys. Rev. B29, R2460 (1984).\n[5] C. Djurberg, J. Mattson and P. Nordblad, J. Appl. Phys.\n75, 5541 (1994); J. Mattson, C. Djurberg and P. Nord-\nblad, Phys. Rev. B61, 11274 (2000).\n[6] J. Kushauer, W. Kleemann, J. Mattsson et al.,\nPhys. Rev. B49, 6346 (1994).\n[7] M. Lederman, J. Hammann and R. Orbach, Physica\nB165-166 , 179 (1990).\n[8] H. Ikeda, J. Phys. C 16, L21 (1983); H. Ikeda, J. Phys. C\n16, L1033 (1983).\n[9] H. Ikeda and K. Kikuta, J. Phys. C 16, L445 (1983) H.\nIkeda and K. Kikuta, J. Phys. C 17, 1221 (1984).\n[10] S. Bastea and P.M. Duxbury, Phys. Rev. E58, 4261\n(1998)\n[11] A. Glaser, A.C. Jones and P.M. Duxbury, Phys. Rev.\nB71, 174423 (2005).\n[12] A.N. Berker and S. Ostlund, J. Phys. C 12, 4961 (1979);\nR.B. Griffiths and M. Kaufman, Phys. Rev. B26, R5022\n(1982).\n[13] P.N. Timonin, Zh. Eksp. Teor. Fiz. 126, 1198 (2004).\n[14] A.R. King, V. Jaccarino, T. Sakakibara et al.,\nPhys. Rev. Lett. 47, 117 (1981).\n[15] D. P. Belanger in Spin Glasses and Random Fields , ed.\nA. P. Young, World Scientific (1997).\n[16] M. Staats, U.NowakandK.D.Usadel, Phase Transitions\n65, 159 (1998).\n[17] F.C. Montenegro, S.M. Rezende and M. D. Coutinho-\nFilho, J. Appl. Phys., 63, 3755 (1988).\n[18] W.C. Barber, F. Ye, D.P. Belanger et al., Phys. Rev.\nB69, 024409 (2004)." }, { "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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Nordlund, Ion and electron irradiation -induced effects in \nnanostructured materials, Journal of applied physics 107, 3 (2010). \n[44] I. Shorubalko, K. Choi, M. Stiefel, and H. G. Park, Ion beam profiling from the interaction with a \nfreestanding 2D laye r, Beilstein journal of nanotechnology 8, 682 (2017). \n[45] P. -H. Lin, B. -Y. Yang, M. -H. Tsai, P. -C. Chen, K. -F. Huang, H. -H. Lin, and C. -H. Lai, Manipulating \nexchange bias by spin –orbit torque, Nature Materials 18, 335 (2019). \n[46] Z. Liu, Z. Luo, I. Shoru balko, C. Vockenhuber, L. J. Heyderman, P. Gambardella, and A. 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": "1908.07003v2.Amorphous_Ferrimagnets__an_Ideal_Host_for_Ultra_Small_Skyrmions_at_Room_Temperature.pdf", "content": " \nAmorphous Ferrimagnets: an Ideal Host for Ultra -Small Skyrmion s at Room \nTemperature \n \nS. Joseph Poon and Chung Ting Ma \nDepartment of Physics, University of Virginia, Charlottesville, Virginia 22904 USA \n \nAbstract \nRecently, magnetic skyrmion ha s emerged as an active topic of fundamental study and applications in \nmagnetic materials research . Magnetic skyrmions are vortex -like spin excitations with topological \nprotection and therefore are more robust to pinning compared with magnetic domain walls. We employ \natomistic simulations to create room -temperature ultra -small Néel skyrmions in amorphous ferrimagnet. \nThe fast propagation and low -dissipation dynamics of ultra-small ferrimagnetic skyrmions make them \nattractive for utilization as an alternative to domain walls in spin-based memory and logic devices. \n \nIntroduction \nIn the 1960s, Professor Theodore (Ted) Geballe and his colleagues pioneered the study of the magnetic \nmoment of 3D transition metal solutes in noble metals and discovered the formation of giant moment in \nthese material systems [1]. The seminal study by Professor Geballe laid the foundation for the fundamental \nresearch on magnetic materials and, equally important, also impacted the study of strong electronic \ncorrelation and orbital effects. To celebrat e his centennial birthday, it is fitting to discuss a new type of \nmagnetic phenomenon called magnetic skyrmion (Figure 1 ), a topological excitation in a magnetic system \nthat could potentially lead to better spin-based memory and logic devices . Originally, skyrmion was named \nfor a certain type of topological soliton invented by T. Skyrme, also in the 1960s, in the study of particle \ncreation in quantum fields. \nBeginning with some background, the d iscovery of giant \nmagnetoresistance (GMR) led to the dawn of spintronic s. \nSpintronics utilize the magnetic moment instead of electric \ncharge to store data in magnetic memory , which potentially can \nhave a significant impact on the future of electronics [2]. Non-\nvolatile magnetic memory is one of the essential approaches in \novercoming the von Neumann bottleneck in computer \narchit ecture. Current magnetic memory relies on the bubble \nand domain wall (DW) to encode information, using DW s in the \nrace-track architecture to encode bits and to drive them with spin \ncurrent. However, there are limitations that impede the advance of \nDW based magnetic memory. Skyrmions can be an alternative to \nDW race -track memory [3]. Being topologically protected and \nmuch smaller than DWs [4-5]., e.g. ~10 nm or smaller in diameter, \nskyrmions can be easier to unpin from lattice defects . Along with \nthe low driving threshold, magnetic skyrmions can form the basis for scalable and high -speed -low-power \nspin-based logic devices . However, such devices can only be possible if the small skyrmions of 10 nm or Fig. 1 Top view of the spin texture \nin a Néel skyrmion from an \natomi stic simulation. Different from \nthe domain wall spins , the spins in a \nskyrmion rotate by 360 degrees from \none side to the other. \nbelow in diameter exist sufficiently long at room temperature. In this article, we will employ atomistic \nsimulations to study a class of materials , namely amorphous ferr imagnetic alloy films t hat are promising in \nhosting ultra -small skyrmions at room temperature . Our s imulation s tudy also showed stable ultra -small \nskyrmions at room temperature in ferromagnetic heterostructures, however, the conditions required are \nmuch more stringent. \nMethod \nSkyrmions are stabilized via the Dzyaloshinskii Moriya \ninteraction (DMI) [6-7], which originated from the interplay of \nspin-orbit effect and inherent chiral asymmetries or interfacial \nsymmetry breaking. Figure 2 illustrates the chiral nature of the \ninterfacial DMI. Intrinsic DMI arises in non -centrosymmetric \ncrystals such as B20 alloy where Bloch skyrmions have been \nfound at low temperature [8-9] Interfacial DMI originates from \ninversion symmetry breaking by a heavy metal interfacial layer \nsuch as Pt and Ir with strong spin -orbit coupling in multilayer \nstacks that contain ferromagnetic Fe and Co. The latter was \nfound to host >40 nm Néel skyrmion s at room temperature [10-\n12]. \nWe have employed the classical atomistic Hamiltonian H to \nperform the simulation of magnetic textures , as shown below: \n \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 (1) \nwhere 𝑠𝑖,𝑠𝑗 are the normalized spins and 𝜇𝑖,𝜇𝑗 are the atomic moments at sites i, and j respectively. The \natomic moment is absorbed into the exchange constant, 𝐽𝑖𝑗=𝜇𝑖𝜇𝑗𝑗𝑖𝑗, and the DMI interaction 𝐷𝑖𝑗=\n𝜇𝑖𝜇𝑗𝑑𝑖𝑗, which is proportional to r i x rj, the position vector between the atoms i, and j and the interface, and \nthe effective anisotropy 𝐾𝑖=𝜇𝑖𝑘𝑖. 𝐻𝑒𝑥𝑡 and 𝐻𝑑𝑒𝑚𝑎𝑔 are the external field and demagnetization field \nrespectively. Only the nearest neighbor interactions are considered. \nThe effective field H eff is calculated using the atomistic Hamiltonian in Eq. (1), and the ground state of the \nmagnetic system is obtained by evolv ing the spins under the following stochastic Landau -Lifshitz -Gilbert \n(LLG) equation, \n𝑑𝑀\n𝑑𝑡=−𝛾\n1+𝛼2𝑀×(𝐻𝑒𝑓𝑓+𝜉)−𝛾𝛼\n(1+𝛼2)𝑀𝑠𝑀×[𝑀×(𝐻𝑒𝑓𝑓+𝜉)] (2) \nwhere 𝛾 is the gyromagnetic ratio, 𝛼 is the Gilbert damping constant, 𝐻𝑒𝑓𝑓 is the effective field, 𝜉 is the \nGaussian white noise term for thermal fluctuations and 𝑀𝑠 is the saturation magnetization. \nAmorphous Ferrimagnet \nAmorphous rare earth transitional metal (RE -TM) ferrimagnets (FiM) is found to provide a favorable \nenvironment to host small skyrmions at room temperature. The magnetic structure of RE -TM FiM consists \nof two sublattices, one occupied by the RE atoms and the other occupied by the TM atoms. The atoms \ncouple ferromagnetically w ithin eac h sublattice and antiferromagnetically between the sublattice s. The \nFig. 2 Schemat ic illustration of \ninterfacial Dzyaloshinskii Moriya \ninteraction . The DMI exchange \ncoupling is given by Dij·(SiSj), \nwhich favors spin canting that \nfacilitates the formation of Néel \nskyrmion . amorphous structure helps to reduce defect pinning, while their intrinsic perpendicular magnetic anisotropy \n(PMA) allows the formation of Néel skyrmion s in thicker films ( e.g. up to 10 nm). Furthermore, a \ndistinctive feature of the ferrimagnet is that the magnetization of RE -TM alloys vanishes at the \nmagnetization compensation temperature due to the cancelation of the magnetization of the two sublattice s. \nFigure 3 shows the simulated ma gnetization of an amorphous Gadolinium -Cobalt ferrimagnet. With near \nzero magnetization , the skyrmion velocity can reach a high speed near ~1,000 m/s [13]. , while near the \nangular -momentum compensat ion temperature, the skyrmion Hall effect is vastly reduced [14]. These \nmaterial advantages make amorphous ferrimagnet a n ideal material for spin -based memory and logic \ndevices. \n \n \n \n \nResults and Discussion \nUsing atomistic LLG simulations, we will now explore the equilibrium state and size of skyrmion in \namorphous Gd 25Co75 film at room \ntemperature by varying the DMI, \nmagnetic anisotropy , and thickness . \nTo capture the unique short -range \norder in amorphous materials, an \namorphous structure of RE -TM \nwas obtained from ab initio \nmolecular dynamics simulation by \nProfessor Howard Sheng using the \nmethod described in ref. 15 [1 5]. \nThe sample size is 50.7 nm x 50.7 \nnm x 5 nm comprising 768000 \natoms . Since the interfacial DMI in \nthese heterostructures originates \nfrom the heavy metal interface , we \nwill use an exponential -decay law to describe \nthe DMI inside the magnetic layer. Indeed, s uch \nrapid decay of DMI has been found in both \ncalculations [16] and experiments [17]. \nFor a 5-nm thick Gd 25Co75 layer, interfacial \nDMI ranges from 0 to 2 mJ/m2 and anisotropy \nranges from 0.05 x 104 J/m3 to 4 x 105 J/m3 are \ninvestigated. From experiment s, the anisotropy \nFig. 3 Simulated saturation magnetization vs. \ntemperature of amorphous Gd 25Co75. The \ncompensation temperature is near 250 K , and \nthe magnetization is small at room \ntemperature. \nFig. 4 Simulated magnetic anisotropy vs. \ninterfacial DMI phase diagram of 5 nm \namorphous Gd 25Co75 at 300 K. Inserted figure \ncorresponds to a 13 nm skyrmions simulated at K \n= 3 x 104 J/m3. The color maps correspond to Co \nsublattice (top) and Gd sublattice (bottom). \nof GdCo was found to be ~ 3 x 104 J/m3 [18] Figure 4 shows the simulated magnetic anisotropy versus \ninterfacial DMI (K-DMI) phase diagram for the 5-nm thick amorphous Gd 25Co75 at 300 K. For a given \nanisotropy, as DMI increases from 0 to 2 mJ/m2, the transition from FiM phase to skyrmions, followed by \nstripe phase occurs . For a given DMI, as the magnetic anisotropy increases, the size of skyrmions decreases , \nand the skyrmions finally collapse into the FiM state with large enough anisotropy . Using experimental \nvalue of K~3 x 104 J/m3, we found skyrmions as small as 13 nm. Such small skyrmion is stabilized with \ninterfacial DMI of ~0.6 mJ/m2. The 2D color map of the out-of-plane reduced magnetization (mz) for the \n13 nm skyrmion is inserted in Figure 4 . In an experiment , Caretta et al.[13] found skyrmion size in the \nrange of 10 nm to 30 nm in \nPt/GdCo/TaO x with an average \nDMI of 0.12 mJ/m2, which \ncorresponds to an interfacial DMI \nof about 0. 9 mJ/m2. With such \ninterfacial DMI, our simulation \nshows a skyrmion size of ~20 nm \nat room temperature , which is in \ngood agree ment with experiment. \nWe further investigate the stability \nof skyrmions at room temperature \nby increas ing the thickness of the \nGdCo layer . Figure 5 shows the \nthickness -DMI phase diagram of \nGd 25Co75 at 300 K. For all \nthicknesses, increase in DMI results in an \nincrease in skyrmion size. The latter can \nbe understood in terms of the \neffectiveness of DMI in spin canting. \nDue to the exponential decay of DMI, a s \nthickness increases , the strength of interfacial DMI required to stabilize skyrmions also increases. Even \nthough the interfacial DMI is less effective in the thicker films , smaller skyrmions, as small as 8 nm, are \nfound. In the 10-nm thick GdCo layer , such sub 10 nm skyrmions are stabilized in the DMI rang e of 1.0 to \n1.2 mJ/m2, which is in the range of measured interfacial DMI in Co/Pt films [17] . A color map of reduced \nmagnetization of a sub 10 nm skyrmion is shown in Figure 5 . Such skyrmion appears to be robust and \ncontain s a well -defined core at the center . \n \nTo further demonstrate the robustness of sub 10 nm skyrmions in GdCo, a numerical tomography plot is \nutilized to reveal the spin texture of the skyrmion in three dimension s. The result is shown in Figure 6 . This \nultra-small skyrmion is found in the 10-nm GdCo film with a n interfacial DMI of 1.1 mJ/m2, The color map \nof m Z is deliberately set to be brighter in order to more clearly show the skyrmions structure. For the Co \nsublattice (left of Figure 6 ), the majority of the spins are pointing down. Near the center, the stripe of green \nand blue corresponds to the center of the skyrmion. One can conclude from the columnar distribution of \ngreen and blue color that this skyrmion is distributed uniformly from top t o bottom. Such columnar \ndistribution of skyrmion is also found in the Gd sublattice (right of Figure 6 ). Such columnar growth of \nskyrmion is favorable for designing skyrmion -based devices. Spin-based logic devices using these \ncolumnar skyrmions will be robust and reliable. Fig. 5 Simulated DMI -thickness phase diagram of \namorphous Gd 25Co75 at 300 K with K = 0.3 x 105 J/m3. The \ninserted figure corresponds to a sub 10 nm skyrmion \nrevealed in 10 nm Gd 25Co75. The c olor maps correspond to \nCo sublattice (top) and Gd sublattice (bottom). \n \nConclusions \nUsing atomistic simulations, we have explored the phase diagram of Néel skyrmion s at room temperature \nin amorphous ferrimagnetic Gd 25Co75 films with interfacial Dzyaloshinskii Moriya interaction (DMI) . Sub-\n10 nm skyrmions are found to form in thick (10 nm) films in the range of DMI values similar to that obtained \nin experi ment . Furthermore, despite the exponential decay of DMI away from the interface, 3D spin texture \nexhibit s a uniform columnar distribution across the film thickness . The present study has revealed the \nrobustness of skyrmions , thus adding to the promise of these topological magnetic entities in spin -based \nnanoelectronics. . \n \nAcknowledgement: \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 \nthe policy of the Government, and no official endorsement should be inferred. Approved for public \nrelease; distribution is unlimited. \n \n \n \nFig. 6 Tomograph of a sub 10 nm skyrmion in 10 nm amorphous Gd 25Co75 at 300 K. Co sublattice \nis on the left, and Gd sublattice on the right. A robust, columnar distribution of a sub 10 nm \nskyrmion is revealed in both sublattices . References \n1. 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Mater. 339, 51-55 (2013) " }, { "title": "2308.03353v1.__textit_In_situ___electric_field_control_of_ferromagnetic_resonance_in_the_low_loss_organic_based_ferrimagnet_V_TCNE____x_sim_2__.pdf", "content": "1 In situ electric-field control of ferromagnetic resonance in the low-\nloss organic-based ferrimagnet V[TCNE] x∼2 \nSeth W. Kurfman1, Andrew Franson1, Piyush Shah2, Yueguang Shi3, Hil Fung Harry \nCheung4, Katherine E. Nygren5, Mitchell Swyt5, Kristen S. Buchanan5, Gregory D. Fuchs4, \nMichael E. Flatté3,6, Gopalan Srinivasan2, Michael Page2, and Ezekiel Johnston-Halperin†1 \n1Department of Physics, The Ohio State University \n2Materials and Manufacturing Directorate, Air Force Research Laboratory \n3Department of Physics and Astronomy, University of Iowa \n4Department of Physics, Cornell University \n5Department of Physics, Colorado State University \n6Department of Applied Physics, Eindhoven University of Technology \n†Corresponding author email: johnston-halperin.1@osu.edu \nWe demonstrate indirect electric-field control of ferromagnetic resonance (FMR) in devices that \nintegrate the low-loss, molecule-based, room-temperature ferrimagnet vanadium \ntetracyanoethylene (V[TCNE] x∼2) mechanically coupled to PMN-PT piezoelectric transducers. \nUpon straining the V[TCNE] x films, the FMR frequency is tuned by more than 6 times the resonant \nlinewidth with no change in Gilbert damping for samples with α = 6.5 × 10−5. We show this tuning \neffect is due to a strain-dependent magnetic anisotropy in the films and find the magnetoelastic \ncoefficient | λs| ∼ (1 − 4.4) ppm, backed by theoretical predictions from DFT calculations and \nmagnetoelastic theory. Noting the rapidly expanding application space for strain-tuned FMR, we \ndefine a new metric for magnetostrictive materials, magnetostrictive agility, given by the ratio of the \nmagnetoelastic coefficient to the FMR linewidth. This agility allows for a direct comparison \nbetween magnetostrictive materials in terms of their comparative efficacy for magnetoelectric \napplications requiring ultra-low loss magnetic resonance modulated by strain. With this metric, we \nshow V[TCNE] x is competitive with other magnetostrictive materials including YIG and Terfenol-\nD. This combination of ultra-narrow linewidth and magnetostriction in a system that can be \ndirectly integrated into functional devices without requiring heterogeneous integration in a thin-\nfilm geometry promises unprecedented functionality for electric-field tuned microwave devices \nranging from low-power, compact filters and circulators to emerging applications in quantum \ninformation science and technology. \nKeywords : magnetostriction, magnonics, molecule-based magnets 2 Introduction \nThe use of electric fields for control of magnetism has been a long-term goal of magnetoelectronics in \nits many manifestations ranging from metal and semiconductor spintronics [1, 2], to microwave electronics \n[3, 4], to emerging applications in quantum information [5]. This interest arises from the potential for clear \nimprovements in scaling, high-speed control, and multifunctional integration. However, while the promise \nof this approach is well established its realization has proven challenging due to the strong materials \nconstraints imposed by the existing library of magnetic materials [3, 4]. The most common approach to \nachieving this local control is through linking piezoelectricity with magnetostriction to achieve electric-\nfield control of magnetic anisotropy, either intrinsically through inherent coupling in multiferroic materials \nor extrinsically through piezoelectric/magnetic heterostructures [3]. Ideally, magnetic materials chosen for \nsuch applications should exhibit large magnetostriction, low magnetic damping and narrow linewidth (high-\nQ) ferromagnetic resonance (FMR), and robust mechanical stability upon strain cycling [4]. While the use \nof multiferroic materials promises relative simplicity in device design, they typically suffer from poor \nmagnetic properties and minimal tunability. The alternative approach of employing heterostructures of \nmagnetic thin films and piezoelectric substrates effectively creates a synthetic multiferroic exploiting the \nconverse magnetoelectric effect (CME) and in principle allows independent optimization of piezoelectric \nand magnetic properties [6, 7, 8, 9, 10, 11]. \nFurther, for applications that rely on magnetic resonance ( e.g., microwave electronics and magnon-based \nquantum information systems), the traditional metrics of magnetostriction, λs, or the CME coefficient, A, \ndo not capture the critical parameters governing damping and loss in this regime ( e.g., linewidth or Gilbert \ndamping coefficient). To date, materials with large magnetoelastic constants and CME coefficients ( e.g., \nTerfenol-D) suffer from high damping, broad magnetic resonance features, and are particularly fragile and \nbrittle [4, 12]. Ferrites such as yttrium iron garnet (YIG), on the other hand, are attractive due to their low-\nloss magnetic resonance properties but typically exhibit minimal to moderate magnetoelastic coefficients. \nFurther, these low-loss ferrites require high growth temperatures (800-900 ◦C) and lattice-matched \nsubstrates to produce high-quality material, which makes integrating these materials on-chip while \nmaintaining low-loss properties challenging [13, 14, 15], and limits their applicability for magnetic \nmicroelectronic integrated circuits (MMIC). Accordingly, alternative low-loss, magnetostrictive materials \nwith facile integration capabilities are desired for applications in electrically-controlled devices. Recently, \na complementary material to YIG, vanadium tetracyanoethylene (V[TCNE] x, x ~ 2), has gained significant \ninterest from the spintronics and quantum information science and engineering (QISE) communities due to \nits ultra-low damping under magnetic resonance and benign deposition characteristics [16, 17, 18, 19, 20, \n21, 22, 23, 24, 25, 26, 27]. 3 Here we present the first systematic experimental study of the magnetostrictive properties of V[TCNE] x. \nComposite heterostructures of V[TCNE] x films and piezoelectric substrates demonstrate shifts in the FMR \nfrequency by 35 – 45.5 MHz, or more than 6 linewidths upon application of compressive strains up to 𝜀=\n−2.4×10ିସ . Further, a systematic analysis shows that the Gilbert damping, α, and inhomogeneous \nbroadening linewidth, 0, are insensitive to strain in this regime and robust to repeated cycling. Density-\nfunctional theory (DFT) calculations provide insight into the elastic and magnetoelastic properties of \nV[TCNE] x and predict a magnetoelastic coefficient 𝜆ሚଵ = −2.52 ppm. Experimental measures of the \neffective magnetoelastic constant, λs, determined by combining optical measurements of the distortion with \nthe corresponding FMR frequency shifts, yield values of λs = −(1 − 4 .4) ppm, which is in good agreement \nwith these DFT predictions. Finally, we define a new figure of merit, magnetostrictive agility, ζ , as the ratio \nof the magnitude of the magnetoelastic coefficient to the FMR linewidth ζ = |λs|/Γ that more closely aligns \nwith the performance requirements for emerging applications of magnetostrictive materials. These results \nestablish a foundation for utilizing strain or acoustic (phononic) excitations for highly efficient strain-\nmodulated magnetoelectronic devices based on V[TCNE] x and other next-generation organic-molecule-\nbased magnetically-ordered materials for coherent information processing and straintronic applications \n[28]. \n \nResults \nV[TCNE] x is a room-temperature, organic-molecule-based ferrimagnet ( Tc ∼ 600 K) that exhibits superb \nlow-damping properties ( α = (3.98 ± 0.22) × 10−5) and high-quality factor FMR ( Q = fR/Γ > 3,000) [17, 18, \n19, 20, 21, 22, 23, 24, 25]. V[TCNE] x thin films are deposited via chemical vapor deposition (CVD) at \nrelatively low temperature and high pressure (50 ◦C and 35 mTorr, respectively) and is largely insensitive \nto substrate lattice constant or surface termination [17, 20]. Further, V[TCNE] x can be patterned via e-beam \nlithography techniques without increase in its damping [19]. The highly coherent and ultra-low loss \nmagnonic properties of V[TCNE] x have driven interest in applications in microwave electronics [26, 29] \nand magnon-based quantum information science and engineering (QISE) [21, 30, 31]. These benign \ndeposition conditions, combined with patterning that does not degrade performance, highlight the versatility \nof V[TCNE] x for facile on-chip integration with pre-patterned microwave circuits and devices [26, 27, 29, \n32, 33, 34]. These excellent magnetic properties are even more surprising given that V[TCNE] x lacks long-\nrange structural order [25]. Early studies indicated that V[TCNE] x films do not exhibit magnetic anisotropy \nbeyond shape effects due to this lack of long-range ordering [19, 20]. However, recent FMR studies on \nV[TCNE] x nanowires, microstructures, and thin films [19, 20, 21], coupled with combined DFT and \nelectron energy loss spectroscopy (EELS) of the crystal structure [25], suggest there is a residual nematic 4 ordering of the c-axis of the V[TCNE] x unit cell, giving rise to an averaged crystal field anisotropy that is \nsensitive to structural and thermally induced strain. However, dynamic measurements of these crystal fields \nand their dependence on strain are currently lacking, preventing a quantitative analysis of the \nmagnetostrictive properties of this material. \nFor this work, PMN-PT/epoxy/V[TCNE] x/glass heterostructure devices are fabricated such that upon \nelectrically biasing the PMN-PT[001] substrate, the piezoelectric effect produces a lateral in-plane strain in \nthe V[TCNE] x thin film, schematically shown in Fig. 1a. PMN-PT is selected for its strong piezoelectric \neffects and high strain coefficients ( d31 ∼ −(500 to 1 ,000) pm/V) to maximize the strain in the devices [3], \nand the epoxy encapsulation layer is selected to allow for device operation under ambient conditions [22]. \nThis device structure allows investigation of the magnetoelastic properties of V[TCNE] x via standard FMR \ncharacterization and analysis. In the main text of this work, measurements on three devices denoted Samples \n1 - 3 are presented. Sample 1 is measured via broadband FMR (BFMR) techniques. Sample 2 is studied via \nX-band (∼9.8 GHz) cavity FMR techniques. Sample 3 is used to directly measure and calibrate strain in the \ndevices via optical techniques. Additional devices characterized via BFMR techniques are presented in the \nSupplemental Information and their characteristics summarized in Table 1. \nThe BFMR response of Sample 1 is described in Fig. 1(b), where individual scans (inset) are fit to a \nLorentzian lineshape to extract the resonance frequency as a function of applied field, HR vs fR. This data \ncan be modeled by considering the V[TCNE] x thin film as an infinite sheet with attendant shape anisotropy \nand with a uniaxial crystal field anisotropy oriented in the out-of-plane direction (as described above). \nAccordingly, the Kittel equation for ferromagnetic resonance reduces to [19, 20, 21] \n (1) \nwhere fR = ω/2π is the FMR resonance frequency, γ is the gyromagnetic ratio, HR is the applied magnetic \nfield at resonance, Heff = 4πMeff =4πMs−H⊥ is the effective magnetization of the V[TCNE] x film with \nsaturation magnetization 4 πMs and uniaxial strain-dependent anisotropy field H⊥, and θ describes the \norientation of the external field as defined in Fig. 1(a). This equation is valid for films where HR ≫ 4πMeff, \nand is appropriate here as the effective magnetization for V[TCNE] x is typically ∼100 G [19] while the \nresonance field is typically between 3500 – 3650 G at X-band frequencies (9.86 GHz) for all magnetic field \norientations. As-grown films exhibit no in-plane anisotropy, consistent with the literature [19, 20, 21], and \nso ϕ-dependences are neglected. When the external magnetic field is held out-of-plane ( θ = 0◦), Eq. (1) \nreduces to \n (2) \n5 Further information about the magnetic damping in thin films can be revealed by comparing the FMR \nfull-width-half-max (FWHM) frequency linewidth, Γ, to the FMR resonance frequency, fR, via (for \n4𝜋𝑀≪𝐻ோ and Γ ≪𝑓ோ [19]) \n Γ = 2αfR + Γ0 (3) \nwhere α is the (dimensionless) Gilbert damping constant and Γ 0 is the inhomogeneous broadening. It \nshould be noted this form for the Gilbert damping utilizing the frequency-swept linewidth is appropriate \ndirectly for out-of-plane magnetized thin films due to symmetry conditions resulting in the linear \nrelationship between 𝐻ோ and 𝑓ோ [19]. Accordingly, Eqs. (1 - 3) show that performing FMR at various \nfrequencies, fields, and magnetization orientations with and without applied strains in V[TCNE] x thin films \nshould provide information regarding the magnetoelastic properties of V[TCNE] x. \nFitting the data from Sample 1 to Eq. (2) reveals an effective magnetization 4 πMeff = 106.2 G and \ngyromagnetic ratio | γ|/2π = 2.756 MHz/Oe, consistent with literature [17, 18, 19, 20, 21, 22]. The Lorentzian \nfits of the FMR response also reveal the linewidth Γ as a function of the resonant frequency, seen in Fig. \n1(c), where fitting to Eq. (3) yields α = 1.02 ± 0.52 × 10−4 and Γ0 = 8.48 ± 1.22 MHz in Sample 1. These \ndamping characteristics are also consistent with literature values for V[TCNE] x [19, 27], and show that the \ndevices incorporate high-quality magnetic films exhibiting superb low-damping properties [32, 33, 34]. \nMoving beyond measurements of the as-grown strain-free sample, the FMR response of the device is \nnow measured while straining the V[TCNE] x film (Fig. 1(d)). Comparing the FMR response of Sample 1 \nwith no applied strain ( EB = 0 kV/cm) and maximum-applied strain ( EB = 13.3 kV/cm) yields a shift in the \nresonance frequency of 45.5 MHz at a resonance frequency fR = 9.8 GHz, corresponding to a CME \ncoefficient A = 3.38 MHz cm/kV (1.23 Oe cm/kV). It is worth noting that while this absolute shift in \nfrequency, and consequent value for CME, is modest when compared to other magnetostrictive materials \n[4], it represents a shift of over 6 magnetic resonance linewidths due to the ultra-low damping and narrow \nFMR linewidths of the V[TCNE] x thin film. This ability to shift cleanly on and off resonance with an applied \nelectric field is central to the functionality of many dynamically tuned MMIC devices, motivating a more \nin depth and systematic investigation of this phenomenon. \nThe magnetostriction in this composite device is explored by biasing the piezoelectric transducer \nbetween 0 kV/cm and 13.3 kV/cm, and the shift in the resonance frequency (for 𝜃= 0∘) tracks the linear \nstrain produced by the transducer [35], as seen in Fig. 2(a). For maximally strained films, fitting to Eq. (2) \nnow reveals 4 πMeff = 122.9 G, a difference of +16 .7 G between EB = 0 kV/cm and EB = 13.3 kV/cm (14% \nchange). Panels (b-d) of Fig. 2 show the FMR linewidth Γ, inhomogeneous broadening Γ 0, and Gilbert \ndamping α, for Sample 1 as a function of applied electric field (strain). These parameters do not vary over \nthe entire tuning range and are robust to repeated cycling ( >300 cycles - see Supplemental Information). 6 This stability indicates that the shift in resonance frequency is due to a true magnetoelastic effect under \nlinear deformation rather than some fatigue induced structure or morphology change in the film, and further \ndemonstrates the potential for device applications. Finally, it is noteworthy that the linewidths and damping \ncoefficients observed in these proof of principle devices are much narrower than typical magnetostrictive \nmaterials, but are roughly twice the value observed in optimized bare V[TCNE] x films (Γ is typically ~3 \nMHz [17, 19, 25]). This suggests that the tuning ratio of 6 times the linewidth may be further extended to \nmore than 10 times the linewidth in fully optimized devices [19, 25]. \nTo confirm that these shifts in the resonance position are due to strain-dependent crystal-field anisotropy \nin V[TCNE] x as prior studies suggest [20, 21], angular-dependent measurements on unstrained and \nmaximally strained films are performed. Sample 2 is mounted in an X-band (∼9.8 GHz) microwave cavity \nso that the structure can be rotated to vary the polar angle, θ. In-plane ( 𝜃= 90∘) and out-of-plane ( 𝜃= 0∘) \nFMR spectra are shown in Supplemental Fig. S1, with FWHM linewidths of 2.17 Oe (5.97 MHz) and 2.70 \nOe (7.45 MHz), respectively. By tracking the resonance field as a function of rotation and fitting to Eq. (1) \nthe effective magnetization Heff = 4πMeff = 74.0 G is extracted for Sample 2, as seen in Fig. 3. The difference \nin 4πMeff between Samples 1 and 2 can be attributed to sample-to-sample variation and remains consistent \nwith literature values [19]. Repeating the measurement with an applied bias of 13.3 kV/cm to the PMN-PT \nreveals an increase of 4 πMeff to 79.4 G, an increase of 5.4 Oe (8% change), which is like the change observed \nin Sample 1. This confirms that strain is modulating the magnetic anisotropy in V[TCNE] x through the \ncrystal field term H⊥ where 4πMeff = 4πMs−H⊥. This strain-dependent crystal field H⊥ is consistent with and \nsupports previous measurements of V[TCNE] x with both thermally and structurally induced strain [19, 20, \n21]. \nAn approximate upper bound to the strain in these devices can be simply calculated through the relation \nε = d31EB = (d31VB)/t ∼ −(6 − 12) × 10−4 for typical PMN-PT d31 piezo coefficients [4, 35]. However, the \naddition of epoxy, V[TCNE] x, and the glass substrate affect the overall stiffness of the device, thereby the \npiezo coefficient changes from d31 of the bare piezo to an effective coefficient deff of the entire stack. This \ndeff is directly measured by exploiting the color change of V[TCNE] x upon laser heating [23] to pattern \nfiducial marks on the samples and monitor their positions under strain using optical microscopy (see \nSupplemental Information). This approach yields an effective piezoelectric coefficient of deff ∼ −180 pm/V \nor strain of ε ∼ −2.4×10−4, reasonable for the PMN-PT heterostructures used here [4, 35]. \nDensity functional theory calculations on the relaxed and strained V[TCNE] x unit cell provide further \ninsight into the elastic and magnetoelastic properties of V[TCNE] x. These properties are calculated using \nthe Vienna ab initio Simulation Package (VASP) (version 5.4.4) with a plane-wave basis, projector-\naugmented-wave pseudopotentials [36, 37, 38, 39], and hybrid functional treatment of Heyd-Scuseria-\nErnzerhof (HSE06) [40, 41]. The experimentally verified [25] local structure of the V[TCNE] x unit cell is 7 found by arranging the central V atom and octahedrally-coordinated TCNE ligands according to \nexperimental indications [42, 43, 44, 45, 46], and subsequently allowing the structure to relax by \nminimizing the energy. These DFT results previously produced detailed predictions of the structural \nordering of V[TCNE] x, along with the optoelectronic and inter-atomic vibrational properties of V[TCNE] x \nverified directly by EELS [25] and Raman spectroscopy [23], respectively. This robust and verified model \ntherefore promises reliable insight into the elastic and magnetoelastic properties of V[TCNE] x. \nThe magnetoelastic energy density for a cubic lattice f = fel + fme = E/V is a combination of the elastic \nenergy density \n (4) \nwhere Cij are the elements of the elasticity tensor and εij are the strains applied to the cubic lattice, and \nthe magnetoelastic coupling energy density \n \nwhere Bi are the magnetoelastic coupling constants and αi where i ∈ {x,y,z} represent the cosines of the \nmagnetization vector [47]. \nThe elastic tensor C = Cij for V[TCNE] x is found by applying various strains to the unit cell and observing \nthe change in the energy. The calculated Cij tensor results in a predicted Young’s modulus for V[TCNE] x YV \n= 59.92 GPa. By directly applying compressive and tensile in-plane strains to the DFT unit cell (i.e. in the \nequatorial TCNE ligand plane [25]), one may calculate the overall change in the total energy density, both \nparallel and perpendicular to the easy axis. The difference between these two, Δ𝐸, is the magnetic energy \ndensity change, which is proportional to the magnetoelastic coupling constant B1 [47] \n ∆E/V = −(ν2D + 1)B1ε|| (6) \nwhere ν2D is the 2-dimension in-plane Poisson ratio and ε|| is the applied in-plane (equatorial TCNE \nplane) epitaxial strain while allowing out-of-plane (apical TCNE direction) relaxation. The elastic and \nmagnetoelastic coefficients are related via the magnetostriction constant λ100 = λs via \n . (7) \nAs a result, the calculated changes of the magnetoelastic energy density with strain provide direct \npredictions of the elasticity tensor ( Cij) and magnetoelastic coefficients ( Bi) for V[TCNE] x. For \npolycrystalline samples of cubic materials, the overall (averaged) magnetoelastic coefficient λs also \nconsiders the off-axis contribution from λ111 such that λs = (2/5)λ100 + (3/5)λ111. However, the off-axis \ncomponent is not considered here for two reasons: (i) the apical TCNE ligands are assumed to align along \n8 the out-of-plane direction ( z-axis, θ = 0◦), and (ii) difficulties in calculating the magnetoelastic energy \ndensity changes upon applying a shear strain that provides the estimate of B2 needed to calculate λ111. The \nformer argument is reasonable as previous experimental results indicate the magnetocrystalline anisotropy \nfrom strain is out-of-plane [21], consistent with the ligand crystal field splitting between the equatorial and \napical TCNE ligands [25]. Further, the lack of in-plane ( ϕ-dependent) anisotropy suggests the distribution \nin the plane averages out to zero. Therefore, the magnetoelastic coefficient calculated here considers an \naverage of the in-plane Cii components in determining λ100. That is, the DFT predicts a magnetoelastic \ncoefficient for V[TCNE] x of \n (8) \nwhere 𝐶𝐼𝑃 = (1/2)(𝐶11 + 𝐶22) = 60.56 GPa and C12 = 37.84 GPa. Accordingly, utilizing the \ncalculated value of B1 = 85.85 kPa (see Supplemental Information) predicts a theoretically calculated 𝜆ሚଵ = \n−2.52 ppm for V[TCNE] x magnetized along the apical TCNE ligand (i.e. θ = 0◦). \nCombining these ferromagnetic resonance, direct strain measurements, and DFT calculations provides \nthe information necessary to determine the magnetoelastic properties of V[TCNE] x. Here, we follow the \nconvention in the literature using the magnetoelastic free energy form from the applied stress σ = Y ε to the \nmagnetostrictive material, Fme = (3/2)λsσ [2, 3, 4]. Accordingly, this free energy yields an expression for the \nstrain-dependent perpendicular (out-of-plane) crystal field [3] \n (9) \nwhere λS is the magnetoelastic coefficient, Y is the Young’s modulus of the magnetic material, d31 is the \npiezoelectric coefficient of the (multiferroic) crystal, and EB is the electric field bias. Here, ε = d31EB is the \nstrain in the magnetic layer obtained based on the assumption that the electrically induced strain is perfectly \ntransferred to the magnetic film. For this study, the direct optical measurement of the strain in the V[TCNE] x \nfilms allows the modification of Eq. 9 by replacing d31EB by the measured ε = deffEB = −2.4 × 10−4 to account \nfor the mechanical complexity of the multilayered device. The magnetoelastic coefficient of V[TCNE] x can \nthen be calculated from Eq. 9 using the values of 4 πMS and H⊥ from FMR characterization, the direct \nmeasurement of ε from optical measurements, and the calculated value of YV = 59.92 GPa from DFT. \nAccordingly, inserting the corresponding values into Eq. 9 yields a magnetoelastic constant for V[TCNE] x \nof λs ∼ −1 ppm to λs ∼ − 4 ppm for the devices measured here. This range shows excellent agreement with \nthe DFT calculations of the magnetoelastic coefficient 𝜆ሚଵ = −2.52 ppm from Eq. 8. This agreement \nprovides additional support for the robustness of the DFT model developed in previous work [23, 25]. \n9 Further, comparison with past studies of the temperature dependence of Heff [21] allows for the extraction \nof the thermal expansion coefficient of V[TCNE] x, 𝛼௧ = 11 ppm/K, at room temperature. \n \nDiscussion \nThe results above compare V[TCNE] x thin films to other candidate magnetostrictive materials using the \nestablished metrics of CME and λs. However, while these parameters are effective in capturing the impact \nof magnetoelastic tuning on the DC magnetic properties of magnetic thin films and magnetoelectric devices, \nthey fail to capture the critical functionality for dynamic (AC) magnetoelectric applications: the ability to \ncleanly tune on and off magnetic resonance with an applied electric field. For example, Terfenol-D is \nconsidered a gold standard magnetostrictive material due to its record large magnetoelastic coefficient λs up \nto 2,000 and CME coefficients 𝐴 as large as 590 Oe cm/kV [3]. However, due to its broad ∼1 GHz FMR \nlinewidths, large 4 πMeff > 9,000 G, high Gilbert damping α = 6×10ିଶ, and brittle mechanical nature, it is \nnot practical for many applications in MMIC. As a result, we propose a new metric that appropriately \nquantifies the capability of magnetostrictive materials for applications in microwave magnonic systems [28, \n33, 34] that takes into account both the magnetostrictive characteristics and the linewidth (loss) under \nmagnetic resonance of a magnetically-ordered material. Accordingly, a magnetostrictive agility ζ is \nproposed here, which is the ratio of the magnetoelastic coefficient λs to the FMR linewidth (in MHz) ζ(fR) \n= |λs|/Γ. For the V[TCNE] x films studied here the magnetostrictive agility at X-band frequencies (9.8 GHz) \nis in the range ζ = {0.164 – 0.660}, comparable to YIG ζ = {0.139 − 0.455} and Terfenol-D ζ = {0.301 – \n0.662} as shown in Table 1. Further, we note that the growth conditions under which high-quality V[TCNE] x \nfilms can be obtained make on-chip integration with microwave devices significantly more practical than \nfor YIG, and that the narrow linewidth (low loss) is more attractive for applications such as filters and \nmicrowave multiplexers than Terfenol-D. \nConclusion \nWe have systematically explored indirect electric-field control of ferromagnetic resonance in the low-\nloss organic-based ferrimagnet V[TCNE] x in V[TCNE] x/PMN-PT heterostructures. These devices \ndemonstrate the ability to shift the magnetic resonance frequency of V[TCNE] x by more than 6 linewidths \nupon application of compressive in-plane strains 𝜀 ~ 10ିସ . Further, we find there is no change in the \nmagnetic damping of the films with strain and that the samples are robust to repeated cycling (> 300 cycles), \ndemonstrating the potential for applications in MMIC without sacrificing the ultra-low damping of \nmagnetic resonance in V[TCNE] x. The changes in the FMR characteristics along with direct optical 10 measurements of strain provide an experimentally determined range for the magnetoelastic coefficient, λS \n= −(1 − 4 .3) ppm, showing excellent agreement to DFT calculations of the elastic and magnetoelastic \nproperties of V[TCNE] x. Finally, we present a discussion on the metrics used in the magnetostriction \ncommunity wherein we point out the shortcomings on the commonly used metrics of the magnetostriction \nand CME coefficients. In this context, we propose a new metric, the magnetostrictive agility, ζ, for use of \nmagnetoelastic materials for coherent magnonics applications. \nThese results develop the framework necessary for extended studies into strain-modulated magnonics \nin V[TCNE] x. Additionally, these magnetoelastic properties in V[TCNE] x suggest that large phonon-\nmagnon coupling in V[TCNE] x might be achieved, necessary and useful for applications in acoustically-\ndriven FMR (ADFMR) or, in conjunction with high-Q phonons, for quantum information applications [28]. \nOther recent work has identified V[TCNE] x as a promising candidate for QISE applications utilizing \nsuperconducting resonators [31] and NV centers in diamond ranging from enhanced electric-field sensing \n[5] to coupling NV centers over micron length scales [30]. These findings lay a potential framework for \ninvestigating the utilization of V[TCNE] x in quantum systems based on magnons and phonons. \n \nAcknowledgments \nS. W. K. developed the project idea, and S. W. K., E. J.-H., P. S., and M. P. developed the project plan for \nexperimental analysis. S. W. K. fabricated the V[TCNE] x heterostructure devices, performed FMR \ncharacterization and analysis, and wrote the manuscript. A. F. developed the analysis software used for \nfitting FMR linewidths and extracting parameter fits. P. S., G. S., and M. P. provided PMN-PT substrates. \nY. S. performed and analyzed DFT calculations of the elasticity tensor and the magnetoelastic coefficients \nfor V[TCNE] x. H. F. H. C. performed and analyzed optical measurements of strain in the devices. K. E. \nN. and M. S. performed BLS measurements on V[TCNE] x/Epoxy devices to extract elastic properties. All \nauthors discussed the results and revised the manuscript. S. W. K., A. F., and E. J.-H. were supported by \nNSF DMR-1808704. P. S., G. S, and M. P. were supported by the Air Force Office of Scientific Research \n(AFOSR) Award No. FA955023RXCOR001. 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Appl. \nPhys. 99, 093909 (2006). \nMethods \nSynthesis of V[TCNE] x and Device Fabrication. \nV[TCNE] x films are deposited via ambient-condition chemical vapor deposition (CVD) in a custom CVD reactor \ninside an argon glovebox (O 2 < 1 ppm, H 2O < 1 ppm) in accordance with literature [17, 18, 19, 20, 21, 22, 23, 24, \n25, 26, 27, 29, 31]. Argon gas flows over TCNE and V(CO) 6 precursors that react to form a V[TCNE] x thin film on \nthe substrates. The pressure inside the CVD reactor for all growths was 35 mmHg, and TCNE, V(CO) 6, and the \nsubstrates are held at 65◦C, 10◦C, and 50◦C, respectively. All substrates were cleaned via solvent chain (acetone, \nmethanol, isopropanol, and deionized (DI) water (×2)) and dried with N 2, followed by a 10 minute UV/Ozone clean \nin a UVOCS T10x 10/OES to remove any residual organic contaminants. \nNominally 400 nm V[TCNE] x films are deposited onto microscope cover glass substrates ( t = 100µm). These \nV[TCNE] x/glass substrates are then mechanically fixed to a PMN-PT transducer (4 mm×10 mm ×0.15 mm) with an \nOLED epoxy (Ossila E130) to create a PMN-PT/Epoxy/V[TCNE] x/Glass heterostructure. The epoxy here not only \nprotects V[TCNE] x from oxidation [22], but also propagates lateral strain into V[TCNE] x film from the piezo \ntransducer upon biasing. While the primary deformation in the piezo transducer is along the poling direction of the 16 PMN-PT ( z), the distortion of the PMN-PT in the thickness direction also produces a lateral in-plane strain in the \nPMN-PT through the Poisson effect (i.e. one must consider here the d31 piezo coefficient of PMN-PT). Therefore, the \nprimary strain experienced by the V[TCNE] x film is in-plane. The PMN-PT electrodes are connected to a Keithley \n2400 voltage source so that electric fields up to EB = VB/tPMN−PT = 13.3 kV/cm can be applied across the PMN-PT layer. \nFerromagnetic Resonance Characterization \nBroadband FMR (BFMR) measurements on Sample 1 and Supplemental Devices A-C were taken using a \ncommercial microstrip (Southwest Microwave B4003-8M-50) and Agilent N5222A vector network analyzer \n(VNA). The devices are mounted so that the magnetic field is normal to the V[TCNE] x film (𝜃 = 0∘ ). S21 \nmeasurements (P = −20 dBm) show the FMR peak upon matched magnetic field and frequency conditions \nin accordance with Eq. 2. A Keithley 2400 Sourcemeter is used to apply up to 200 V to the piezoelectric \ntransducers – accordingly, the maximum-applied strain in the 150 μm PMN-PT corresponds to an electric \nfield 𝐸 = 13.3 kV/cm as mentioned in the main text. \nAll angular-dependent FMR measurements (Sample 2) were performed in a Bruker X-band (~9.6 GHz) \nEPR (Elexsys 500) spectrometer. The frequency of the microwave source is tuned to match the resonant \nfrequency of the cavity before each scan to ensure optimal cavity tuning. All scans had a 0.03 G modulation \nfield at 100 kHz modulation frequency and were performed at the lowest possible microwave power (0.2 \nμW) to prevent sample heating and non-linear effects distorting the FMR lineshape. The V[TCNE] x/PMN-PT \ndevices are mounted on a sapphire wafer and loaded into glass \ntubes for FMR measurements such that the samples can be rotated in-plane (IP: 𝜃 = 90∘ ) to out-of-plane \n(OOP: 𝜃 = 0∘ ) for FMR measurements in 10 degree increments, where resonance occurs upon matched field \nand frequency conditions according to Eq. 1. \nDensity Functional Theory Calculations \nThe pseudopotentials used are default options from VASP’s official PAW potential set, with five valence electrons \nper vanadium, four per carbon and five per nitrogen [36, 37, 38, 39]. For the rest of the calculation we used 400 eV \nfor the energy cutoff and a Γ centered 5x5x3 k-mesh sampling. From these results, the elastic tensor Cij for V[TCNE] x \nis calculated. Using the elastic tensor, the Young’s modulus for V[TCNE] x is averaged over the C11, C22, and C33 \ncomponents to yield YV = 59.92 GPa. From the DFT calculations, the full elastic matrix from the Cij is given by (in \nunits of GPa) \n𝐶=\n⎣⎢⎢⎢⎢⎡66.4437.847.96\n37.8454.683.79\n7.96 3.79 58.641.38−0.200.53\n0.09−1.55−0.37\n−0.69 0.76 0.31\n1.38 0.09 −0.69\n−0.20 −1.55 0.76\n0.53 −0.37 0.3135.16 0.25 −0.95\n0.25 6.65 −0.17\n−0.95 −0.17 9.94 ⎦⎥⎥⎥⎥⎤\n \n \n 17 Optical Measurements of Strain in V[TCNE] x \nV[TCNE] x films can be patterned via laser heating techniques, whereupon the material changes color when heated \nabove its thermal degradation temperature ( ∼ 370 K) [16, 23]. To more appropriately calibrate strain in the \nV[TCNE] x films versus applied bias, we directly measure the deformation in the films by exploiting the color change \nof V[TCNE] x upon laser heating [23] and optical microscopy techniques. Fresh V[TCNE] x/PMN-PT devices are \nexposed to a focused laser spot to create ad hoc fiducial marks on the film in Sample 3 (Supplementary Fig. S5) in \na 50 µm × 50 µm square. By measuring the distance between these laser-written structures with and without applied \nstrain, we can precisely and directly measure the strain in the V[TCNE] x films upon electric bias thus allowing a \nmore precise calculation of magnetoelastic coefficients. Using these methods, we apply a bias of 13.3 kV/cm on \nSample 3 and find a strain ε ∼ 2.4×10−4 which is in reasonable agreement with estimated values of strain using the \nthickness of the PMN-PT (150 µm) and typical piezo coefficient d31 ∼ 500 − 1000 pm/V). 18 \n \n \n \nFigure 1: (a) Effective device schematic, coordinate system, and wiring diagram for V[TCNE] x/PMN-PT \nheterostructures. (b) Ferromagnetic resonance frequency fR vs external field Hext with the field held OOP ( θ \n= 0◦) measured via BFMR. The external field is held constant as the microwave frequency is swept. (Inset) \nRepresentative BFMR scan at f0 = 9.8 GHz, Hext = 3,660 Oe. (c) FMR linewidth Γ versus FMR frequency \nfor OOP field ( θ = 0◦). A linear fit (red line) extracts the dimensionless Gilbert damping parameter α = 1.02 \n± 0.52 × 10−4 and the inhomogeneous broadening Γ 0 = 8.48 ± 1.22 MHz. (d) BFMR scans for unstrained (0 \nkV/cm – black) and maximally strained (13.3 kV/cm – red). The shift in the FMR frequency is ∼45 MHz, \na shift ∼4 linewidths. \n 19 \nFigure 2: V[TCNE] x damping analysis with applied strain: (a) Plot showing differential (shifted) \nFMR resonance position f R−f0 where f 0 is the resonance at 9.8 GHz, (b) FWHM linewidth Γ, (c) \ninhomogeneous broadening Γ 0, and (d) Gilbert damping α versus applied electric field bias E B. \nWhile the resonance position shifts by multiple linewidths, there is negligible effect in the \nlinewidth or damping of the material. 20 \n \n \n \n \n \n \nFigure 3: Cavity X-band FMR measurements on V[TCNE] x/PMN-PT devices. Angular dependence of \nFMR resonant field H R at 𝑓ோ ∼ 9.6 GHz is measured without (black squares) and with (red circles) strain. \nIn-plane and out-of-plane peak-to-peak (FWHM) linewidths are 1.25 (2.16) Oe and 1.56 (2.70) Oe, \nrespectively. \n 21 \n \n \n \n \n \n \n \n \nTable 1: Extracted parameters from V[TCNE] x strain devices compared to YIG, Terfenol-D, and \nother magnetostrictive materials. Asterisk indicates the frequency-equivalent linewidth calculated \nfrom the field-swept FMR linewidth and accounts for the ellipticity of FMR precession for in-plane \nmagnetized materials following the method in Ref. [56]. \n22 Supplemental Information: In situ electric-field control of \nferromagnetic resonance in the low-loss organic-based ferrimagnet \nV[TCNE] x∼2 \n \n \nCavity X-band FMR of Sample 2 \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S1: X-band (9.8 GHz) cavity FMR scans of Sample 2 for DC magnetic field in-plane ( 𝜃=90°) and out-\nof-plane ( 𝜃 = 0° ). Peak-to-peak (p2p) linewidths and resonance positions determined from a fit to a \nLorentzian derivative, from which the full-width-half-max (FWHM) linewidth Γ is found by multiplying by \n√3. 23 V[TCNE] x Resonance Frequency with Piezo Switching Strain \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S2: V[TCNE] x out-of-plane magnetized differential resonance frequency 𝛿𝑓ோ=9.8 𝐺𝐻𝑧−𝑓ோ \nas a function of applied bias voltage to 150 µm PMN-PT . The “butterfly” hysteresis arises from the \nhysteretic behavior of the piezo strain upon the polarization direction switching. The inset shows \nfrequency-swept FMR spectra and fits at maximum and minimum frequency shift. 24 \n \n \n \n \nFigure S3: Differential resonance frequency ( 𝑓ோ,=9.8 GHz) in the same device from Fig. S2 switching \nbetween 𝑉= ±25 V (𝐸= ±1.67 kV/cm) as a function of the number of the number of switches \nbetween positive and negative applied strain. The FWHM linewidth of the V[TCNE] x remains \neffectively constant for over 300 positive/negative (tensile/compressive) strain applications, and the \nresonant frequency for the respective compressive and tensile additionally remains effectively constant. \nThese conditions were selected based on the linewidth and resonance frequency tuning such that the \nresonance features do not overlap, thereby demonstrating a means to electrically-bias a device on and \noff resonance. \n25 V[TCNE] x Density Functional Theory Calculations of Strain-Dependent \nMagnetoelastic Energy \n \nV[TCNE] x Optical Strain Characterization \nFigure S5: Positions of fiducial marks “burned” onto the V[TCNE] x film measured via optical techniques \nupon biasing a 150 µm PMN-PT piezo transducer. The extracted strain in x and y is averaged to 𝜀 =\n2.4×10ିସ and is used to calculate the magnetoelastic coefficients 𝜆ௌ presented in the text. \nFigure S4: DFT-calculated magnetic energy difference of the V[TCNE] x unit cell upon manipulating \nthe applied strain. The orange line is a tangential linear fit at 𝜀 = 0 to solve for 𝛥𝐸/𝑉 in the main \ntext that provides the magnetoelastic coupling 𝐵ଵ. 26 \nAdditional V[TCNE] x Device Strain Characterizations \n \n \n \nFigure S6: Supplemental devices measured via BFMR techniques ( 𝜃=0°). Supplemental device C varies from the \nothers only by the thickness of the PMN-PT piezo ( 𝑡= 500 𝜇𝑚 ), so that the electric field across the device \n(hence the strain) is adjusted accordingly. 27 Additional Linewidth and Damping Analysis: Supplemental Device C ( 𝒕𝑷=\n𝟓𝟎𝟎 𝝁𝒎) \n \n \n \n \nFigure S7: Gilbert analysis of Supplemental Device C as a function of applied electric-field bias. (a) Resonance \nfrequency for an out-of-plane magnetization orientation and applied external field 𝐻ோ= 3,658.8 G (𝑓ோ,=\n9.83 GHz). (b) FWHM linewidth corresponding to the resonance frequencies in panel (a). (c) \nInhomogeneous broadening and (d) Gilbert damping parameters. Note there is negligible change in \nlinewidth, inhomogeneous broadening, and Gilbert damping for positive and negative bias up to the piezo \nswitching fields at ±3.2 kV/cm. The error bars in (a) and (b) are smaller than the markers used for the data \npoints. " }, { "title": "2108.10881v1.The_domain_wall_motion_driven_by_a_rotating_field_in_a_ferrimagnet.pdf", "content": "The domain-wall motion driven by a rotating \feld in a ferrimagnet\nMunsu Jin,1,\u0003Ik-Sun Hong,2,\u0003Duck-Ho Kim,3Kyung-Jin Lee,1and Se Kwon Kim1,y\n1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n2KU-KIST Graduate School of Converging Science and Technology,\nKorea University, Seoul 02841, Republic of Korea\n3Center for Spintronics, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea\n(Dated: August 25, 2021)\nWe theoretically study a ferrimagnetic domain-wall motion driven by a rotating magnetic \feld.\nWe \fnd that, depending on the magnitude and the frequency of the rotating \feld, the dynamics\nof a ferrimagnetic domain wall can be classi\fed into two regimes. First, when the frequency is\nlower than a certain critical frequency set by the \feld magnitude, there is a stationary solution\nfor the domain-wall dynamics, where a domain-wall in-plane magnetization rotates in-phase with\nthe external \feld. The \feld-induced precession of the domain wall gives rise to the translational\nmotion of the domain wall via the gyrotropic coupling between the domain-wall angle and position.\nIn this so-called phase-locking regime, a domain-wall velocity increases as the frequency increases.\nSecond, when the frequency exceeds the critical frequency, a domain-wall angle precession is not\nsynchronous with the applied \feld. In this phase-unlocking regime, a domain wall velocity decreases\nas the frequency increases. Moreover, the direction of the domain-wall motion is found to be reversed\nacross the angular compensation point where the net spin density of the ferrimagnet changes its\nsign. Our work suggests that the dynamics of magnetic solitons under time-varying biases may serve\nas platform to study critical phenomena.\nI. INTRODUCTION\nSpintronics is the \feld which aims at advancing in-\nformation technology beyond what has been achievable\nwith charge-based electronics by exploiting spin degree of\nfreedom [1]. A natural venue to look for spin-based func-\ntionality is magnet materials, which are known to exhibit\nvarious excitations that can be used for information car-\nriers such as spin waves and topological solitons [2, 3].\nIn particular, a magnetic domain wall, which is a proto-\ntypical soliton in easy-axis magnets, has been a subject\nof intensive studies in spintronics due to its technological\nutilities as topologically robust information carriers as\nwell as intriguing physics [4, 5]. For example, a domain-\nwall racetrack memory, where a series of domain walls are\nmoved along the one-dimensional racetrack while carry-\ning information, has been shown to have potential for\nfast, nonvolatile, and three-dimensional solid-state mem-\nory architecture [5, 6]. In addition to practical utilities,\na domain wall is known to exhibit various fundamentally\ninteresting nonlinear phenomena. One example is given\nby the so-called Walker breakdown, which refers to a phe-\nnomenon of sudden drop of the domain-wall velocity at\ncertain critical strength of the driving force due to the\nonset of the precession motion of domain wall [7]. The\nWalker breakdown in the \feld-driven domain-wall motion\nhas been experimentally demonstrated in ferromagnetic\nwires, e.g., in Ref. [8].\nIn e\u000borts to expand material platforms for spintronics\nfrom ferromagnets that have been conventional material\n\u0003These authors contributed equally to this work.\nysekwonkim@kaist.ac.krplatform for spintronics, antiferromagnets have been re-\nceiving much attention in spintronics as alternative ma-\nterial choices due to their certain advantages over ferro-\nmagnets [9{11]. For example, the dynamics of antifer-\nromagnets are known to exhibit THz intrinsic frequency,\nwhich is generally faster than ferromagnetic dynamics\nwhich is on the order of GHz. Also, the absence of the\nequilibrium magnetization of antiferromagnets allows for\nthe development of denser spintronic devices compared to\nferromagnet-based devices which su\u000ber from strong cross-\ndevice interactions mediated by the stray \feld. In partic-\nular, antiferromangetic domain wall has been shown to be\nfundamentally di\u000berent from ferromagnetic counterparts\nand thus has been studied intensively in the last decade.\nFor example, antiferromagnetic domain wall is shown to\nnot exhibit the Walker breakdown unlike a ferromagnetic\ncase and thus can be driven with higher velocities [12].\nAlso, when the antiferromagnetic domain-wall velocity is\nclose to the maximum magnon group velocity, it has been\nexperimentally demonstrated to exhibit the Lorentz-like\ncontraction by shrinking its width according to the rel-\nativistic kinematics [13{18]. Despite the fundamental\ninterest and technological potentials, however, it is still\nexperimentally challenging to detect and control antifer-\nromagnetic dynamics due to its zero net magnetization,\nalthough there have been some progress enabled by x-ray\nabsorption spectroscopy [19{21], spin-polarized scanning\ntunneling microscopy [10, 22], and quantum sensing with\nsingle spins [23, 24].\nRecently, ferrimagnets, which consist of two or more\ninequivalent magnetic sublattices that are coupled anti-\nferromagnetically, have emerged in spintronics as mate-\nrial platforms that can o\u000ber advantages of both ferro-\nmagnets and antiferromagnets [25]. They generally have\na small, but \fnite magnetization and thus can be de-arXiv:2108.10881v1 [cond-mat.mes-hall] 24 Aug 20212\nH\nAAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaJUY9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1mu1C9K1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOiDjQM=xAAAB7HicbVBNS8NAEJ3Ur1q/qh69BItQLyUpoh6LXjxWMLXQhrLZbtqlm03YnQgl9Dd48aCIV3+QN/+NmzYHbX0w8Hhvhpl5QSK4Rsf5tkpr6xubW+Xtys7u3v5B9fCoo+NUUebRWMSqGxDNBJfMQ46CdRPFSBQI9hhMbnP/8YkpzWP5gNOE+REZSR5yStBIXreO55VBteY0nDnsVeIWpAYF2oPqV38Y0zRiEqkgWvdcJ0E/Iwo5FWxW6aeaJYROyIj1DJUkYtrP5sfO7DOjDO0wVqYk2nP190RGIq2nUWA6I4Jjvezl4n9eL8Xw2s+4TFJkki4WhamwMbbzz+0hV4yimBpCqOLmVpuOiSIUTT55CO7yy6uk02y4l43m/UWtdVPEUYYTOIU6uHAFLbiDNnhAgcMzvMKbJa0X6936WLSWrGLmGP7A+vwBiGKN2g==X(t)\nAAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeiF49V7Ae0oWy2m3bpZjfsboQQ+g+8eFDEq//Im//GTZuDtj4YeLw3w8y8IOZMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WiCG0TyaXqBVhTzgRtG2Y47cWK4ijgtBtMb3O/+0SVZlI8mjSmfoTHgoWMYGOlh7QyrNbcujsHWiVeQWpQoDWsfg1GkiQRFYZwrHXfc2PjZ1gZRjidVQaJpjEmUzymfUsFjqj2s/mlM3RmlREKpbIlDJqrvycyHGmdRoHtjLCZ6GUvF//z+okJr/2MiTgxVJDFojDhyEiUv41GTFFieGoJJorZWxGZYIWJseHkIXjLL6+STqPuXdYb9xe15k0RRxlO4BTOwYMraMIdtKANBEJ4hld4c6bOi/PufCxaS04xcwx/4Hz+AB6bjRg=y\nAAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaJUY9ELx4hkUcCGzI79MLI7OxmZtYECV/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1mu1C9K1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOuLjQU=zAAAB9HicbVDLSgNBEJyNrxhfUY9eBoMQL2E3iHoM6sFjBPOA7BJmJ73JkNmHM72BEPIdXjwo4tWP8ebfOEn2oIkFDUVVN91dfiKFRtv+tnJr6xubW/ntws7u3v5B8fCoqeNUcWjwWMaq7TMNUkTQQIES2okCFvoSWv7wdua3RqC0iKNHHCfghawfiUBwhkby3DuQyNz6QJTxvFss2RV7DrpKnIyUSIZ6t/jl9mKehhAhl0zrjmMn6E2YQsElTAtuqiFhfMj60DE0YiFobzI/ekrPjNKjQaxMRUjn6u+JCQu1Hoe+6QwZDvSyNxP/8zopBtfeRERJihDxxaIglRRjOkuA9oQCjnJsCONKmFspHzDFOJqcCiYEZ/nlVdKsVpzLSvXholS7yeLIkxNySsrEIVekRu5JnTQIJ0/kmbySN2tkvVjv1seiNWdlM8fkD6zPH+kHkYs=\u0000\u0000(t)\nAAAB7nicbVBNS8NAEJ34WetX1aOXxSLUS0mKqMeiF48V7Ae0oWy2m3bpZhN2J0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekEhh0HW/nbX1jc2t7cJOcXdv/+CwdHTcMnGqGW+yWMa6E1DDpVC8iQIl7ySa0yiQvB2M72Z++4lrI2L1iJOE+xEdKhEKRtFK7V5jJCp40S+V3ao7B1klXk7KkKPRL331BjFLI66QSWpM13MT9DOqUTDJp8VeanhC2ZgOeddSRSNu/Gx+7pScW2VAwljbUkjm6u+JjEbGTKLAdkYUR2bZm4n/ed0Uwxs/EypJkSu2WBSmkmBMZr+TgdCcoZxYQpkW9lbCRlRThjahog3BW355lbRqVe+qWnu4LNdv8zgKcApnUAEPrqEO99CAJjAYwzO8wpuTOC/Ou/OxaF1z8pkT+APn8weDtI8J\u0000(t)\nFIG. 1. Schematic of the magnetization con\fguration of a fer-\nrimagnetic domain wall, where blue and red arrows represent\nthe magnetic moments of two sublattices of the ferrimagnet.\nThe domain-wall position is denoted by X(t) and the in-plane\ndomain-wall magnetization angle is denoted by \b( t). The ex-\nternal \feld H(t) rotates within the xyplane and the domain-\nwall in-plane magnetization lags behind the external \feld by\n\u0001\b(t).\ntected and controlled by conventional methods used for\nferromagnets. Also, under suitable conditions, their dy-\nnamics resembles the dynamics of antiferromagnets since\ntheir magnetic sublattices are antiferromagnetically cou-\npled similarly to antiferromagnets [26]. In other words,\nantiferromagnet-like dynamics of ferrimagnets is control-\nlable and detectable due to its \fnite magnetization. This\nfeature of ferrimangets, typi\fed by rare-earth transition-\nmetal (RE-TM) ferrimagnets, allows for the fast domain-\nwall motion [27{30] and ultrafast magnetization switch-\ning [31{33]. In this work, we are interested in the dy-\nnamics of a domain wall motion in a ferrimagnet.\nMost of the studies on a domain-wall motion have fo-\ncused on the e\u000bects of DC biases such as an external \feld\nand a current. In searching for novel magnetic phenom-\nena, domain-wall motion by oscillating biases have been\nreceiving increasing attention in the \feld. For example,\nthe motion of an antiferromagnetic domain wall by a ro-\ntating \feld has been studied in Refs. [34, 35]. Also, it\nhas been shown that a ferromagnetic domain-wall mo-\ntion driven by time-periodic \feld or current can exhibit\na sudden drop of the domain-wall velocity akin to the\nWalker breakdown [36, 37]. In Ref. [37], this Walker-like\nbreakdown of an AC-bias-driven ferromagnetic domain-\nwall motion has been explained by phase-locking and\nphase-unlocking transition, which mimics an analogous\nphenomenon in an electric RLC circuit discovered by\nAdler [38]. In spintronics, the phase-locking of the spin-\ntorque oscillator to an AC current has also been explained\nby invoking its analogy to the Adler equation [39{41]. Al-\nthough the AC-bias-driven domain-wall motion has been\nstudied for ferromagnets and antiferromagnets, the cor-\nresponding problem for a ferrimagnetic domain wall has\nnot been studied.\nIn this paper, we theoretically investigate a ferrimag-\nnetic domain-wall motion driven by a rotating \feld,\nwhich is schematically illustrated in Fig. 1. We \fnd that,\nwhen the frequency is below a certain critical frequency,\nthe precession of the in-plane magnetization inside the\ndomain wall is synchronized with the applied rotating\n\feld. In this low-frequency regime, the domain-wall ve-\nlocity increases linearly as the frequency increases. Whenthe frequency exceeds the critical frequency, on the other\nhand, the domain-wall motion cannot keep pace with the\nrotating \feld, making its motion asynchronous with the\nrotating \feld. In this case, the domain-wall velocity de-\ncreases as the frequency increases. We refer the former\nand the latter regimes as the phase-locking regime and\nthe phase-unlocking regime, respectively. The analyti-\ncal solutions are checked by performing the numerical\nsimulations, which show good agreement in both phase-\nlocking and phase-unlocking regimes. The unique feature\nof the ferrimagnetic domain-wall motion occurs in the\nvicinity of angular momentum compensation point ( TA):\nthe direction of the domain-wall motion is reversed as the\nferrimagnet passes across TAdue to the sign \rip of the\nnet spin density at TA. For experimental feasibility of us-\ning a rotating \feld to drive a domain wall, we would like\nto mention that there has already been an experimental\ndemonstration of the chirality reversal of a vortex domain\nwall induced by a rotating magnetic \feld [42].\nThis paper is organized as follows. In Sec. II, we de-\nvelop a theory for the dynamics of a ferrimagnetic do-\nmain wall in the presence of a rotating \feld within the\nLandau-Lifshitz-Gilbert-like equations of motion for fer-\nrimagnets. The main analytical results are the critical\nfrequency [Eq. (8)] that separates between the phase-\nlocking and the phase-unlocking regimes, the domain-\nwall velocity as a function of the frequency in the phase-\nlocking regime [Eq. (13)] and the velocity in the phase-\nunlocking regime [Eq. (15)]. In Sec. III, we present our\nnumerical simulation results and compare them with the\nanalytical solutions. In Sec. IV, we summarize our work.\nII. THEORY FOR DOMAIN-WALL DYNAMICS\nDRIVEN BY A ROTATING FIELD\nIn this section, we develop a theory for the dynam-\nics of a ferrimagnetic domain wall driven by a rotating\n\feld in close connection to the existing theory for the dy-\nnamics of a ferromagnetic domain wall driven by an AC\n\feld [37]. For concreteness, we consider a RE-TM ferri-\nmagnets, where RE magnetic moments and TM magnetic\nmoments are exchange-coupled antiferromagnetically.\nA. Analytical model\nThe dynamics of ferrimagnets is described by the\nLandau-Lifshitz-Gilbert (LLG)-like equation, which is\ngiven by [43{49]\n\u000es_n\u0000\u000bsn\u0002_n\u0000\u001an\u0002n=\u0000n\u0002he\u000b; (1)\nwhere nis the unit vector in the direction of the mag-\nnetization of the TM sublattice, \u000es=sTM\u0000sREis the\nnet spin density of a ferrimagnet along n,s=sTM+sRE\nis the sum of spin densities of two sublattices, \u000b > 0\nis the Gilbert damping constant, he\u000b\u0011\u0000\u000eU=\u000enis the3\n\feld conjugate to the order parameter n, and\u001ais the mo-\nment of inertia of the antiferromagnetic dynamics for the\nstaggered magnetization n[11], which is inversely pro-\nportional to the microscopic exchange energy between\nthe two magnetic sublattices.\nWe consider a quasi-one-dimensional ferrimagnet in a\nrotating \feld, which can be modeled by the potential\nenergyU=R\ndx[fA(@xn)2\u0000K(nz)2+Ky(ny)2g=2\u0000\nMH\u0001n], whereAis the exchange coe\u000ecient, K > 0 is the\neasy-axis anisotropy (also called perpendicular magnetic\nanisotropy), Ky>0 is hard-axis anisotropy that captures\nthe shape anisotropy induced by the magnetostatic inter-\naction,M=MTM\u0000MREis the net magnetization of the\nferrimagnet, H=H(cos(!t);sin(!t);0) represents the\nrotating \feld about the zaxis at the frequency given by\n!. Without loss of generality, we consider the cases with\nH > 0 and! >0. In this work, we neglect the nonlocal\ndipolar interaction since, due to the antiferromagnetic\nalignment of the two magnetic sublattices, the net mag-\nnetization of ferrimagnets is orders of magnitude smaller\nthan that of ferromagnets.\nDue to the easy-axis anisotropy, the ferrimagnet sup-\nports a stable nonlinear soliton solution with boundary\ncondition n(x!\u00061 ) =\u0006^z, which is a called a domain\nwall. An equilibrium domain-wall solution is given by\nthe following Walker ansatz [7]:\nn=\u0012\ncos \b sechx\u0000X\n\u0015;sin \b sechx\u0000X\n\u0015;tanhx\u0000X\n\u0015\u0013\n;\n(2)\nwhere\u0015=p\nA=K is the parameter for the domain-wall\nwidth,Xrepresents the domain-wall position, and \b\nis the in-plane angle of the domain-wall magnetization.\nSee Fig. 1 for the schematic illustration of the domain\nwall. The domain-wall position Xrepresents a zero-\nenergy mode associated with the spontaneous breaking\nof the translational symmetry of the system. By plug-\nging the domain-wall solution to the potential energy U,\nwe obtain the following energy of the domain wall:\nU(\b) =\u0000\u0019\u0015MH cos(\b\u0000!t) +\u0015Kysin2\b:(3)\nThis result indicates that when the external \feld is suf-\n\fciently strong, H\u001dKy=M, the domain-wall angle \b\nwill follow!t, i.e., the phase of the external \feld, closely\nto minimize the Zeeman energy. When the anisotropy\ndominates the external \feld Ky\u001dMH, the domain-wall\nangle will be kept closely to 0 or \u0019and there would be no\nappreciable e\u000bect of the external \feld on the domain-wall\ndynamics.\nThe low-energy dynamics of the domain wall can be\ndescribed the dynamics of the two collective coordinates,\nX(t) and \b(t). Within the collective-coordinate ap-\nproach, we can derive the following coupled equations\nfrom the LLG-like equation [50{53]\n\u00002\u000bs_X+ 2\u000es\u0015_\b\u00002\u001aX= 0; (4)and\n\u00002\u000bs\u0015_\b\u00002\u000es_X\u00002\u001a\u0015\b\n= 2\u0015Kysin \b cos \b + \u0019\u0015MsHsin (\b\u0000!t):(5)\nIn this work, we are interested in the time-averaged dy-\nnamics of the domain wall over su\u000eciently long time. By\ntaking time-average of Eq. (4), we obtain the following\naverage domain-wall velocity:\nh_Xi=\u0015\u000es\n\u000bsh_\bi; (6)\nwherehXiandh\biare set to be zero by assuming that the\ndomain-wall dynamics is periodic such that the velocity\n_X(t) and the angular velocity _\b(t) are periodic functions\nof timet. The domain-wall velocity h_Xiis linearly pro-\nportional to the angular precession of the magnetization\nh_\bi, which is rooted in the gyrotropic coupling between\nXand \b [52]. Note that the net spin density \u000esappears in\nthe proportionality constant. In ferrimagnets, the value\n\u000esvaries when the temperature changes. In particular,\nit changes sign across the angular momentum compensa-\ntion pointTA, which will be invoked below to argue that,\nfor the given rotating \feld, the sign of the domain-wall\nvelocity \rips as the temperature varies across TA.\nThe coupling [Eq. (6)] between h_Xiandh_\bienables us\nto drive the domain wall by a rotating \feld. For exam-\nple, when the \feld magnitude His su\u000eciently strong and\nthe \feld rotation is su\u000eciently slow, the in-plane mag-\nnetization inside the domain wall will be mostly parallel\nto the \feld direction H(t) =H(cos(!t);sin(!t);0). This\nmeans that the domain-wall angle \b follows !tclosely,\nleading toh_\bi\u0019!. Then, when the net spin density is\n\fnite\u000es6= 0, the domain wall should move at average\nvelocity given by h_Xi\u0019\u0015\u000es!=(\u000bs). Understanding the\ndomain-wall dynamics for general situations, e.g., with\nhigher frequencies, requires more sophisticated analysis,\nwhich we present below.\nThe time-averaged Eq. (5) can be solely written in\nterms of the domain-wall angle \b by replacing h_Xiby\n\u0015\u000es\n\u000bsh_\bi, which results in\nh_\bi=\u0000!Hhsin (\b\u0000!t)i\u0000!Khsin 2\bi; (7)\nwhere\n!H\u0011\u000bs\u0019MsH\n2f(\u000bs)2+\u000e2sg; (8)\nis the characteristic frequency determined by the external\n\feld and\n!K\u0011\u000bsKy\n2f(\u000bs)2+\u000e2sg; (9)\nis the characteristic frequency determined by the hard-\naxis anisotropy. This equation describes the dynamics\nof the domain-wall angle \b driven by a rotating \feld\nin thexyplane. In this work, we are interested in4\nthe e\u000bect of the rotating \feld on the domain-wall dy-\nnamics. Therefore, we will restrict our attention to the\nsituations where the external \feld dominates the hard-\naxis anisotropy term so that we can set Ky= 0 and\n!K= 0. With this approximation, Eq. (7) is reduced\ntoh_\bi=\u0000!Hhsin (\b\u0000!t)i. Instead of solving this av-\neraged version, we will present an exact solution of the\nfollowing equation\n_\b =!Hsin (!t\u0000\b); (10)\nand will use the solution to obtain the domain-wall ve-\nlocityh_Xias a function of the \feld magnitude Hand\nthe \feld frequency !. Our analysis results, which will\nbe obtained below, will be compared with the simulation\nresults in Sec. III.\nB. Phase-locking and phase-unlocking regimes\nTo solve Eq. (10), let us introduce a new parameter\n\u0001\b\u0011!t\u0000\b. The physical meaning of \u0001\b is the phase\ndi\u000berence between the domain-wall angle and the rotat-\ning \feld. The equation of motion for \u0001\b is given by\nd\u0001\b\ndt=!\u0000!Hsin \u0001\b: (11)\nThis equation has been studied in the \feld of nonlin-\near dynamics [54]. Note that two frequencies appear\nin the equation: the \feld rotation frequency !and the\nmagnitude-related frequency !H. Depending on the rel-\native magnitude of these two frequencies, the dynamics\nis divided into two regimes.\nFirst, let us consider the cases where !H>!, i.e., the\ncases where the \feld is su\u000eciently strong or the frequency\nis su\u000eciently slow. In this regime, the equation permits\na steady-state solution given by\nsin\u00001!\n!H= \u0001\b;for!!H, where\nthe rotation frequency is large compared to !H. In this\ncase, Eq. (11) does not possess a steady-state solution.\nIt still permits an exact solution given implicitly by\ntan\u0001\b\n2=!H\n!+r\n1\u0000!2\nH\n!2tanp\n!2\u0000!2\nH(t\u0000t0)\n2;\n(14)\nwheret0is an arbitrary constant. Note that the period\nof the solution is given by T= 2\u0019=p\n!2\u0000!2\nH, which\nis longer than the period of the applied \feld 2 \u0019=!, im-\nplying that the evolution of the domain-wall angle \b is\nnot in-phase with the rotating \feld. For this reason, the\ndomain-wall dynamics with ! > !His referred to be in\nthe phase-unlocking regime. The averaged angular ve-\nlocity is given by h\u0001_\bi= 2\u0019=T =p\n!2\u0000!2\nH, and thus,\nfrom Eq. (6), the averaged domain-wall velocity is given\nby\nh_Xi=\u0015\u000es\n\u000bs\u0012\n!\u0000q\n!2\u0000!2\nH\u0013\n;for!>!H;(15)\nwhich is a decreasing function of !. This is our second\nmain result: In the phase-unlocking regime ( ! > !H)\nwhere the external \feld rotates too fast for the domain\nwall to keep pace with it, the average domain-wall veloc-\nity decreases as the frequency increases.\nIII. NUMERICAL ANALYSIS\nTo con\frm the analytical results, particularly the\ndomain-wall velocity in the phase-locking regime\n[Eq. (13)], and one in the phase-unlocking regime\n[Eq. (15)], we performed the atomistic spin simula-\ntions by solving the two coupled Landau-Lifshitz-Gilbert\nequations for two antiferromagntically-coupled sublat-\ntices representing TM and RE magnetizations. The\nmaterial parameters that used in the simulations are\nA= 2:5\u000210\u00007erg/cm,K= 9:5\u0002107erg/cm3, and\nKy= 3\u0002103erg/cm3, and\u000b= 0:002. The used gyro-\nmagnetic ratios of the TM and the RE sublattices are\n\rTM= 1:936\u0002107s\u00001Oe\u00001and\rRE= 1:76\u0002107\ns\u00001Oe\u00001, respectively. The cell size was 0 :4\u000250\u00021\nnm3, corresponding to x,y, andzaxis, respectively.\nThe system size is 400 \u000250\u00021 nm3. Table I shows\nthe magnetization and the spin density parameters that\nwe used to model the e\u000bect of the temperature. The\ntemperature T4corresponds to the magnetization com-\npensation point TMwhere the magnetizations of the two\nsublattice are equal and thus the net magnetization van-\nishes. The temperature T7corresponds to the angular\nmomentum compensation point TAwhere the spin den-\nsities of the two sublattices are equal and thus the net\nspin density vanishes. The applied \feld strengths are\n3000 Oe for T1;T2;T3, 1000 Oe for T4;T5, and 200 Oe\nforT6;T7;T8;T9. The \feld magnitude is chosen for each5\nTABLE I. Material parameters used for simulations. MTM,MRE,sTM,sRE,\u000es, andsare the magnetization of TM elements,\nthe magnetization of RE elements, the spin density of TM elements, the spin density of RE elements, the net spin density, and\nthe total spin density, respectively. T4andT7representTMandTA, respectively.\nIndex T1 T2 T3T4(TM)T5 T6T7(TA)T8 T9\nMTM(emu/cm3) 1170 1140 1110 1080 1050 1020 990 960 930\nMRE(emu/cm3) 1260 1200 1140 1080 1020 960 900 840 780\nsTM(erg\u0001s/cm3) 6:04\u000210\u000055:89\u000210\u000055:73\u000210\u000055:58\u000210\u000055:42\u000210\u000055:27\u000210\u000055:11\u000210\u000054:96\u000210\u000054:8\u000210\u00005\nsRE(erg\u0001s/cm3) 7:16\u000210\u000056:82\u000210\u000056:48\u000210\u000056:14\u000210\u000055:8\u000210\u000055:45\u000210\u000055:11\u000210\u000054:77\u000210\u000054:43\u000210\u00005\n\u000es(=sA\u0000sB)\u00001:1\u000210\u00005\u00009:3\u000210\u00006\u00007:4\u000210\u00006\u00005:6\u000210\u00006\u00003:7\u000210\u00006\u00001:9\u000210\u000060 1:86\u000210\u000063:72\u000210\u00006\ns(=sA+sB) 1:3202\u000210\u000041:2707\u000210\u000041:2211\u000210\u000041:1715\u000210\u000041:1219\u000210\u000041:0723\u000210\u000041:0557\u000210\u000049:7314\u000210\u000059:2355\u000210\u00005\ntemperature such that the resultant domain-wall veloci-\nties are comparable.\nFigure 2(a) shows the domain-wall velocity h_Xias a\nfunction of the frequency of the rotating \feld for various\ncon\fgurations. The lines represent the analytical results,\nwhich are given by Eq. (13) for !!Hwith the critical frequency !Hgiven by Eq. (8).\nThe dots represent the simulation results. The analytical\nresults and the simulation results agree with each other\nreasonably well. Several features are noteworthy. First,\natTMwhere the magnetization is zero, the domain-wall\nvelocity vanishes, which is due to the absence of the cou-\npling of the external \feld and the domain wall. Second,\natTAwhere the net spin density is zero, the velocity\nvanishes. In our analytical results, the domain-wall ve-\nlocity is proportional to \u000es, and thus it is expected to\nvanish atTA. Physically, this is due to the absence of\nthe gyrotropic coupling between the domain-wall posi-\ntionXand the domain-wall angle \b at TA. Thirdly, the\nsign of the domain-wall velocity, i.e., the direction of the\ndomain-wall motion depends on the sign of the net spin\ndensity\u000es. ForT1;T2;\u0001\u0001\u0001;T6where\u000es<0, the sign of\nthe velocity is negative, and for T8andT9where\u000es>0,\nthe sign of the velocity is positive. This means that for\nthe given rotating \feld, if we vary the temperature of the\nferrimagnet across TA, the direction of the domain-wall\nmotion should reverse exactly at TA, which may be ex-\nploited to detect TAexperimentally. Figure 2(b) shows\nthe domain-wall velocity as a function of the net spin den-\nsity\u000eswithin the phase-locking regime for the frequency\nf=!=2\u0019= 90 MHz. The analytical solutions [Eq. (13)]\nand the simulation results are depicted by square and\ntriangle symbols, respectively. Note that the sign of the\nvelocity changes, i.e., the direction of the domain-wall\nmotion reverses, as the net spin density changes its sign.\nThe results with all the temperatures except the magne-\ntization compensation point T4(TM) are used [55].\nFigure 3(a) shows the domain-wall velocity as a func-\ntion of the frequency for the con\fguration T5(see Ta-\nble I for the de\fnition). The critical frequency !H\nobtained from Eq. (8) is shown as a vertical dashed\nline. Figure 3(b) and (c) show the evolution of my(t),\nthey-component of the magnetization evaluated at the\ndomain-wall center and Hy(t), they-component of the\nAAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyWpoB6LXjxWsB/QhLLZbtqlm03YnYgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5QSK4Rsf5tgpr6xubW8Xt0s7u3v5B+fCoreNUUdaisYhVNyCaCS5ZCzkK1k0UI1EgWCcY3878ziNTmsfyAScJ8yMylDzklKCRPA/ZEwZhVg3Op/1yxak5c9irxM1JBXI0++UvbxDTNGISqSBa91wnQT8jCjkVbFryUs0SQsdkyHqGShIx7Wfzm6f2mVEGdhgrUxLtufp7IiOR1pMoMJ0RwZFe9mbif14vxfDaz7hMUmSSLhaFqbAxtmcB2AOuGEUxMYRQxc2tNh0RRSiamEomBHf55VXSrtfcy9rFfb3SuMnjKMIJnEIVXLiCBtxBE1pAIYFneIU3K7VerHfrY9FasPKZY/gD6/MH0+eRjA==(b)AAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyWpoB6LXjxWsB/QhLLZbtqlm03YnYgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5QSK4Rsf5tgpr6xubW8Xt0s7u3v5B+fCoreNUUdaisYhVNyCaCS5ZCzkK1k0UI1EgWCcY3878ziNTmsfyAScJ8yMylDzklKCRPA/ZEwZhViXn03654tScOexV4uakAjma/fKXN4hpGjGJVBCte66ToJ8RhZwKNi15qWYJoWMyZD1DJYmY9rP5zVP7zCgDO4yVKYn2XP09kZFI60kUmM6I4EgvezPxP6+XYnjtZ1wmKTJJF4vCVNgY27MA7AFXjKKYGEKo4uZWm46IIhRNTCUTgrv88ipp12vuZe3ivl5p3ORxFOEETqEKLlxBA+6gCS2gkMAzvMKblVov1rv1sWgtWPnMMfyB9fkD0mGRiw==(a)\n50100150200250300-600-400-2000200 T9 T8 T7 (TA) T6 T5 T4 (TM) T3 T2 T1Domain wall velocity (m/s)\nFrequency of the field (MHz)\n-1.0x10-5-5.0x10-60.05.0x10-6-400-2000200 analytical solution simulationDomain wall velocity (m/s)\nNet spin density (erg⋅s/cm3)FIG. 2. (a) Domain-wall velocity h_Xias a function of the\nrotating-\feld frequency f=!=2\u0019for various con\fgurations\nwith the parameters shown in Table I. The lines are analyt-\nical solutions, Eq. (13) for ! < ! H(phase-locking regime\nwhere a domain-wall magnetization precesses at the same fre-\nquency as the external \feld) and Eq. (15) for !>! H(phase-\nunlocking regime where a domain-wall magnetization rotates\nslower than the external \feld). The dots represent simulation\nresults. Note that the domain-wall velocity changes its sign\nas the temperature varies across the angular momentum com-\npensation point TA. (b) Domain-wall velocity as a function\nof the net spin density \u000eswithin the phase-locked regime for\nthe frequency f= 90 MHz. The data from all the temper-\naturesT1;T2;\u0001\u0001\u0001;T9except the magnetization compensation\npointT4(TM), where the frequency f= 90 MHz belongs to\nthe phase-unlocking regime, is used. The squares and the tri-\nangles represent the analytical solutions [Eq. (13)] and the\nsimulation results, respectively.6\n(a)(b)(c)0501001502002503000-50-100-150-200Domain wall velocity (m/s)\nRotating field frequency (MHz) T5Phase-locking regimePhase-unlocking regime\n!!/2$70 MHz260 MHz051015202530-1.0-0.50.00.51.0Normalized amplitude\nTime (ns) my Hy my (analytical solution)\n051015202530-1.0-0.50.00.51.0Normalized amplitude\nTime (ns) my Hy my (analytical solution)\nFIG. 3. (a) Domain-wall velocity h_Xias a function of the rotating-\feld frequency f=!=2\u0019for the case of T5(de\fned in\nTable I). The lines and the dots represent the analytical solutions and the simulation results, respectively. The critical frequency\n!Hwhich separates the two distinct regimes of domain-wall dynamics is calculated from Eq. (8) and is shown as the vertical\ndashed line. When the frequency is below the critical frequency, the dynamics of the domain wall is in the phase-locking\nregime, where the domain wall precesses at the same frequency of the rotating \feld and thus the velocity increases linearly\nas a function of the frequency. When the frequency is above the critical frequency, the dynamics of the domain wall motion\nis in the phase-unlocking regime and the resultant velocity decreases as the frequency increases. (b, c) Time evolution of the\ny-component of the magnetization myat the domain-wall center and the external \feld Hyfor the rotating-\feld frequency of\n(b) 70 MHz and (c) 260 MHz. The black solid line and the red dashed line are obtained from the simulations. The blue dotted\nlines are the analytical solutions obtained from (b) Eq. (12) and (c) Eq. (14).\nrotating \feld at the frequencies of 70 MHz and 260 MHz,\nrespectively. The black solid lines and the red dashed\nlines represent the simulation results for myandHy, re-\nspectively. The dashed blue lines show the analytical so-\nlutions given by (b) Eq. (12) and (c) Eq. (14). In the\nphase-locking regime shown in Fig. 3(b), the domain-\nwall magnetization precesses at the same frequency of\nthe external \feld and thereby the domain-wall velocity\nincreases linearly as the frequency increases as expected\nfrom the analytical solution [Eq. (13)]. In the phase-\nunlocking regime shown in Fig. 3(c), the time duration\nformyto change between 1 and \u00001 is much longer than\nthe period of Hy, meaning that the domain-wall preces-\nsion is much slower than the applied \feld. In this case,\nthe domain-wall velocity decreases as the frequency in-\ncreases [Eq. (15)]. There are some deviations between the\nanalytical solution and the simulation result in Fig. 3(a,\nb), which are presumably due to the approximations that\nwe take to obtain the analytical results such as neglecting\nXand\b in Eq. (4) and Eq. (5).\nIV. SUMMARY\nWe have studied the dynamics of a ferrimagnetic do-\nmain wall driven by a rotating \feld analytically by using\nthe Landau-Lifshitz-Gilbert-like equations for ferrimag-\nnets and also by numerically solving the coupled LLG\nequations. We have found that, depending on the fre-\nquency of the \feld rotation, there are two distinct regimes\nof the dynamics of a domain wall. In the phase-lockingregime, where the frequency is below the critical fre-\nquency, the domain-wall velocity is proportional to the\nfrequency of the rotating \feld. In the phase-unlocking\nregime where the frequency is above the critical fre-\nquency, the domain-wall velocity decreases as the fre-\nquency increases. In addition, we have found that the\ndirection of the domain-wall motion depends on the sign\nof the net spin density of the ferrimagnet. This results\nin the reversal of the domain-wall velocity sign as the\ntemperature varies across the angular momentum com-\npensation point TA.\nACKNOWLEDGMENTS\nThis work was supported by Brain Pool Plus Pro-\ngram through the National Research Foundation of Ko-\nrea funded by the Ministry of Science and ICT (Grant\nNo. NRF-2020H1D3A2A03099291), by the National Re-\nsearch Foundation of Korea funded by the Korea Govern-\nment via the SRC Center for Quantum Coherence in Con-\ndensed Matter (Grant No. NRF-2016R1A5A1008184).\nK.J.L. was supported by the National Research Foun-\ndation of Korea (Grant No. NRF-2015M3D1A1070465).\nD.H.K was supported by the POSCO Science Fellowship\nof POSCO TJ Park Foundation, by the Korea Institute\nof Science and Technology (KIST) institutional program\n(No. 2E31032), and by the National Research Coun-\ncil of Science & Technology (NST) grant (Project No.\n2N45290) funded by the Korea government (Ministry of\nScience and ICT).\n[1] I. \u0014Zuti\u0013 c, J. 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Here, based on symmetry analysis and the first-\nprinciples electronic structures calculations, we predict that the LiFe 2F6is ad-wave altermagnetic\nand charge-ordering-mediated ferroelectric material. Moreover, the LiFe 2F6transforms into a fer-\nrimagnetic and ferroelectric phase with strong magnetoelectric coupling under biaxial compressive\nstrain. Interestingly, the spins of the valence band and the conduction band are opposite in fer-\nrimagnetic LiFe 2F6, which facilitates a simutaneously spin-triplet excitonic insulator phase. More\nimportantly, the spin triplet excitons with spin 1 and -1 can be switched by electric fields in ferri-\nmagnetic LiFe 2F6due to strong magnetoelectric coupling. Due to the abundance of novel physical\nproperties, LiFe 2F6will certainly attract a wide range of theoretical and experimental interest.\nIntroduction. Multiferroics with the coexistence of fer-\nroelectric order and magnetic order are one of the cores of\ncondensed matter physics and may be used in the next\ngeneration electronic devices [1–5]. Usually, ferromag-\nnetism takes place in metals, while ferroelectric materials\nare insulators. Thus, multiferroic materials with coexist-\ning magnetization and polarization are very rare [6, 7].\nMost of the multiferroic materials have coexisting anti-\nferromagnetic and ferroelectric phases. In addition, there\nare a few ferrimagnetic materials with coexisting magne-\ntization and polarization [8, 9].\nVery recently, altermagnetism as a new magnetic phase\nhas been theoretically proposed and experimentally ver-\nified to be distinct from ferromagnetism and conven-\ntional antiferromagnetism [10–15]. Due to the absence\nof both spin symmetry {T∥IT}and{C⊥\n2∥t}, the al-\ntermagnetic materials have k-dependent spin polariza-\ntion, which can lead to d-wave, g-wave, and i-wave mag-\nnetic phase depending on the spin group symmetry [10].\nHere, the symmetry operations at the left and right of\nthe double vertical bar act only on the spin space and\nlattice space, respectively; the notation C⊥\n2represents\nthe 180 degrees rotation perpendicular to the spin di-\nrection; the notations I,Tandtdenote space-inversion,\ntime-reversal, and fractional translation operations, re-\nspectively. The k-dependent spin polarization in alter-\nmagnetic materials can result in many novel physical\neffects, such as the unique spin current [12, 13, 16],\nthe giant magnetoresistance, the tunneling magnetoresis-\ntance [17] and nontrivial superconductivity [18]. More-\nover, like ferromagnetic materials, altermagnetic mate-\nrials also break time-reversal symmetry. Therefore, the\ntime-reversal symmetry-breaking macroscopic phenom-\nena, such as quantum anomalous Hall [19], anomalous\nHall [14, 15, 20–23], anomalous Kerr effects [24] and so\n∗guopengjie@ruc.edu.cn\n†guyuhao0709@gmail.com\n‡zlu@ruc.edu.cnon, can be also realized in altermagnetic materials. But\ndifferent from ferromagnetism, altermagnetism may exist\nin an insulating state with high a N´ eel temperature. Ac-\ncordingly, altermagnetism and ferroelectricity can be re-\nalized in a single material, which will open up new oppor-\ntunities to achieve spin polarization and ferroelectric po-\nlarization simultaneously in a single material. However,\nso far, only one candidate material CaMnO 3hasd-wave\naltermagnetism and ferroelectricity [11]. Thus, predict-\ning more materials with coexisting altermagnetism and\nferroelectricity is very urgent for the study of their novel\nphysical properties.\nOn the other hand, excitonic insulators need to sat-\nisfy that the excitonic bonding energy is larger than the\nsingle-particle energy gap and the excitons have spon-\ntaneous Bose condensation [25, 26]. Although much\nprogress has been made in this area [27–29], compelling\nexperimental evidence is still lacking. This is because\nexcitons are electrically neutral, resulting in no suitable\nexperimental means to detect the exciton condensation\nfor nonmagnetic excitonic insulators. Recently proposed\nspin-triplet excitonic insulators have spin signal, which\ncan be detected by spin transport experiments. The\nspin-triplet excitonic state requires that the spins of the\nconduction band and the valence band of ferromagnetic\ninsulator are opposite and with weak spin-orbit coupling\n(SOC) [30]. Moreover, due to the spin selection rule,\nthis type of ferromagnetic insulators favor excitonic in-\nsulators [30]. Unfortunately, so far, this type of ferro-\nmagnetic insulators have not yet been discovered exper-\nimentally. The altermagnetism and ferrimagnetism with\nopposite spins arrangement may provide new directions\nfor finding magnetic materials with opposite spin of the\nconduction band and the valence band.\nIn this work, based on symmetry analysis and the first-\nprinciples electronic structures calculations, we predict\nthat the LiFe2F6is a multiferroic material with d-wave\naltermagnetism and ferroelectricity. Then, we investigate\nthe physical properties of LiFe2F6under biaxial strain.\nWe find that the LiFe2F6changes from d-wave alter-arXiv:2312.13911v1 [cond-mat.mtrl-sci] 21 Dec 20232\nc\nd\nA+\nF-fe\nA-\nF+a\nb\nFe\nLi\nF\nz\ny\nFe2\nFe1Fe3Fe4\nFIG. 1. Crystal structure, Brillouin zone (BZ) and four most\nrelevant collinear magnetic structures of LiFe 2F6.a, Crystal\nstructure of Li2FeF6 with high symmetry P42−mnm (136).\nb, the corresponding BZ with high-symmetry point and line.\nc–f, The A+, A-, F- and F+ collinear magnetic structure\nof LiFe 2F6, respectively. The red and blue arrows represent\nspin-up and spin-down magnetic momentums, respectively.\nmagnetic ferroelectric phase to ferrimagnetic ferroelec-\ntric phase. Interestingly, the spins of the valence band\nand the conduction band are opposite in ferrimagnetic\nLiFe2F6, which facilitates a spin-triplet excitonic insu-\nlator phase. More importantly, the spins of the bot-\ntom conduction band and the top valence band can be\nswitched by electric field due to strong magnetoelectric\ncoupling.\nMethod. Our electronic structure calculations em-\nployed the Vienna ab initio simulation package (VASP)\ncode [31] with the projector augmented wave (PAW)\nmethod [32]. The Perdew-Burke-Ernzerhof for solids\n(PBEsol) [33] exchange-correlation functional and the\nGGA plus on-site repulsion Umethod (GGA+ U) in the\nformulation of Liechtenstein et al. [34] were used in our\ncalculations. Here, the effective on-site exchange inter-\naction Jwas fixed to U/5. Moreover, the screened hy-\nbrid functional [35, 36] introduced by Heyd, Scuseria,\nand Ernzerhof (HSE) with the HSE06 version [37] were\nalso used to calculate the electronic structure. We used\nthe standard Berry phase method to estimate the ferro-\nelectric polarization P[38, 39]. We adopted the nudged\nelastic band (NEB) [40] method to simulate the flipping\nofPand estimate the energy barrier in this process. The\nkinetic energy cutoff was set to be 650 eVfor the expand-\ning the wave functions into a plane-wave basis and the\nenergy convergence criterion was 10−7eV. The crystal\nstructures were fully relaxed until the force on each atom\nwas less than 0.001 eV/˚A. The Γ-centered k-mesh was set\nas 16×16×8. In the calculations of strained LiFe2F6,\nthe in-plane lattice constants were fixed while the length\nof the caxis and the atomic positions were optimized.\nResults and discussion. LiFe2F6has a tetragonal crys-\ntal structure (Fig. 1 a). At high temperature, LiFe2F6\nhasP42/mnm (136) space group symmetry. The cor-\na b c\nd e f\nC4zt\nFe2.5+\nC4zt\nFe2+\nC4zt\nFe3+04080120\nA+\nA-\nF+\nF-\n0 1 5 4 3 2 6\nU (eV)\nA+\nA-\nF+\nF-\n0 1 5 4 3 2 6\nU (eV)04080120\n0 1 5 4 3 2 6\nU (eV)00.40.8\n D4h\nC4vFIG. 2. Relative energy of different magnetic states with the\nvariation of correlation interaction Uand polarization charge\ndensity for LiFe 2F6.a, The relative energy of different mag-\nnetic states with the variation of correlation interaction Ufor\nhigh-symmetry P42−mnm (136) phases. b, The relative\nenergy of different magnetic states with the variation of cor-\nrelation interaction Ufor low-symmetry P42nm(102) phases.\nc, The relative energy of different crystal phase with the varia-\ntion of correlation interaction U.d–f, The polarization charge\ndensity of Fe2.5+, Fe2+andFe3+, respectively. The red and\nblue represent spin-up and spin-down charge density, respec-\ntively. The polarization charge densities are calculated under\ncorrelation interaction U= 4eVand exchange interaction\nJ= 0.8eV. The trepresents a fractional translation with\n(1/2, 1/2, 1/2).\nresponding Brillouin zone (BZ) is shown in Fig. 1 band\nthe high-symmetry lines and points are labeled. Due to\nthe nonsymmorphic space symmetry, there are four Fe\natoms in the primitive cell of LiFe2F6. From Fig. 1 a, the\nfourFeatoms can be divided into two classes according\nto different orientations of Fe−Foctahedrons, exam-\nple for Fe1 and Fe2. According to the four Featoms\nin primitive cell, there are the most relevant collinear\nmagnetic structures including three collinear antiferro-\nmagnetic states A+, A-, F- and ferromagnetic state F+\n(Fig. 1 c–f). Since the angles of Fe1−F−Fe2 and\nFe2−F−Fe3 are respectively 133 and 95 degrees, the\nsuperexchange interactions may lead to Fe2 (Fe3) and\nFe1(Fe4) with opposite spin arrangement and Fe1(F2)\nandFe4(Fe3) with the same spin arrangement, which\ncorresponds to A+ antiferromagnetic state. To deter-\nmine the magnetic ground state of the high-symmetry\nLiFe2F6, we calculate relative energies of four different\nmagnetic states with the variation of correlation interac-\ntionU. From Fig. 2 a, the A+ antiferromagnetic state is\nthe most stable, which is consistent with our theoretical\nanalysis.\nFor A+ antiferromagnetic state, the Feions with the\nsame spin arrangement are connected by I, thus the spin\nsymmetry {C⊥\n2∥I}is broken. Moreover, the spin symme-\ntry{C⊥\n2∥t}is also broken due to nonmagnetic Fanions.\nConsidering that the Feions with opposite spin arrange-\nment can be connected by the spin symmetry {C⊥\n2∥C4zt},\nthe A+ antiferromagnetic state is a d-wave altermag-\nnetic state. In order to show the d-wave altermagnetic3\ncharacteristics more intuitively, we calculate the polariza-\ntion charge density of the high-symmetry LiFe2F6. From\nFig. 2 d, spin-up and spin-down Feions have anisotropic\npolarization charge density deriving from the Fe−Foc-\ntahedrons of different orientations. Obviously, the Fe\nions with opposite spin polarization are not connected\nby the Iorttransformation, but connected by the C4zt\ntransformation.\nBy analyzing the chemical formula of LiFe2F6, all\nFeions should be of 2.5 valence. Indeed, our calcu-\nlations show that all Feions are of 2.5 valence in the\nhigh-symmetry LiFe2F6. On the other hand, M¨ ossbauer\nexperiment found that there are Fe2+andFe3+in\nLiFe2F6[41] and X-ray diffraction experiment revealed\na low-symmetry P42nm(102) phase above room tem-\nperature [42]. Thus, LiFe2F6may exist a ferroelectric\nphase transition induced by charge order, which has\nbeen demonstrated by Dong’s theoretical calculations [9].\nSince there is only a slight difference between the high-\nsymmetry and the low-symmetry structures, the A+ al-\ntermagnetic state may be still most stable in the low-\nsymmetry LiFe2F6, which has been confirmed by previ-\nous neutron scattering experiment [43]. Thus, LiFe2F6\nis a multiferroic material with d-wave altermagnetism\nand ferroelectricity. In order to show the altermagnetic\nand charge ordering characteristics more intuitively, we\nalso calculate the polarization charge density of the low-\nsymmetry LiFe2F6, as shown in Fig. 2 eandf. The large\ndifference of polarization charge density of Fe2+and\nFe3+reflects the charge order (Fig. 2 eandf). Compar-\ning the polarization charge density of Feions with differ-\nent valence states, the anisotropy of polarization charge\ndensity of Fe2+is the strongest, while that of Fe3+is the\nweakest. Obviously, the Fe2+(Fe3+) ions with opposite\nspin polarization are not connected by the Iorttransfor-\nmation, but connected by the C4zttransformation, which\nreflects d-wave altermagnetic characteristic.\nIn order to further investigate the properties of alter-\nmagnetism and ferroelectricity, we need to determine a\nsuitable correlation interaction U. This suitable corre-\nlation interaction Uvalue can be determined by the ex-\nisting experimental results and our calculation results.\nThen, we calculate relative energies of four different mag-\nnetic states with the variation of correlation interaction\nUfor the low-symmetry LiFe2F6. Different from the\nhigh-symmetry LiFe2F6, the most stable magnetic state\nchanges from A+ altermagnetic state to A- ferrimagnetic\nstate with increasing of correlation interaction U. Since\nthe magnetic ground state is A+ altermagnetic state, the\ncorrelation interaction Uis less than 4.6 eV(Fig. 2 b).\nOn the other hand, when the correlation interaction Uis\nless than 3 eV, the charge order is not stable (Fig. 2 c).\nThus, the correlation interaction Uis between 3 eVand\n4.6eV. In the following calculation, we choose the cor-\nrelation interaction Uto be equal to 4 eV.\nThe electronic band structures were calculated for the\nlow-symmetry and high-symmetry LiFe2F6, which are\nshown in Fig. 3 aandb. From Fig. 3 aandb, both the\n1.0\n-0.50.5\n0.0\n-1.0\nГXM ZRA Z Г\n1.0\n-0.50.5\n0.0\n-1.0\nГXM ZRA Z Г\n1.0\n-0.50.5\n0.0\n-1.0\nГ M* M\n1.0\n-0.50.5\n0.0\n-1.0\nГ M* M\n9.0\n0.06.0\n3.0\nP↑ P↓\na b\nc d eC4v D4hFIG. 3. The electronic band structure without SOC and fer-\nroelectric properties of LiFe 2F6.a, The electronic band struc-\ntures along the high-symmetry directions of low-symmetry\nLiFe 2F6, respectively. b, The electronic band structures along\nthe high-symmetry directions of high-symmetry LiFe 2F6, re-\nspectively. candeare the electronic band structures with\nopposite ferroelectric polarization for low-symmetry LiFe 2F6,\nrespectively. d, The ferroelectric polarization simulated by\nthe NEB method. Insets: Initial and final structures. The\nred arrows represent spin magnetic momentum. The long ar-\nrow is Fe3+ and the short arrow is Fe2+. The red and blue\nlines represent spin-up and spin-down bands.\nhigh-symmetric and low-symmetric phases are semicon-\nductors. Ferroelectric phase transition induced by charge\norder increases bandgap of LiFe2F6from 311 meV to\n474meV. Due to the absence of spin symmetry {C⊥\n2∥I}\nand{C⊥\n2∥t}, altermagnetic LiFe2F6hask-dependent spin\nsplitting, example for the Γ −Mdirection (Fig. 3 aand\nb). In fact, due to the spin symmetry {C⊥\n2∥Mxt}and\n{C⊥\n2∥Myt}, spin-up and spin-down bands are degener-\nate in these four cyan faces (Fig. 1 b). Except for these\nfour cyan faces, spin-up and spin-down bands are split at\ngeneral kpoints in the BZ. However, ferroelectric phase\ntransition has only a small effect on spin splitting of the\nbands (Fig. 3 aandb). Meanwhile, we also calculate the\nelectronic band structure along the M∗−Γ−Mdirec-\ntions as shown in Fig. 3 c. From Fig. 3 c, the spin-up\nbands on the M∗−Γ axis change into spin-down bands\non the Γ −Maxis reflecting the characteristics of the\nd-wave altermagnetism. Considering the d-wave alter-\nmagnets favor unique spin current by electrical means,\nthe low-symmetry LiFe2F6may have spintronic, transis-\ntor and ferroelectric functionalities simultaneously.\nDifferent from traditional ferroelectrics, the ferro-\nelectrics in LiFe2F6origins from charge order, as Fe2+\nandFe3+alternatively arrange. We used Berry phase\nmethod to estimate the ferroelectric P of LiFe2F6, which\ngives 15.1 µC/cm2along the zaxis for the d-wave alter-\nmagnetic state. This is basically consistent with the in-\ntuitive charge order result, 12.3 µC/cm2. Meanwhile, we\nalso adopted NEB method to simulate ferroelectric polar-\nization of LiFe2F6. The energy barrier is only 9.9 meV\nperFeatom (Fig. 3 d), which is substantially smaller4\n0203040\n10\nP↑ P↓\n1.0\n-0.50.5\n0.0\n-1.0\nГXM ZRA Z Г\n1.0\n-0.50.5\n0.0\n-1.0\nГXM ZRA Z Г\na b\nc d0204060\n0.960.98 1.04 1.02 1.00\nStrain\nA+\nA-\nF+\nF-\nFIG. 4. The results of low-symmetry LiFe 2F6under biaxial\nstrain. a, The relative energy of different magnetic states as\na function of biaxial strain. b, The NEB method is used to\nsimulate ferroelectric polarization of ferrimagnetic LiFe 2F6.\nInsets: initial and final structures. canddare the elec-\ntronic band structures with opposite ferroelectric polarization\nof ferrimagnetic LiFe 2F6, respectively. The red and blue ar-\nrows represent spin-up and spin-down magnetic momentum,\nrespectively. The long arrow is Fe3+ and the short arrow\nisFe2+. The red and blue lines represent spin-up and spin-\ndown bands.\ncompared to other ferroelectric materials. This seems\nnatural as the main process of ferroelectric switching in\nLiFe2F6is the charge transfer in Fe2+-Fe3+pair, rather\nthan the moving of the ions. On the other hand, the\nFe2+andFe3+ions in altermagnetic state have the same\nspin arrangement, the charge transfer in Fe2+-Fe3+pair\nhas no effect on the A+ altermagnetic state. Thus, the\nelectronic band structures of polarized/antipolarized are\nthe same, reflecting the weak magnetoelectric coupling\nin altermagnetic LiFe2F6(Fig. 3 cande). However, in\nA- ferrimagnetic and F- altermagnetic state, the switch\nbetween positive and negative ferroelectric polarization\ncan cause obvious change for the electronic structure as\nthe spins in Fe2+andFe3+pair are opposite. Note: The\nF- magnetic state in high-symmetry phase is a conven-\ntional collinear antiferromagnetic state, but the charge-\norder-induced ferroelectric phase transition will trans-\nform the F- from the conventional collinear antiferromag-\nnetic state to an altermagnetic state, which is the reason\nof its strong magnetoelectric coupling.\nPossible spin-triplet excitonic insulator phase. Inter-\nestingly, the low-symmetry LiFe2F6can change from the\nA+ altermagnetic state to A- ferrimagnetic state under\ncompressive biaxial strain [9]. We also calculate relative\nenergies of four different magnetic states as functions of\nbiaxial strain. Indeed, the altermagnetic state change\nto ferrimagnetic state under compressive biaxial strain\n(Fig. 4 a). Likewise, we also used Berry phase method to\nestimate the ferroelectric P which is 13.4 µC/cm2along\nthezaxis for the ferrimagnetic state, which is basically\nconsistent with the previous calculation. The energy bar-rier is 41.5 meV per Fe atom for the ferrimagnetic state\n(Fig. 4 b). For the positive ferroelectric polarization, the\nFe2+andFe3+ions have opposite spin arrangement,\nwhich makes the magnetic moment of a primitive cell to\nbe -2 µB. When the ferroelectric polarization is reversed,\nthe magnetic moment of a primitive cell changes to 2 µB\n(Fig. 4 bInsets: final structure). Thus, the ferrimag-\nnetic LiFe2F6has very strong magnetoelectric coupling.\nFurthermore, the switch between positive and negative\nferroelectric polarization may cause huge change in the\nelectronic band structure.\nThen, we calculate the electronic band structures of the\npositive and negative ferroelectric polarization for ferri-\nmagnetic LiFe2F6. Comparing the positive and negative\nferroelectric polarization, their electronic band structures\nare the same but the spin of bands is reversed (Fig. 4 cand\nd), which corresponds to the reversal of the spin mag-\nnetic moment before and after the ferroelectric polariza-\ntion reversal. Interestingly, the spins of the valence band\nand the conduction band are opposite (Fig. 4 candd).\nSince Li,Fe, and Fare all light elements, the LiFe2F6\nhas weak SOC. In the ferrimagnetic LiFe2F6with weak\nSOC, the transition of electrons obeys the spin selec-\ntion rule. Thus, electrons transition from the bottom\nvalence band to the top conduction band need spin flip,\nthe electron-hole excitations give rise to spin-triplet exci-\ntons. Due to the spin selection rule, spin-triplet excitons\nmay be very stable in ferrimagnetic LiFe2F6. Moreover,\nthe spin-triplet excitons have spin to be 1 and -1 for the\npositive and negative ferroelectric polarization, respec-\ntively. More importantly, the spin-triplet excitons with\nS=1 and -1 can be switched by electric field due to strong\nmagnetoelectric coupling. If the spin-triplet excitons can\ncondense into superflow, the two spin superfluid phases\nwith S=1 and -1 can be switched by electric field, which\nis very important for theory, experiment, and the design\nof new devices. Finaly, we also calculate the electronic\nstructure of the ferrimagnetic LiFe2F6by hybrid func-\ntional. The bandgap of the ferrimagnetic LiFe2F6is in-\ncreased to 1.08 eV, but the spins of the valence band\nand the conduction band are still opposite. Thus, the\nferrimagnetic LiFe2F6may still possess the spin-triplet\nexcitonic phase in the framework of hybrid functional.\nIn summary, based on symmetry analysis and the first-\nprinciples electronic calculations, we predict the LiFe2F6\nis ad-wave altermagnetic, charge-ordering-mediated fer-\nroelectric material. Under biaxial compressive strain, the\nLiFe2F6transforms into charge-ordering-mediated ferri-\nmagnetic, ferroelectric phase with strong magnetoelec-\ntric coupling. Interestingly, the spins of the valence band\nand the conduction band are opposite in ferrimagnetic\nLiFe2F6, which facilitates the spin-triplet excitonic in-\nsulator phase. More importantly, If the spin-triplet ex-\ncitons condense into superflow, the two spin superfluid\nphases with S=1 and -1 can be switched by electric\nfield. 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B 6, 1968 (1972)." }, { "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\nkφ��k(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 .10.01 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": "2207.10775v1.Unusual_ferrimagnetism_in_CaFe2O4.pdf", "content": "1 Unusual ferrimagnetism in CaFe2O4 Hiroki Ueda1,†,‡,*, Elizabeth Skoropata1,†,*, Cinthia Piamonteze1, Nazaret Ortiz Hernández1, Max Burian1, Yoshikazu Tanaka2, Christine Klauser3, Silvia Damerio4,#, Beatriz Noheda4,5, and Urs Staub1,* 1 Swiss Light Source, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland. 2 RIKEN SPring-8 Center, Sayo, Hyogo 679-5148, Japan. 3 Laboratory for Neutron and Muon Instrumentation, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland. 4 Zernike Institute for Advanced Materials, University of Groningen, 9747AG- Groningen, Netherlands. 5 CogniGron Center, University of Groningen, 9747AG- Groningen, Netherlands. Abstract: Incomplete cancellation of collinear antiparallel spins gives rise to ferrimagnetism. Even if the oppositely polarized spins are owing to the equal number of a single magnetic element having the same valence state, in principle, a ferrimagnetic state can still arise from the crystallographic inequivalence of the host ions. However, experimental identification of such a state as “ferrimagnetic” is not straightforward because of the tiny magnitude expected for M and the requirement for a sophisticated technique to differentiate similar magnetic sites. We report a synchrotron-based resonant x-ray investigation at the Fe L2,3 edges on an epitaxial film of CaFe2O4, which exhibits two magnetic phases with similar energies. We find that while one phase of CaFe2O4 is antiferromagnetic, the other one is “ferrimagnetic” with an antiparallel arrangement of an equal number of spins between two distinct crystallographic sites with very similar local coordination environments. Our results further indicate two distinct origins of an overall minute M; one is intrinsic, from distinct Fe3+ sites, and the other one is extrinsic, arising from defective Fe2+ likely forming weakly-coupled ferrimagnetic clusters. These two origins are uncorrelated and have very different coercive fields. Hence, this work provides a direct experimental demonstration of “ferrimagnetism” solely due to crystallographic inequivalence of the Fe3+ as the origin of the weak M of CaFe2O4. † These authors contributed equally to this work. ‡ Present address: SwissFEL, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland. # Present address: Institut de Ciència de Materials de Barcelona, ICMAB-CSIC, Campus UAB, 08193 Bellaterra, Spain. * Correspondence authors: hiroki.ueda@psi.ch, elizabeth.skoropata@psi.ch, and urs.staub@psi.ch 2 Ferrimagnetism is a type of magnetic order in which the populations or amplitudes of oppositely-polarized spins are different, resulting in net magnetization (M) [1]. Antiparallel spins from different constituent magnetic elements [2], different valences of a magnetic element [1], and/or different crystallographic sites unbalancing the numbers of antiparallel spins [3] cause a ferrimagnetic order with a relatively large M. However, what if the number of parallel spins and antiparallel spins, both from a single magnetic element with the same valence, is the same but the magnetic sites that host them are crystallographically inequivalent? From the symmetry point of view, such a state breaks the time-reversal symmetry as ferrimagnets. Similar but distinct local coordination environments can result in different sizes of magnetic moments, even if the host ions have the same valence. However, experimental demonstration of such a state as being ferrimagnetic is not straightforward, as it requires independently quantifying two almost equivalent sites and grasping evidence that the sublattices host M, which is comparably as small as a potential impurity contribution. Tiny M displays a weak response to applied magnetic fields (H) and thus, naturally results in a large coercive field. Hence, such a “ferrimagnetic” state possesses two reversible states but can be robust towards H. CaFe2O4 is a unique material exhibiting two distinct magnetic phases with a collinear spin arrangement, both recognized as antiferromagnetic (AFM) [4,5]. The overall magnetism comes from two Fe sites, Fe1 and Fe2, having octahedral coordination with six surrounding O2-. Both sites possess the formal oxidation state of 3+ (S = 5/2) and are located on Wyckoff positions type 4c of the orthorhombic space group Pnma (#62), which does not have a space-inversion center. Namely, the two Fe sites have nominally the same valence, and there are exactly the same number of atoms of each type in a unit cell. Fe1 has a more distorted octahedral coordination than Fe2 [6]. The sites form a zigzag chain along [001] in a sequence of Fe1-Fe1-Fe2-Fe2 as seen in Fig. 1(a) or 1(b), and thus, two intra-chain magnetic interactions exist: between two different Fe sites and between two Fe of the same type. The super-exchange interactions among Fe3+ depend on the bond angles of Fe-O-Fe. The magnetic interactions between the same sites of Fe3+ along the [001] chain are weak because the bond angles are close to 90°. The weak intra-chain interactions result in two energetically almost degenerate spin configurations that have either ferromagnetic (FM) or AFM patterns along [001] for Fe3+ located at the same site [see Figs. 1(a) and 1(b), respectively]. The former phase is called A phase [magnetic space group: Pn’ma’ (#62.448)] while the latter is called B phase [magnetic space group: Pnma’ (#62.445)] with spins pointing along [010] but a different stacking sequence along [001]: up-up-down-down and up-down-up-down, respectively. The B phase is the magnetic ground state since the exchange interactions are AFM, while the A phase is metastable due to single-ion anisotropy emerging by mixing higher-energy multiplet states [7]. The local FM spin configuration in the A phase (up-up) can be found as the antiphase domain boundary of the AFM B phase [up-down-up (domain 1) and up-down-up (domain2)]. CaFe2O4 shows a phase transition to the B phase at TNB ≈ 200 K upon cooling and coexistence of the A and B phases on further cooling below TNA ≈ 175 K [8]. The temperature range of the phase coexistence persists down to the lowest temperatures for single-crystal samples [8] while remains down to T* ≈ 130 K for powder samples, below which only the A 3 phase is present [9], as visualized in Fig. 1(c). It is proposed that the A phase is more stable in powder samples than in single-crystals samples because of the strain introduced by grinding [7]. Interestingly, despite the believed AFM nature, a small remanent M of ~0.02 μB, which points along [010], summed over eight Fe3+ in the unit cell [10] is consistently reported in CaFe2O4, which onsets at around 185 K. Despite more than a half-century of investigation [4,5], the origin of the small M remains unclear. Previous studies based on magnetometry and neutron scattering proposed contributions to M (i) from Fe2+-induced ferrimagnetic clusters due to oxygen vacancies [11] and (ii) from domain boundaries of an AFM magnetic structure that can be locally FM [12]. Magnetic diffuse scattering experiments have attributed M to the presence of FM domain boundaries which also display an enhancement in H resulting from an increase in domain density [8]. Recently, uncoupled spins from the AFM domains have been discussed as a possible origin, based on spin Hall magnetoresistance measurements [13]. Note that the small M is not due to a spin canting since M and the Néel vector are both along [010] [12]. As found in Fig. 1(a), all Fe1 (Fe2) spins in the A phase point in the same direction but are antiparallel between the two sites. Hence, whereas the B phase is AFM, the A phase with the up-up-down-down configuration matching the periodicity of the lattice displays a “ferrimagnetic” state due to the crystallographic inequivalence of the two Fe3+ sites despite the same numbers of antiparallel spins. Different local coordination environments with distinct orbital hybridization with surrounding O2- could provide the finite orbital angular momentum of Fe3+ or a different expectation value of the spin moment leading to different sizes of antiparallel magnetic moments even though the resulting slight difference between the two Fe3+ magnetic moments may not easily be detectable by neutron scattering. Note that such orbital hybridization can correlate to the reported mixing of the higher-energy multiplet states [7] as observed in cobaltites [14,15]. This orbital mixing is predicted to stabilize the A phase as discussed above [7] and can create a magnetic anisotropy even though the high-spin state of Fe3+ generally exhibits a negligible magnetic anisotropy due to the absence of orbital angular momentum. X-ray magnetic circular dichroism (XMCD) is a powerful and beneficial spectroscopic technique for the study of magnetism. It is possible to measure element- and site-specific magnetic hysteresis curves [16] and to extract orbital angular momentum from the overall magnetic moment of an atom by applying the sum rule [17]. Even if the finite orbital angular momentum is too small to be detected, distinct local coordination environments between the two sites can be reflected in the spectrum, enabling us to independently quantify the two almost equivalent magnetic sites. Here, we report our investigation of the origin of M in epitaxial CaFe2O4 films by means of synchrotron-based x-ray techniques, resonant elastic x-ray scattering (REXS) and XMCD. These two techniques are complementary since REXS on the (001) reflection, which is forbidden due to the 21' symmetry along [001], probes the AFM behavior from the B phase without any contamination from the A phase because the A phase does not break the 21' symmetry along [001], while XMCD probes M that could originate from the potentially “ferrimagnetic” A phase. Our results indicate that the B phase remains down to the lowest temperature with a developing population of the A phase upon cooling, which resembles the 4 behavior of single crystals. XMCD spectra show three contributions in M; two of them representing antiparallel Fe3+ moments from the Fe1 and Fe2 sites, as expected for the A phase, with the third one originating from Fe2+, likely arising from defects, e.g., oxygen vacancies. Even though the orbital angular momentum that differentiates the size of the magnetic moments between the two sites is too small to be detected within the experimental precision, the measured magnetic hysteresis curves evidently attest to the ferrimagnetic nature of the A phase. \n Fig. 1 Magnetic structures of CaFe2O4 in (a) the A phase and (b) the B phase, where orange and blue spheres represent Fe3+ having a magnetic moment pointing to +b and –b, respectively, and (c) its phase sequence for powder samples (up) [9] and for single-crystal samples (down) [8]. A number written on the spheres denotes the Fe sites, either Fe1 or Fe2. (a) and (b) were drawn by VESTA [28]. Epitaxial films of CaFe2O4 with a thickness of ~100 nm were grown on a TiO2 (110) substrate by the pulsed laser deposition method. The films contain complex needle-like crystallines that organize forming domains because of the strain release processes and the epitaxial relation with the substrate. Details are described in Ref. [10]. Essentially for our study, there is a significant fraction of [001]-oriented domains along the surface normal, as displayed in Fig. 2(a). REXS experiments were performed at the RESOXS end-station [18] of the X11MA beamline [19] in the Swiss Light Source (Switzerland) and at the BL17SU [20] in the SPring-8 (Japan), and XMCD experiments were performed at the X07MA beamline [21] of the Swiss Light Source. The incident photon energy was set around the Fe L2,3 edges for both types of experiments, whose setups are shown in Figs. 2(b) and 2(c). XMCD spectra were measured at several temperatures below TNB in H up to ±6 T. The grazing incidence of x rays (~30°) allows us to detect M along [010], which is along the surface plane. We employed the total electron yield (TEY) and x-ray excited optical luminescence (XEOL) [22] detection modes, which provide surface- (~5-10 nm) and complete film depth sensitivity, respectively. To suppress charging affecting the TEY signals, 1.9 nm of Pt were deposited on the film surface. Representative spectra comparing TEY and XEOL data show excellent agreement (Supplemental Fig. S1). Due to the relatively large sample thickness, apparent self-absorption \nabc11112222Ca2+O2-11112222Fe3+(S// +b)Fe3+(S// -b)(a) (b) \nT BA+BATNB(≈ 200 K)TNA(≈ 175 K)T*(≈ 130 K )M. Songvilayet al. BA+BC. Stock et al. (c) Aphase Bphase 5 effects [22] hinder quantitative analysis of the XEOL spectra and hence, we show here only data taken with the TEY method. \n Fig. 2 (a) Schematic representation of the CaFe2O4 film orientation grown on a TiO2 substrate, adapted from Ref. [10]. There are different crystallographic domains present, and the surface normal is either [001] or [302] oriented. Experimental setup for (b) REXS and (c) XMCD measurements. The (001) reflection appears around ~55° of θ at the Fe L3 edge. For the XMCD measurements, we employed grazing incidence of x rays around 30°, enabling us to examine magnetization along [010]. Figure 3(a) shows (00L) REXS profiles around L = 1 at various temperatures. A small intensity observed even above TNB is due to resonantly-allowed scattering from aspheric electron distribution ascribed to Fe 3d orbitals (see Supplemental Material for details). The presence of aspheric electron distribution is also confirmed from the observed quadrupole splitting in Mössbauer spectra [10]. The increase in the (001) reflection intensities for decreasing temperatures below TNB [see Fig. 3(b)] reflects the growth of the B phase. Note that the A phase does not contribute to the (001) reflection because of the 21' symmetry along [001]. The distinct origins of the scattering below or above TNB are also reflected in the differences in the energy spectra, displayed in Fig. 3(c). The presence of the B phase down to the lowest achieved temperatures resembles the behavior of single-crystal samples, which show the coexistence of the A and B phases below TNA [see Fig. 1(c)] [8]. \n6 Fig. 3 (a) REXS profiles of the (001) reflection taken at various temperatures. (b) Temperature dependence of (001) integrated intensities, and (c) photon-energy dependence while maintaining the diffraction condition for the (001) reflection at various temperatures. (a) and (b) were taken at the photon energy of 710.8 eV. To visualize how the B phase develops upon cooling, we created two-dimensional maps of the (001) intensities at several temperatures across TNB with an x-ray beam size of ~15 μm ´ 30 μm scanning with a step size of ~7 μm and ~15 μm for horizontal and vertical directions, respectively, as displayed in Figs. 4(a)-4(d). Because of the various crystallographic domains in the CaFe2O4 film [10], we can only sample the crystallographic domain fraction that has the [001] axis out-of-plane. Therefore, the intensities are spatially inhomogeneous even at 200 K in the absence of magnetic order, as seen in Fig. 4(d). The irregular shape of the intensity distribution is consistent with the previously reported structural domain morphology [10]. To better examine the developing magnetic contributions, we normalized the intensities taken at each temperature to those taken at 200 K. Figures 4(e)-4(g), obtained from the maps in Figs. 4(a)-4(c), establish that the magnetic contribution develops everywhere over the scanned region but is not uniform. The nonuniform magnetic profiles can be due to (i) the different amplitudes of the magnetic contributions from in-plane crystallographic domains [10] and/or (ii) the spatially inhomogeneous evolution of the B phase. The latter implies the evolution of the A phase, which does not contribute to the (001) intensities, and is consistent with the XMCD data shown later. Note that (ii) is not caused by inhomogeneously distributed crystallographic domains with a different out-of-plane orientation because of the normalization by the 200 K data. \n(a) \n(b) \n705710715720725024 30 K 100 K 150 K 200 K 300 KIntensity [arb. units]\nEnergy [eV](c) 7 Fig. 4 Two-dimensional maps of (001) intensities taken at (a) 40 K, (b) 100 K, (c) 150 K, and (d) 200 K, with the photon energy of 710.25 eV. Intensities are normalized by those of the 200 K data at (e) 40 K, (f) 100 K, and (g) 150 K. These maps were taken in the region surrounded by a white box in (h). Black areas in (a)-(g) lie outside of the sample. The appearance of the “ferrimagnetic” A phase can be investigated by XMCD. Figure 5 displays XAS and XMCD spectra taken at T = 30 K and 150 K in H = 6 T after zero-field cooling (ZFC) from room temperature. The overall XAS with two well-resolved peaks (eg and t2g) at the Fe L3 edge is typical of a predominant Fe3+ character. As the two different Fe3+ sites are AFM coupled in the A phase and are expected to have slightly different spectral shapes due to the distinct local coordination environments, the two peaks with opposite signs (β and γ) in the XMCD can be assigned to these two sites. While the amplitude of γ is similar between 30 K and 150 K, that of β is different at the two temperatures and is comparable with the γ peak at 30 K, further supporting their origins from distinct Fe-sites. Note that we found a history dependence on XMCD spectra for the two peaks, consistent with the fact that the A phase is in competition with the B phase and thus the A-phase population can differ by the T and H history conditions. Shown in Fig. 6 are the XMCD spectra taken at the same measurement condition, at 150 K in 6 T, but after approaching with different T and H evolution; after cycling H (±6 T) at 2 K for the black curve, compared with no prior magnetic cycle for the green curve. Both the peaks are weaker in the green curve, indicating a smaller M and less population of the A phase with respect to the B phase. The history dependence is consistent with the inhomogeneous development of the B phase implied by the REXS data. \n8 Fig. 5 (a) XAS and (b) XMCD taken at 30 K (red) and 150 K (black). XAS is obtained as the sum of two data with opposite helicity of circular polarization (C+/C-) and is normalized to its highest peak, whereas XMCD is obtained as the difference between C+ and C-. \n Fig. 6 History dependence of XMCD spectra taken at 150 K after ZFC directly from room temperature (green) compared with the XMCD at 150 K after ZFC to 2 K, magnetic cycling (±6 T), and warming in +6 T (black). The black data is the same as Fig. 5(b) and Fig. 7(b). For the assignment of the XMCD spectral features, multiplet simulations of the Fe sites were performed using the MultiX code [23]. Crystal field multiplet parameters of a typical Fe Oh site were used [24], but the local structure of the Fe1 and Fe2 sites was obtained from the bulk structure [25] in order to account for the differing octahedral distortions of the two crystallographic sites. Figure 7 shows the results of the multiplet simulations for the XMCD data shown in Fig. 5. The calculated site-selective XMCD spectra are shown in Figs. 7(c) and 7(d). Based on these results, we find that the Fe2 and Fe1 sites display a relative energy shift of ~0.45 eV accounting for the well-resolved β and γ peaks, respectively. On the other hand, the feature labeled as α is not reproduced well by the two Fe3+ sites. Such a \n9 lower-energy peak compared to that of Fe3+ is often assigned to Fe2+. However, multiple possible contributions from anisotropic Fe2+ in differently oriented domains hindered us to reproduce the feature uniquely with multiplet simulations. Nevertheless, the independent behavior of the peak α from the Fe3+ features is clearly visible in the experiment, as shown in Fig. 6. Thus, it is reasonable to attribute the peak α to Fe2+. Our observation of an XMCD signal from Fe2+ suggests a high-spin state, unlike the proposed low-spin state (S = 0) for the Fe2+ from magnetic susceptibility measurements of bulk samples [11]. Fe2+ can originate from oxygen vacancies as discussed previously [11] and can form a ferrimagnetic cluster with surrounding Fe3+. Note that the population of Fe2+ must be very small as the XAS is dominated by the typical Fe3+ spectra, which is also consistent with Mössbauer spectroscopy reports of a single order parameter of the Fe3+ sites below TNB [10,26] and no observable Fe2+ (detectable typically within few percent). The peak β reflecting the Fe3+ moments of the Fe2 site is larger at 30 K than at 150 K while the peak γ reflecting the Fe3+ moments of the Fe1 site is almost the same between the two temperatures, indicating different temperature evolution of the Fe1 and Fe2 sites in the A phase. At +6 T and 150 K, we find the antiparallel orientation of the Fe1 and Fe2 sites with a clear imbalance of the site-specific magnetization with MFe1 > MFe2. At +6 T and 30 K, MFe2 becomes larger and both Fe3+ sites contribute approximately equally to the XMCD. This can be understood by the growth of the A phase when lowering the temperature, as reported in single-crystal samples; for higher T the Fe2-site moments (β) are significantly smaller than the Fe1-site moments (γ) in thin domains of the A phase because of the imbalanced population, whereas for lower T the population gets more balanced [see the sketch in Fig. 7(e)]. In other words, it suggests that the intrachain exchange interaction between two Fe1 is likely more FM than that between two Fe2. The favorable FM interaction between two Fe1 might correlate with more distortion in the octahedral coordination than Fe2, which can mix the higher-energy multiplet states into the ground state and stabilize the A phase with intrachain FM coupling between the same sites of Fe. The view in the sketch explains larger M at 150 K than 30 K [10] as two net M from the sublattices represented by the peaks β and γ are antiparallel to each other and β develops more and gets comparable with γ at lower temperatures. Since the local FM spin configuration in the A phase is the same as that in the antiphase domain boundary of the AFM B phase, the slightly different onset temperature of M (~185 K) compared to TNA might imply the short-range order of the A phase, which is likely present as the antiphase domain boundary of the B phase. 10 Fig. 7 XMCD spectra taken at (a) 30 K and 6 T, and (b) 150 K and 6 T, which are decomposed into three contributions, as found in (c) and (d). Broken curves correspond to calculated respective contributions to the spectra while a red curve corresponds to the sum of the contributions. (e) A sketch representing the growth of the A phase and resultant decrease in total magnetization. We aimed to obtain direct evidence that the A phase is ferrimagnetic. The reversal of β and γ XMCD peaks should take place by sweeping H. The same fitting procedure as done for the data in Fig. 7 enables us to extract site-specific magnetic hysteresis curves. Figures 8(a) and 8(b) display XMCD spectra taken at 150 K and 30 K, respectively, in various H. Extracted magnetic hysteresis curves of the two Fe sites are shown in Figs. 8(c) and 8(d). It is observed that the XMCD peaks reverse their signs by sweeping H at 150 K, directly \nT= 150 KH= + 6 TT= 30 KH= + 6 T(a)(b)\n(c)(d)\n(e)High TMtot111111112222222222BAB\nLow TMtot111111112222222222BAB11 evidencing that these two AFM-coupled Fe3+ sites are responsible for M, namely the A phase of CaFe2O4 is ferrimagnetic. Another feature is that M from the two Fe3+ sites does not reverse at 30 K even when going from +6 T to –6 T, in contrast to the data taken at 150 K. This observation is counterintuitive as the high-spin configuration of Fe3+ commonly exhibits a small anisotropy leading to small coercive fields due to the equal filling of the orbitals. However, the tiny overall magnitude of M from the two imperfectly compensated sublattice magnetizations leads to a strongly reduced force on M from H, naturally explaining an enlarged coercive field. In addition, the significant distortion of the FeO6 octahedron that could mix higher-energy multiplet states into the ground state through charge transfer (or orbital hybridization) with the surrounding O2- [7] may generate magnetic anisotropy that results in an enlarged coercive field. The minor steps observed at H = 0 T in the 30 K Fe3+ data of Figs. 8(c) and 8(d) can be due to (i) the flipping of Fe2+ XMCD signals that spectrally overlap on the peaks β and γ and/or (ii) the flipping of Fe3+ moments coupling to Fe2+. Both possibilities imply a small coercive field of Fe2+ moments. Therefore, the uncoupled spins from the AFM phase, i.e., from the main Fe3+ spins, reported previously [13] should be ascribed to defective Fe2+ that can easily respond to H. \n Fig. 8 Magnetic-field dependence of the XMCD signals at (a) 150 K and (b) 30 K. Hysteresis curve for Fe3+ (1) and Fe3+ (2) at (c) 30 K and (d) 150 K obtained from fits with calculated spectra to (a) and (b). In conclusion, we have investigated the origin of the tiny remanent magnetization reported in nominally antiferromagnetic CaFe2O4 by synchrotron-based spectroscopic techniques. Although the numbers of antiparallel spins, up and down both hosted by Fe3+ with octahedral coordination, are exactly the same in the A phase, we clarified that the A \nT= 150 K+ 6 T0 T-1 T-6 T0 T+ 1 TT= 30 K30 K\n150 K(b)(c)\n(d)+ 6 T0 T-1 T-6 T0 T+ 1 T(a)12 phase is ferrimagnetic due to the difference in the local coordination environments of the two sites by measuring site-specific magnetic hysteresis loops. The magnetization magnitude is comparably small with an independent defect contribution (Fe2+) having a different coercive field. Such a ferrimagnetic state may be useful for future spintronics as (i) the coercive field is much larger than normal ferromagnets or ferrimagnets due to small overall magnetization, as evident in our study on Fe3+, which usually exhibits small coercive fields, demonstrating the robustness of its magnetic state under an external magnetic field, (ii) there are two switchable ferroic states (±M), and (iii) dominating antiferromagnetic interactions, allowing magnetic resonance frequencies in the sub-THz range as antiferromagnets. Our discovery may open up a new direction of material design for future devices. Data availability Experimental data are accessible from the PSI Public Data Repository [27]. References 1. M. L. Néel, Propriétés magnétiques des ferrites; ferrimagnétisme et antiferromagnétisme, Ann. Phys. 12, 137-198 (1948). 2. I. S. Jacobs, High field magnetization study of ferrimagnetic arrangements in chromite spinels. J. Phys. Chem. Solids. 15, 54-65 (1960). 3. S. J. Pickart and H. A. Alperin, Magnetic structures of ferrimagnetic RbNiF3 and CsFeF3. J. Appl. Phys. 42, 1617-1618 (1971). 4. H. Watanabe, H. Yamauchi, M. Ohashi, M. Sugimoto, T. Okada, and M. Fukase, Neutron diffraction study on CaFe2O4. J. Phys. Soc. Jpn. 22, 939 (1967). 5. L. M. Corliss, J. M. Hastings, and W. Kunnmann, Magnetic phase equilibrium in chromium-substituted calcium ferrite. Phys. Rev. 160, 408-413 (1967). 6. B. F. Decker and J. S. Kasper, The structure of calcium ferrite. Acta Cryst. 10, 332-337 (1957). 7. H. Lane, E. E. Rodriguez, H. C. Walker, Ch. Niedermayer, U. Stuhr, R. I. Bewley, D. J. Voneshen, M. A. Green, J. A. Rodriguez-Rivera, P. Fouquet, S.-W. Cheong, J. P. Attfield, R. A. Ewings, and C. Stock, Metastable antiphase boundary ordering in CaFe2O4. Phys. Rev. B 104, 104404 (2021). 8. C. Stock, E. E. Rodriguez, N. Lee, M. A. Green, F. Demmel, R. A. Ewings, P. Fouquet, M. Laver, Ch. Niedermayer, Y. Su, K. Nemkovski, J. A. Rodriguez-Rivera, and S.-W. Cheong, Solitary magnons in the S = 5/2 antiferromagnet CaFe2O4. Phys. Rev. Lett. 117, 017201 (2016). 9. M. Songvilay, S. Petit, M. Koza, S. Rols, E. Suard, V. Skumryev, C. Martin, and F. Damay, Disorder and magnetic excitations in CaCrxFe2–xO4 (x = 0, 0.5). Phys. Rev. B 101, 014407 (2020). 13 10. S. Damerio, P. Nukala, J. Juraszek, P. Reith, H. Hilgenkamp, and B. Noheda, Structure and magnetic properties of epitaxial CaFe2O4 thin films. npj Quantum Materials 5, 33 (2020). 11. R. Das, S. Debnath, G. Narsinga Rao, S. Narasimhan, and F. C. Chou, Ferrimagnetic cluster formation due to oxygen vacancies in CaFe2O4-δ. Phys. Rev. B 98, 144404 (2018). 12. C. Stock, E. E. Rodriguez, N. Lee, F. Demmel, P. Fouquet, M. Laver, Ch. Niedermayer, Y. Su, K. Nemkovski, M. A. Green, J. A. Rodriguez-Rivera, J. W. Kim, L. Zhang, and S.-W. Cheong, Orphan spins in the S = 5/2 antiferromagnet CaFe2O4. Phys. Rev. Lett. 119, 257204 (2017). 13. S. Damerio, A. A. Kaverzin, V. Ocelík, G. R. Hoogeboom, B. J. van Wees, and B. Noheda, Antiferromagnetic ordering and uncoupled spins in CaFe2O4 thin films probed by spin Hall magnetoresistance. Adv. Electron Mater. 8, 2100963 (2021). 14. R. H. Potze, G. A. Sawatzky, and M. Abbate, Possibility for an intermediate-spin ground state in the charge-transfer material SrCoO3. Phys. Rev. B 51, 11501-11506 (1995). 15. P. Ravindran, H. Fjellvag, A. Kjekshus, P. Blaha, K. Schwarz, and J. Luitz, Itinerant metamagnetism and possible spin transition in LaCoO3 by temperature/hole doping. J. Appl. Phys. 91, 291-303 (2002). 16. G. van der Laan and A. I. Figueroa, X-ray magnetic circular dichroism –A versatile tool to study magnetic. Coord. Chem. Rev. 277-278, 95-129 (2014). 17. P. Carra, B. T. Thole, M. Altarelli, and X. Wang, X-ray circular dichroism and local magnetic fields. Phys. Rev. Lett. 70, 694-697 (1993). 18. U. Staub, V. Scagnoli, Y. Bodenthin, M. García-Fernández, R. Wetter, A. M. Mulders, H. Grimmer, and M. Horisberger, Polarization analysis in soft X-ray diffraction to study magnetic and orbital ordering. J. Synchrotron Rad. 15, 469-476 (2008). 19. U. Flechsig, F. Nolting, A. Fraie Rodríguez, J. Krempaský, C. Quitmann, T. Schmidt, S. Spielmann, and D. Zimoch, Performance measurements at the SLS SIM beamline. AIP Conf. Proc. 1234, 319-322 (2010). 20. T. Takeuchi, A. Chainani, Y. Takata, Y. Tanaka, M. Oura, M. Tsubota, Y. Senba, H. Ohashi, T. Mochiku, K. Hirata, and S. Shin, An ultrahigh-vacuum apparatus for resonant diffraction experiments using soft x rays (hυ = 300-2000 eV). Rev. Sci. Instrum. 80, 023905 (2009). 21. C. Piamonteze, U. Flechsig, S. Rusponi, J. Dreiser, J. Heidler, M. Schmidt, R. Wetter, M. Calvi, T. Schmidt, H. Pruchova, J. Krempasky, C. Quitmann, H. Brune, and F. Nolting, X-Treme beamline at SLS: X-ray magnetic circular and linear dichroism at high field and low temperature J. Synchrotron Rad. 19, 661-674 (2012). 14 22. C. Piamonteze, Y. W. Windsor, S. R. V. Avula, E. Kirk, and U. Staub, Soft X-ray absorption of thin films detected using substrate luminescence: a performance analysis. J. Synchrotron Rad. 27, 1289-1296 (2020). 23. A. Uldry, F. Vernay, and B. Delley, Systematic computation of crystal-field multiplets for x-ray core spectroscopies. Phys. Rev. B 85, 125133 (2012). 24. B. Liu, C. Piamonteze, M. U. Delgado-Jaime, R.-P. Wang, J. Heidler, J. Dreiser, R. Chopdekar, F. Nolting, and F. M. F. de Groot, Sum rule distortions in fluorescence-yield x-ray magnetic circular dichroism. Phys. Rev. B 96, 054446 (2017). 25. I. O. Galuskina, Y. Vapnik, B. Lazic, T. Armbruster, M. Murashko, and E. V. Galuskin, Harmunite CaFe2O4: A new mineral from the Jabel Harmun, West Bank, Palestinian Autonomy, Israel, Am. Mineral. 99, 965-975 (2014). 26. H. Yamamoto, T. Okada, H. Watanabe, and M. Fukase, Mössbauer effect study of spin relaxation in CaFe2O4. J. Phys. Soc. Jpn. 24, 275-279 (1968). 27. https://doi.org/10.16907/4a134d5f-29be-45c8-8c3c-b86065c34703 28. K. Momma and F. Izumi, VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272-1276 (2011). Acknowledgements The resonant x-ray diffraction experiments were performed at the X11MA beamline in the Swiss Light Source under proposal No. 20191307 and at the BL17SU in the SPring-8 under proposal No. 20200012. The x-ray magnetic circular dichroism measurements were performed at the XTreme beamline in the Swiss Light Source during in-house access. H.U. acknowledges the National Centers of Competence in Research in Molecular Ultrafast Science and Technology (NCCR MUST-No. 51NF40-183615) from the Swiss National Science Foundation and from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 801459 – FP-RESOMUS. E.S. is supported by the NCCR Materials’ Revolution: Computational Design and Discovery of Novel Materials (NCCR MARVEL No. 182892) from Swiss National Foundation and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 884104 (PSI-FELLOW-III-3i). N. O. H. acknowledges financial support of the Swiss National Science Foundation, No. 200021_169017. M.B. is supported by the Swiss National Science Foundation through project Nos. 200021-196964 and 200021_169698, respectively. Financial support by the Groningen Cognitive Systems and Materials Center (CogniGron) and the Ubbo Emmius Funds of the University of Groningen is also gratefully acknowledged. Additional information Competing interests: The authors declare no competing interests. " }, { "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. Table 1 summarizes the parameters used for modeling magnetization dynamics, which were obtained from Ostler\net al. and Radu et al.14, 42. The anisotropy (K u) is along the z-direction. Initially, we applied an out-of-plane magnetic field to\nalign the spins in both sublattices in the out-of-plane direction, with the Gd sublattice pointing parallel to the + z direction. To\ncreate the initial state with skyrmion, a 0.01 T magnetic field is applied41. After initialization, the magnetic field is set to zero\nthroughout the calculation.\n6/9References\n1.Diao, Z. et al. Spin-transfer torque switching in magnetic tunnel junctions and spin-transfer torque random access memory.\nJ. Physics: Condens. Matter 19, 165209, DOI: 10.1088/0953-8984/19/16/165209 (2007).\n2.Manchon, A. et al. Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems. Rev. Mod. Phys.\n91, 035004, DOI: 10.1103/RevModPhys.91.035004 (2019).\n3.Grimaldi, E. et al. 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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": "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. Mater. Sci. 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Junction size effect on switching current and thermal stability in \nCoFeB/MgO perpendicular magnetic tunnel junctions. Applied Physics Letters 99, \n042501 (2011). https://doi.org:10.1063/1.3617429 \n51 Hu, G. et al. in 2019 IEEE International Electron Devices Meeting (IEDM). 2.6.1 -\n2.6.4. \n52 Rehm, L., Wolf, G., Kardasz, B., Pinarbasi, M. & Kent, A. D. Sub -nanosecond spin -\ntorque switching of perpendicular magnetic tunnel junction nanopillars at cryogenic \ntemperatures. 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": "1305.3256v1.Ferrimagnetic_Spin_Wave_Resonance_and_Superconductivity_in_Carbon_Nanotubes.pdf", "content": "arXiv:1305.3256v1 [physics.gen-ph] 14 May 2013Ferrimagnetic Spin Wave Resonance and Superconductivity i n Carbon Nanotubes\nIncorporated in Diamond Matrix\nDmitri Yerchuck (a), Yauhen Yerchak (b), Vyacheslav Stelmakh ( b), Alla Dovlatova (c), Andrey Alexandrov (c)\n(a) - Heat-Mass Transfer Institute of National Academy of Scien ces of RB, Brovka Str.15, Minsk, 220072, dpy@tut.by\n(b) - Belarusian State University, Nezavisimosti Avenue 4, Minsk, 2 20030, RB\n(c) - M.V.Lomonosov Moscow State University, Moscow, 119899\n(Dated: April 15, 2019)\nThe phenomenon of ferrimagnetic spin wave resonance [uncom pensated antiferromagnetic spin\nwave resonance] has been detected for the first time. It has be en observed in carbon nanotubes,\nproduced by high energy ion beam modification of diamond sing le crystals in /angbracketleft100/angbracketrightdirection. Pe-\nculiarities of spin wave resonance observed allow to insist on the formation in given nanotubes\nofs+-superconductivity at room temperature, coexisting with u ncompensated antiferromagnetic\nordering.\nPACS numbers: 71.10.-w, 73.63.Fg, 78.30.-j, 76.30.-v, 76. 50.+g, 78.67.-n\nKeywords: magnetism, nanotubes, superconductivity\nI. INTRODUCTION AND BACKGROUND\nA number of theoretical and experimental works have\nbeen devoted to the studies of ferromagnetic and an-\ntiferromagnetic spin waves, including resonance behav-\nior. Theoretical works are starting in 1930 from pio-\nneering work of Bloch [1], where the ferromagnetic spin\nwaves were fundamentally studied. It has been found in\nparticular, that ferromagnetic spin waves obey the k2-\ndispersion law. At the same time, it was established,\nthat antiferromagnetic spin waves have quite other - lin-\neark-dispersion law. It was done by Hulthen. Hulthen\nhas carried through the quantization of the spin waves\nin an antiferromagnet and he has found for the first\ntimek-dispersion law for antiferromagnetic spin waves\n[2] in 1936. The study of antiferromagnetic spin waves\nwas extended by Anderson [3] by inclusion of zero-point\nspin wave energy. Anderson has shown, that the ex-\nact ground-state energy eigenvalue of antiferromagnet\nis close to the energy of its approximate two-sublattice\nmodel. Anderson has found, that the frequencies of the\nspin waves fall into two branches, and they are\nωk=dJS/radicalbig\n1−γk2, (1)\nwhereγkis\nγk=d/summationdisplay\ni=1coski\nd, (2)\nki,i=1,dare the components of the wave vector /vectork,d\nis dimensionality of the lattice, Sis spin value, Jis ex-\nchange coupling parameter. The equation (1) is reduced\nin the case of small amplitude values of vector /vectorkto the\nfollowing form\nωk∼dJSk, (3)\nThisdispersionlawiscoincidingwiththelaw,obtainedin\n[2], that is it really quite different from the ferromagneticcase. Anderson draws attention, that the differerence in\ndispersion laws is not the most significant difference be-\ntween the two types of spin waves; it is the difference in\namplitude per quantum of excitation which leads to the\nmorestrikingeffects. It means, inparticular, thatit isre-\nquired much more energy to excite an antiferromagnetic\nspin wave, than to excite a ferromagnetic spin wave.\nSpin wave theory was devoloped, for instance, in the\nworks [4], [5], [6], [7], [8] and in many others. It has\nbeen shown in [4], [5], that the transformation to nor-\nmal spin wave modes, used by Anderson for the case of\nthe absence of external magnetic field is not suitable in\nthe case ofits presence. Independently, normalspin wave\nmodes were derived using a different formalism by Ziman\n[6] and by Nakamura [7]. A brief outline is given in [5] for\nthe spin-wave approximation to the near ground states\nof ferromagnetism and antiferromagnetism. Simple pic-\ntorial models of spin waves were introduced. These mod-\nels clarify the striking difference between ferromagnetic\nspin waves, which obey the k2-dispersion law, and an-\ntiferromagnetic spin waves, which obey the k-dispersion\nlaw. The frequency spectrum and the damping constant\nofthe spinwavesin thetwo-sublatticeantiferromagnetics\nwere investigated in [8] on the quantum-statistical basis\nby the use of the relaxation function method. The inter-\nest for nanophysics represents the study of uniform spin\nwave modes in antiferromagnetic nanoparticles with un-\ncompensated moments, performed in the work by Bahl\net al [9]. According to [9], in magnetic nanoparticles the\nuniform precession (q = 0 spin wave) mode gives the pre-\ndominant contribution to the magnetic excitations. The\nauthors have calculated the energy of the uniform mode\nin antiferromagnetic nanoparticles with uncompensated\nmagnetic moments, using the coherent potential approx-\nimation. They have shown, that in the simple uniaxial\ncase, the uncompensated moment has a profound effect\non the excitation energy, but in the planarcase it is much\nless significant. In fact it has been shown, that spin wave\nmodes in antiferromagnetics and ferrimagnetics are obe-2\ning qualitatively to the same laws, the difference is the\nonly quantitative.\nEspecially interesting seem to be the resonance phe-\nnomena on spin waves. The phenomenon of ferromag-\nnetic spin wave resonance (FMSWR) was theoretically\npredicted by Kittel [10] and it was experimentally con-\nfirmed by Seavey and Tannenwald [11]. On the ob-\nservation of antiferromagnetic spin wave resonance was\nclaimed in [12], at that the authors have insisted, that\ntheir work is the first work in given field. The experi-\nments were performed on epitaxial films of MnF2. Let\nus remark, that the interpetation of rather interesting\nexperimental results, proposed in [12], seems to be incor-\nrect. The authors describe antiferromagnetic spin wave\nresonance in the frame of the k2-dispersion law, that\nis, in full contradiction with general spin wave theory,\nbriefly above rewieved. The analysis of their results is\nembarrassed by inaccurate representation of experimen-\ntal data in [12]. The authors insist, that several SWR\nmodes are unresolved, however concrete number of unre-\nsolved modes is not indicated. Taking into account the\ndistance between the first and the second lines in low in-\ntensive line sequence in the spectrum of the the sample\nwith the thickness in 0.98 µm, equaled to ≈50 G and\nthe distance of the first line in given sequence from the\nposition ofmain resonancemode equaled to ≈83G, then\naccording to k2-dispersion law authors’ concept, the first\nline has to be the seveth mode. It is not in agreement\nwith correspondingdistance between the the seventh and\nthe ninth modes in the spectrum of the sample with the\nthickness in 0.23 µm, since to the splitting in ≈50 G\nin the spectrum of the sample with the thickness in 0.98\nµmhastocorrespondthe splitting ≈(0.98/0.23)2×50G,\nthat is 907.4 G. At the same time the evaluation of anal-\nogous distance, that is the distance between the seventh\nand the ninth modes from the spectrum of the sample\nwith the thickness in 0.23 µmgives the value in ≈575\nG. We see, that the descrepance is large. Moreover, tak-\ning into account the linewidth of the modes, equaled to\n8 G, the only the first and the third modes can be unre-\nsolved between themselves, but they must give the fea-\nture (shoulder or not very pronouced peak) on the wing\nof much more broad main resonance mode, all the more\nthe mode with number 5 has to be seen. To give the cor-\nrect explanation for the spectra observed, the additional\nexperimental data have to be represented - the values of\nratio of amplitudes of magnetic component of microwave\nfield and the ratio of Q-factors, gain factors for intensity\nin the spectra of both the samples and also modulation\namplitude and modulation frequency used. It can be, in\nparticular, new quantum effect.\nTherefore,itisfollowedfromabovegivenanalysis,that\nthework[12]cannotbereferredtothebibliographyofthe\nworks, describing the phenomenon of AFMSWR. At the\nsame time the spin-wave spectrum in antiferromagnets\nwas studied experimentally already in 60th years of the\nlast century, at that the theory of AFMSWR was simul-\ntaneously developed. For instance, the role of uniaxialtension on the spin-wave spectrum in easy-plane antifer-\nromagnets was studied, [13], [14], [15]. It was shown that\nthe effect of uniaxial tension in the basal plane of the\nantiferromagnet crystal can be described by an effective\nmagnetic field, at that the additional gap arising in the\nspin-wave spectrum has to be taken into consideration.\nThe e���ect of uniaxial tension is rather strong. It has\nbeen shown, that even weak distortions can significantly\nmodify the spin wave spectrum.\nVery interesting results on AFMSWR are reported in\n[16]. Spinwaveresonancelineswithextremelylargewave\nnumbers corresponding to wave vectors kin the range\n105−106cm−1were observed in thin plates of FeBO 3.\nSpin-wave resonance was clearly observed in the temper-\nature interval 30-250 K It was the low frequency branch\nof the spin-wave spectrum, which was analysed at static\nmagnetic field H, directed transverselyto C3crystal axis\nby the relation, taking additionally into account, in dis-\ntinction from relation (1), Dzyaloshinsky field and mag-\nnetoelastic coupling\nω1,k=γ/radicalBig\nH(H+HD)+H2\n∆+α2\n1k2\n1+α2\n2k2\n2,(4)\nwhereγis the gyromagnetic ratio, HDis the Dzyaloshin-\nsky field, H∆is a parameter determined by magnetoe-\nlastic coupling, α1andα2are non-uniform exchange\nconstants in the basal plane and along the C3-axis re-\nspecively, k1andk2are wave-vector components anal-\nogously in the basal plane and along the C3-axis re-\nspecively, external field His applied in the basal plane\nof the crystal.\nWe have to remark, that there is optical analogue of\nAFMSWR, that is antiferroelectric spin wave resonance\n(AFESWR), which was theoretically described and ex-\nperinmentally discovered for the first time by infrared\n(IR) spectroscopy study of carbynes in [17]. Especially\nsignificant was the observation of AFESWR with linear\nk-dipersion law, where kis magnitude of wave vector /vectork,\nthat is the general spin wave theory, [2], [3], [4], [5], [6],\n[7], wasexperimentallyconfirmed forAFESWR-casetoo.\nIt is in agreementwith the theoreticalresults represented\nin [17], from which is followed, that the conclusionsofthe\nantiferroelectric spin wave theory and antiferromagnetic\nspin wave theory are qualitatively identical, in particu-\nlar, the same linear k-dipersion law is taking place (at\nlowk-values).\nDiscovery of new types of superconducting materials\nhas accelerated in 21th century. Especially interestig was\nthe discovery of superconductivity coexisting with anti-\nferromagnetic ordering in the iron-based layered pnictide\ncompound LaFeAsO (that is, in material with prevailed\n2D-dimensional strucure). It was repoted in [18]. Next,\nthesuperconductivityhasbeendiscoveredinbothoxygen\ncontaining RFeAsO (R = La, Nd, Sm) compounds and\nin oxygen free AFe2As2(A = Ba, Sr, Ca) compounds.\nIt is interesting, that the superconductivity occurs upon\ndoping into the FeAs layers of either electrons or holes.3\nLet us remark, that owing to the highly two-dimensional\nstructure the pnictides are like to the cuprates. It gave\nrise to the viewpoint that the physics of the pnictides is\nsimilar to the cuprates. However, there is at present the\ndominating viewpoint that Mott-transition physics does\nnotplayasignificantroleforthe ironpnictides, andthere\nare strong indications, that antiferromagnetic ordering is\ndetermined by the formation of the spin-density wave\n(SDW), that is quite another type of antiferromagnetism\nin comparison with Heisenberg antiferromagnetism of lo-\ncalized spins takes place.\nAt the same time on the coexistence of the super-\nconductivity with spin wave resonance has not been re-\nported. It wiil be reported in given work on the observa-\ntion of uncompensated antiferromagnetic spin wave reso-\nnance, that is, in fact, ferrimagnetic spin wave resonance\nin carbon NTs, which is coexisting with the supercon-\nductivity for the first time.\nOn the experimental revealing of magnetic ordering\nat all in carbon structurally ordered systems was re-\nported for the first time during the IBMM-Conference\nin Knoxville, TN, USA [19] and on E-MRS Conference\nin Strasbourg, France [20]. Let us remark, that almost in\nthe sametime wasreporedonmagneticorderingin struc-\nturally non-ordered carbon materials in the work [21],\nwhere ferromagnetic ordering in pyrolytic carbon, pro-\nduced by chemical vapour deposition method was found.\nLet us also remark, that simultaneously, the reports [19],\n[20] were the first reports on the formation by high en-\nergy ion beam modification (HEIBM) of diamond single\ncrystals structurally and magnetically ordered quasi-one-\ndimensional (quasi-1D) system along ion tracks, that is,\non the formation of new carbon allotropic form, which\nwas identified with nanotubes (NTs), incorporated in di-\namond matrix in direction, which is precisely coinciding\nwith ion beam direction. Given NTs possess by a num-\nber of very interesting physical properties [22], [25], [24].\nWhen concern the only magnetic ordering, it was estab-\nlished, that, for instance, incorporated nanotubes, pro-\nduced by neon HEIBM of diamond single crystal along\n/angbracketleft100/angbracketrightcrystallographic direction, possess by weak antifer-\nromagnetic ordering [22], [25], [24]. At the same time,\ncopper HEIBM with implantation direction along /angbracketleft111/angbracketright\ncrystal axis, nickel HEIBM with implantation direction\nalong/angbracketleft110/angbracketrightaxis [22], [25], [24] and boron HEIBM of\npolycrystalline diamond films with implantation direc-\ntion transversely to film surface [23] lead to formation\nof NTs, incorporated in diamond matrix, which possesss\nby ferromagnetic ordering. It was established directly by\nobservation of ferromagnetic spin wave resonance [23],\n[25], [24]. It was found, that magnetic ordering is in-\nherent property for given carbon electronic system and\nit is not connected with magnetic impurities. Very re-\ncently [26], antiferroelectric ordering has been found in\nthe same pure carbon allotropic form - quasi-1D carbon\nzigzag-shaped nanotubes (CZSNTs), obtained by boron-\nand copper-HEIBM of diamond single crystals in /angbracketleft111/angbracketright-\ndirection. The proof for antiferroelectric ordering wasobtained directly by means of the detection of the new\noptical phenomenon - antiferroelectric spin wave reso-\nnance(AFESWR),whichwastheoreticallydescribedand\nexperinmentally confirmed for the first time in [17]. Re-\ncently, the physical origin of the mechanism of the for-\nmation of ferromagnetic ordering in carbon nanotubes\nproduced by high energy ion beam modification of di-\namond single crystals in /angbracketleft110/angbracketrightand/angbracketleft111/angbracketrightdirections has\nbeen established. It is determined by asymmetry of spin\ndensity distribution in Su-Schrieffer-Heeger topological\nsoliton lattice formed in 1D Fermi quantum liquid state\nof the only π-electronic subsystem of given NTs [27]. It\nwas experimentally proved, that σ-electronic subsystem\ndoes not give any contribution to the mechanism of the\nformation of ferromagnetic ordering in given NTs.\nQuite other picture was observed in carbon NTs, pro-\nduced by high energy ion beam modification of diamond\nsingle crystals in /angbracketleft100/angbracketrightdirection. The strong uncom-\npensated antiferromagnetic ordering magnetic strength\ncharacteristics of which are comparable with magnetic\nstrength characteristics of magnetic ordering in atomic\nsystems with unfilled or partly filled inner atomic shells]\ncoexisting with superconductivity at room temperature\nis argued in the work [28]. The mechanisms of supercon-\nducting state formation are proposed to be the following.\nOn the one hand, s-wave mechanism, mediated by the\ncoupling of charge carriers with stretched phonon modes\nlike to those ones, taking place in MgB2[29], heavily\nboron doped diamond [30], [31], [32], [33] and sandwich\nS-Si-QW-S structures [34] seems to be realised. More-\nover, just crimped cylindrical shape allows to increase\nthe strength of C-C bonds by preservation of high den-\nsity of the states on Fermi surface, resulting from low di-\nmensionality (quasi-1D) of NTs. On the ‘other hand, the\nmultiband structure of valence and conductivity bands\nallows to realise the formation of joint antiferromagnetic-\nsuperconducting state by means of the s±-wave and p-\nwave formation like to pnictides. Taylor expansion over\ndimerization coordinate of electron-electron interaction\nand electron-phonon interaction indicate on the possibil-\nity of the formation of superconducting state by differ-\nent channels. The independent on dimerization coordi-\nnate (which can be both in static and dynamic states)\nelectron-electron repulsion terms can give the contribu-\ntion to antiferromagnetic-superconducting s+-state for-\nmation like to pnictides. The attractive terms, which are\nproportional to dimerization coordinate, can lead to for-\nmation of superconducting s-state by two mechanisms.\nThe first mechanism is like to BCS-superconductivity\nmechanism [35]. The second mechanism is new. It is me-\ndiated by the coupling of charge carriers with stretched\nphonon modes in C-C bonds.\nEspecially interesting seems to be the role of external\nquantized EM-field, since the only in resonance condi-\ntions the switch to superconducting state was realised.\nIt was explained by appearance by resonance of long-\nlived coherent system of resonance hypersound phonons.\nIt means, that quantized radiospectrospy-rangeEM-field4\nhas to be working constituent for realisation of oom tem-\nperature superconducting state. It was suggested, that\nthe room temperature superconducting state in /angbracketleft100/angbracketright-\nNTs, incorporated in diamond matrix is the superposi-\ntion of a number of superconducting-wave states above\nindicated.\nThemainaimoftheworkpresentedistoobtainexperi-\nmentallythe exactexperimentalproofforthe conclusions\nin [28].\nII. EXPERIMENTAL TECHNIQUE\nThe same samples, that in [28], that is the samples of\ntype IIa natural diamond, implanted by high energy ions\nof nickel (the energy of ions in ion beam was 335 MeV)\nhave been studied. The absolute spin number in each\nof the samples studied did not exceed before implanta-\ntion the value ≈1012spins. Therefore, the samples were\nmagnetically pure samples. Ion implantation (ion beam\ndose was 5 ×1013cm−2) was performed along /angbracketleft100/angbracketrightcrys-\ntal direction, that is, transversely to sample (100)-plane\nuniformly along all the plane surface. The temperature\nof the samples during the implantation was controlled\nand it did not exceed 400 K.\nX-band ESR-spectrometer ”Radiopan” was used for\nthe registration of magnetic resonance spectra. They\nwere registered by using of TE102mode rectangular cav-\nity at room temperature. The ruby standard sample was\npermanently placed in the cavity on its sidewall. One\nof the lines of ESR absorption by Cr3+point paramag-\nneticcenters(PC)inrubycrystalwasusedforthecorrect\nrelative intensity measurements of resonance absorption,\nfor the calibration of the amplitude value of magnetic\ncomponent of the microwave field and for precise phase\ntuning of modulation field. The correct relative intensity\nmeasurements become to be possible owing to unsaturat-\ning behavior of ESR absorption in ruby in the range of\nthe microwave power applied, which was ≈100 mW in\nthe absence of attenuation. Unsaturable character of the\nabsorption in a ruby standard was confirmed by means\nof the measurements of the absorption intensities in two\nidentical ruby samples in dependence on the microwave\npower level. The first sample was standard sample, per-\nmanently placed in the cavity, the second sample was\nplaced in the cavity awayfrom the loop of magnetic com-\nponentofmicrowavefieldso,thatitsresonancelineinten-\nsity was about 0.1 of the intensity of corresponding line\nof the standard sample. The slightly different orienta-\ntion of the samples allowed the simultaneous registration\nof both the samples without overlapping of their absorp-\ntion lines. The foregoing intensity ratio about 0.1 was\nprecisely preserved for all microwave power values in the\nrange used. Consequently, ruby samples are realy good\nstandard samples by ESR spectroscopy studies.III. RESULTS\nTheESRspectrum observedincarbonnanotubes, pro-\nduced by nickel high energy /angbracketleft100/angbracketrightion beam modifica-\ntion of natural diamond single crystals, is presented in\nFigure 1 in crystal direction [111]. It was reported in\n[28], that the spectrum observed was excited sponta-\nneously the only by very precise orientation of external\nstatic magnetic field /vectorH0in the 011) plane of the sam-\nple and that resulting spectrum was consisting of three\nclearlypronouncedlines, atthattwofromgivennewlines\nhave rather large anisotropic linewidths and from two\nvery broad lines, which were registered only partly in\nthe range of magnetic fields 0-4000 G. We will preserve\nthe designationsforthree clearlypronouncedlines, which\nwere used in [28], that is Rbfor the relatively right broad\nline and by L for the more broad left line. The right\nbroad line was overlapped with relatively narrow almost\nisotropic line, designated by Rn(given line was observed\nby usual(that is not very precise) sample orientation).\nVery broad strongly intensive anisotropic absorption can\nbe characterisedby two dip positions (in integrated spec-\ntrum) at ∼2410 G and ∼2892 G (corresponding to to\ndifferent lines by spectrum registration in the direction\ncoinciding with [111] diamond lattice direction, Figure\n1. It was established in [28], that two dip positions for\ngiven background absorption were coinciding by static\nmagnetic field direction in 60 degreesfrom [100]diamond\ncrystal direction, that was argued to be the display of\nthe fact, that the symmetry of the interaction, leading\nto the appearance of very strong background absorption\nis determined by inherent magnetic symmetry of NTs,\nproduced by [100] HEIBM, which is not coinciding with\nstructure symmetry of given NTs.\nLet us summarise to better understanding of sub-\nsequent discussion the experimental result,described in\n[28]. Dependencies of absorption amplitudes of L-line\nandRbline on magnetic component of microwave field\nat fixed orientation of polarising magnetic field /vectorH0||[100]\ncrystal axis have been studied. Given dependencies were\nquite different for L-line and Rbline. The dependence\nfor L-line is superlinear. It is similar to the dependen-\ncies, which were earlier observed in the samples, modi-\nfied by HEIBM with copper, neon, nickel ions (however\nwith dose 5 ×1014) [22], [25], [24], that is, in the case of\nentire modification of diamond layer, which is localised\nnear surface. In that case the layer consists the only of\nNTs, which seem to be interacting each other, that in its\nturn leads to more short spin-lattice relaxation time T1\nfor individual spin carrier (see for more detailed expla-\nnation the papers [22], [25], [24]. In the studied sample\n(integral dose is 5 ×1013), individual NTs are isolated by\ndiamond structure, nevertheless the superlinear depen-\ndence is taking place, indicating on another mechanism\nofT1shortening. Dependence of absorption amplitude\nof the right broad line Rbin ESR spectrum of NTs on\nmagnetic component of microwave field was found to be\nstrongly nonlinear. It is characterised for the values of5\nrelative magnetic component of microwave field H1/H(0)\n1\nin the range (0-0.75) by usual saturating law, but in the\nrange (0.75-1) it acquires prominent superlinear nonsat-\nurating character [28]. Given dependence was observed\nin ESR-spectroscopy for the first time. Angular depen-\ndence of g-factor of the line L consists of two branches.\nOne branch is in the angle range 0-60 degrees from [100]\ncrystallattice direction(which is coinciding with NT axis\ndirection), the second branch is in the angle range 60-90\ndegrees. It was remarked in [28], that the connection\npoint of two branches, equaled to 60 degrees for the g-\nvalues of L-line is coinciding with the point of the junc-\ntion of two dips in the very broad (and consequently very\nintensive) absorption, testifying on the same (or related)\nnature of the resonance processes, which are responsi-\nble for the appearance of L-line and very broad lines. It\nwas also found, that the deviation of g-values from free\nelectron value g = 2.0023 is very large, at that the min-\nimal value is achieved in the range 16-20 degrees from\nthe [011] direction in diamond lattice and it is equal to\n≈2.0719, maximal g-value corresponds to NT axis di-\nrection, that is to [100] crystal lattice direction and it is\nequal to≈2.3120. Given values are characteristic for the\nsystems with the strong magnetic ordering. Given data\nwere interpreted to be the direct proof of the sponta-\nneous transition of NT system, incorporated in diamond\nlattice, in the state with the strong magnetic ordering.\nAngular dependence of ESR absorption intensity of the\nline L has qualitatively opposite character to g-factor de-\npendence. The maximal absorption value corresponds to\nthe direction, being to be transversal to NT axis, which\nis coincides with [011] direction in diamond lattice. Ad-\nditional maximum is observed at 60 degrees from given\ndirection. It is characteristic, that both the maxima in\nangular dependence of absorption intensity of the line L\nare observed also in angular dependence of its linewidth.\nIt is indication, that the main features in angular depen-\ndence of ESR absorption intensity are governed by angu-\nlar dependence of linewidth. Especially interesting, that\nthe line L is asymmetric with values of the ratio A/B of\nthe asymmetry extent, which are quite different in com-\nparisonwiththoseonesbyusualDysonshape[36]. Letus\nremark, that if Dysoneffect istaking place, then by usual\ntuning of microwavephase on absorption registration the\nvalue A/B is equal to 2.55 for derivative of the responce\nsignal in the case of static (immobile) paramagnetic cen-\nters(PC)inconductivemediawhenthesamplesarethick\nin comparison with the scin depth. It is determined by\nthe space dispersion contribution [37], which is appeared\nby ESR detection in conductive media. It corresponds\nto the ratio of space dispersion contribution and absorp-\ntion contribution to resulting ESR response equaled to\n(1 : 1) [37]. The value A/B for absorption derivative\nis increasing from 2.55 to more than 19 for mobile PC\nin dependence on velocity of the spin diffusion [38]. In\nthe case of thin samples the ratio A/B has intermediate\nvalues, between 1 and above indicated, depending on the\nthickness of the samples. It was found, that the ratioA/B is less than 1 and it is strongly anisotropic. The\nmaximal A/B value is near [111] crystal lattice direction\nand it is equal to 0.83. The minimal A/B value is near\n60 degrees from [011] crystal lattice direction and it is\nequal to 0.49. It was also remarked, that by Dyson ef-\nfect in conducting samples (in particular in the samples\nwith metallic NTs, producing the network) the maximal\ndeviation from the ratio A/B = 1 has to be observed by\nmicrowave field propagation direction along the sample\nside with maximal size, that is by H0along [100] crystal\ndirection, in the case, when the network is opaque for\nmicrowave field in direction, transverse to NT axis direc-\ntion, or by H0along [011] in the case, when the network\nis opaque the only for microwave field propagation in di-\nrection, coinciding with NT axes. The observed maximal\ndeviation of ratio A/B from A/B = 1 at ≈60 degrees\nfrom [011] confirms the conclusion on nontrivial nature\nof Dyson-like effect in the case studied.\nIt has been found to be substantial, that Q-factoris in-\ncreasingin the ranges, where deviation of ratioA/B from\nA/B = 1 is also increasing, that is, increase is starting\nnear 60 degrees from [011] crystal lattice direction and\nincrease takes place in the range near 10-30 degrees from\nthe same [011] crystal direction. For usual Dyson effect\nit has to be conversely, Q-factor has to be minimal in the\ndirection of maximal deviation of ratio A/B from A/B =\n1, that is near 30 degrees from [100] crystal lattice direc-\ntion. However, it was found, that Q-factor has in given\ndirection the maximal value.\nIt was argued in [28], that the results above described\nare agreed with spontaneous transition of the system to\nthe state which characterises by coexistence simultane-\nously of antiferromagnetic (AFM) ordering and super-\nconductivity, which is realized in electron spin resonance\nconditions and it is absent without resonance. It was\nalso concluded, that the nature of given state and mech-\nanisms, leading to its formation cannot be entirely coin-\ncidingwith all knownones. Really, the suggestionon just\nAFM ordering (but not ferromagnetic) is in agreement\nwith observation of two both very broad and moderately\nbroad lines. The appearance of two resonance lines (if\nlinearly polarised microwave field is used by detection)\nwas established by Kittel in the work [39], which was the\nfirst work on the theory of AFM-resonance. We have\nfound, that magnetic moments of two sublattices being\nto be opposite directed are uncompensated in their mag-\nnitude, that is, strongly speaking, we are dealing with\nuncompensated AFM-resonance or in other words with\nferrimagneticresonance. Thisissoindeed, sincetheratio\nof intensities of the absorption, corresponding to L and\nRb-lines is equal to ≈3.5.\nThe experimental spectra were analysed more care-\nfully. It has been found, that along with main un-\ncompensated AFM-modes the uncompensated AFM spin\nwave resonance (SWR) modes are also excited. They are\nclearlyregisteredonrightside frommainuncompensated\nAFM-modes, see Figure 2, where the modes with the\nnumbers m=1,3 are represented. The splitting between6\n/s49/s54/s48/s48 /s50/s52/s48/s48 /s51/s50/s48/s48 /s52/s48/s48/s48/s53/s48/s54/s48/s55/s48/s56/s48/s65/s66/s83/s79/s82/s80/s84/s73/s79/s78/s32/s40/s114/s46/s117/s46/s41\n/s77/s65/s71/s78/s69/s84/s73/s67/s32/s70/s73/s69/s76/s68/s32/s40/s71/s97/s117/s115/s115/s41\nFigure 1: Spectral distribution of ESR absorption intensit y\nin diamond single crystal, implanted by high energy nickel\nions by beam direction transversely (100) sample plane, the\nsample was rotated in (0 11) plane, /vectorH0||[111] crystal axis\n/s51/s48/s48/s48 /s51/s50/s48/s48 /s51/s52/s48/s48 /s51/s54/s48/s48 /s51/s56/s48/s48 /s52/s48/s48/s48/s53/s53/s54/s48/s54/s53\n/s109/s32/s61/s32/s49\n/s109/s32/s61/s32/s50/s65/s66/s83/s79/s82/s80/s84/s73/s79/s78/s32/s40/s114/s46/s117/s46/s41\n/s77/s65/s71/s78/s69/s84/s73/s67/s32/s70/s73/s69/s76/s68/s32/s40/s71/s97/s117/s115/s115/s41/s109/s32/s61/s32/s51\nFigure 2: Detailed spectral distributionofmagnetic reson ance\nabsorption intensity in the range 3000 - 4100 Gauss in dia-\nmond single crystal, implanted by high energy nickel ions by\nbeam direction transversely (100) sample plane, the sample\nwas rotated in (0 11) plane, /vectorH0||[111] crystal axis\nthe modes is the following: between the center of ”grav-\nity” of two main AFM-modes and the first SWR-mode\nthe splitting isequalto246 ±5G, betweenthe firstSWR-\nmode and the secondSWR-mode it is equal to 228 ±10G\nand the distance between the second and the third SWR-\nmodes is equal to 251 ±15G. Therefore, we have nearly\nequidistant SWR splitting, that is SWR with practicaly\nlinear dispersion law and consequently the direct proof\nfor the formation of uncompensated AFM ordering./s48 /s49 /s50 /s51 /s52/s48/s49/s48/s48/s48/s48/s50/s48/s48/s48/s48/s51/s48/s48/s48/s48/s121/s32/s61/s32/s65/s49/s42/s101/s120/s112/s40/s120/s47/s116/s49/s41/s32/s43/s32/s121/s48\n/s32/s32\n/s121/s48 /s57/s54/s46/s50/s54/s53/s51/s54\n/s65/s49 /s50/s52/s53/s46/s54/s50\n/s116/s49 /s48/s46/s54/s49/s51/s49/s54/s73/s78/s84/s69/s78/s83/s73/s84/s89/s32/s40/s114/s46/s117/s46/s41\n/s77/s79/s68/s69/s32/s78/s85/s77/s66/s69/s82\nFigure 3: Dependence of AFMSWR mode intensity on mode\nnumber in diamond single crystal, implanted by high energy\nnickel ions by beam direction transversely (100) sample pla ne,\nthe sample was rotated in (0 11) plane, /vectorH0||[111] crystal axis\nThe shape of SWR-modes and their intensity distri-\nbution were analysed. Taking into account the exper-\nimental distribution for the values of amplitudes Ai,\ni=1,3, of SWR-modes A1:A2:A3= 1.06 : 2.65 : 1.83\nand the linewidth ∆ Hj,j=1,3, ∆H1= 35.7±3G,\n∆H2= 49.6±5G, ∆H3= 134±8G, we obtain the in-\ntensity distribution, represented in Figure 3. It is seen,\nthat the strong growth of intensity is observed with the\nmode number increasing. So, the intensity of the third\nmode is greater by a factor of 24.3 of the intensity of\nthe first mode. Given growth can be approximated by\nexponential dependence, although it is evident, that the\nnumber of experimental points is insufficient upto insist,\nthat given law is really takes place. Nevertheless, the\nstrong growth of intensity with mode number is unusual\nfor all earlier known magnetic and electrical SWR reso-\nnances, for which the intensity of the mode with greater\nnumber is not exceeding the intensity of the mode with\nlower number, see, for instance, [11], [40], [17]. It seems\ntobe veryuseful to analysethe asymmetryextent A/Bof\nthe modes observed. It was done for the only second and\nthe third modes. It is interesting, that the deviation of\nasymmetry extent from 1 is positive in distinction from\nthe caseofthe main AFM-modes. TheratioA/Bis equal\n1.07±0.03for the second mode and it is equal 1 .25±0.05\nfor the third mode. Therefore, there is clear tendency to\nthe increase of asymmetry extent with mode number in-\ncrease.\nIV. DISCUSSION\nIt will be further argued, that the additional results\nabove described confirm the preliminary conclusions of\nthe work [28], that is spontaneous transition of the sys-7\ntem studied to the state which characterises by coexis-\ntence simultaneously of uncompensated AFM ordering\nand superconductivity is really takes place in magnetic\nresonance conditions. We will show, that the peculiari-\nties of angular dependences of cavity Q-factor and A/B\nratio, the numerical values of A/B ratio for both main\nuncompensated AFM-modes and AFMSWR modes are\nnot connected with usual Dyson effect and they are de-\ntermined by quite other mechanism.\nIt is known, that the dynamical spin susceptibility of\na superconductor with s+cannel is given by an formula,\nobtainedbyrandomphaseapproximation(RPA) method\nin [41]. Within RPA the spin response has an operator\nform. It is\nˆχs(/vector q,Ω) = [ˆI−ˆΓˆχ0\ns(/vector q,Ω)]−1ˆχ0\ns(/vector q,Ω),(5)\nwhereˆIis the unit operator, ˆΓ is the coupling operator,\nˆχ0\ns(/vector q,Ω) is the operator, formed by the interband and\nintraband bare susceptibilities. Given relation was anal-\nysed in the work [42] in application to iron based super-\nconductors. However, the result obtained will be quali-\ntatively true for any superconductor with s+cannel. It\nhas been found, that in the normal state, the observable\nquantity, corresponding to dynamical spin susceptibility\noperator is only weakly logarithmically depends on fre-\nquency. In a superconducting state, it has a resonance\nbehavior. In the absence of SDW instability the authors\nof [42] have obtained the following expression for reso-\nnance frequency\nΩ =/radicalBig\nv2(/vector q−/vectorQ)2−(Ω0)2, (6)\nwherev=vF√\n2,vFis Fermi velocity, /vectorQis momentum, cor-\nresponding to point in the folded Brillouin zone, around\nwhich electron pockets are centered. Ω 0in (6) is\nΩ0= 2∆/radicalbigg\n(Γr\nSDW)−1−logEF\nE0, (7)\nwhereEFis Fermi energy, Γr\nSDWis factor, character-\nising interband interaction and E0is the largest of su-\nperconducting gap ∆ and the cutoff energy associated\nwith nonequivalence of the Fermi surfaces for electrons\nand holes. The shape of given resonance was repre-\nsented in Figure 3 of [41], It is seen, that the shape is\nstrongly asymmetric. The authors have also found that\nthe resonance peak is confined to the AFM wave vector\n/vectorQand disappears rapidly for |/vector q|<|/vectorQ|. So already at\n|/vector q|= 0.995|/vectorQ|the susceptibility is much smaller than its\nvalue at |/vectorQ|. Physically the appearance of magnetic res-\nonance behavior in s+superconductors is determined by\nthe presence of the magnetic fluctuation spectrum con-\nsiststing of the continuum of the AFM spin fluctuations\npeaked at /vectorQand which arise to be the consequence of the\ninterband scattering.Spin resonance in s+superconductor was compared in\n[41] and in [42] with the spin resonance in dx2−y2su-\nperconductors. Both the resonances have the similarities\nand differences. On the one hand, they are excitonic\nresonances, and they occur because the superconducting\ngap changes sign between the Fermi surface points with\nmomenta /vectorkand/vectork+/vectorQ. On the other hand, the resonance\nfrequency in a dx2−y2superconductors disperses down-\nward because of the nodes, while for a nodeless s+su-\nperconductor,the resonancedispersesupward, with large\nvelocity. Concerning our data, it is seen from Figure 1 in\ngiven paper and from Figures 1 to 3 and Figure 9 in [28],\nthat symmetry extent of resonance line L is in agreement\nwith the formation of s+superconducting state in the\nsample studied. Really, to more intensive right side of\nresonance line by its registration with the frequency scan\nwill correspond the line with more intensive left side by\nits registration with the fixed frequency and by scan of\nstatic magnetic field.\nTherefore, asymmetry of spin resonance in supercon-\nductors has quite other origin in comparison with Dyson\neffect in metals (or other conducting media). We will\nshow that AFMSWR resonance in s+superconductors\nhas also peculiarities, which allow to detect the super-\nconducting state. They are the following.\n1.The change of asymmetry extent of resonance modes\nin comparison with main AFM modes (positive devia-\ntion from A/B = 1) and its increase with mode number\nincrease.\nIt can be explained by the fact, that AFMSWR modes\nhave nodes, that, like to the resonance lines in a dx2−y2\nsuperconductorsbecometheoppositeasymmetryincom-\nparison with s+nodeless wave registered directly by\nAFM resonance. It is clear, that positive deviation from\n1 will growth with increse of node number, that is, with\nmode number increase, that actually takes place. 2.The\nsubstantial growth of the intensity of the AFMSWR\nmodes with mode number increase, see Figure 3.\nLet us remark that intensity conservationlaw for SWR\nmodes was found for NTs incorporated in diamond ma-\ntrixwith otherimplantationdirections[24], carbynesand\nfor some organic quasi-1D substances (polyvinylidene-\nhalogenides - PVDF) [17]. At that, there are similarities\nin spectroscopic properties between FMSWR, observed\nby ESR in ferromagnetically ordered 1D-lattices of topo-\nlogical solitons (SSH-solitons or those ones, belonging to\nSSH-class) and antiferroelectric SWR (AFESWR). So,\namplitude of FMSWR modes are decreasing with mode\nnumber mproportionally1\nm. At the same time, the\nlinewidths are increasing by such a way, that intensity\nof the modes is practically conserved [24]. It holds by\nAFESWR also true. So, in [17] is given the concrete ex-\nample - the ratio of relative amplitudes of the first to the\nsecond AFESWR modes in PVDF sample, is 1 .9(±0.2)\ncm−1. The linewidths are 19 and 37( ±1.5)cm−1, which\ngives the same ratio for the linewidths of the second to\nthe first modes. Given property was considered to be\nsufficient to insist, that topological quasiparticles are re-8\nsponsible for SWR. In the other earlier known cases, for\ninstance by FMSWR in ferromagnetic metals, the inten-\nsity of SWR modes is decreasing with mode number in-\ncreasing, see, for instance, Figure 1 in [11].\nThe substantial increase of the intensity of AFMSWR\nmodes with mode number increasing becomes to be un-\nderstandable, if to take into account the presence of the\nmagnetic fluctuation spectrum consisting of the contin-\nuum of the AFM spin fluctuations peaked at /vectorQ. For\nAFMSWR modes |/vector q| /negationslash= 0 and |/vector q|is increasing with\nmode number increasing coming near to the value of /vectorQ.\nThen the dynamicalmagnetizationwill be determined by\nFourier component of the magnetic fluctuation field on\nthe operating microwave frequency of the spectrometers,\nwhich is added to dynamical magnetization produced by\nmagnetic component of microwave field used.\nTherefore, we obtain the direct experimental proof of\nthe formation of superconducting s+cannel in the sam-\nple studied. The possibility of the realization of another\ncannelsandthe roleofresonanceconditionsarediscussed\nin [28].\nV. CONCLUSIONS\nThe phenomenon of ferrimagnetic spin wave resonance\n[uncompensated antiferromagnetic spin wave resonance]\nhas been detected for the first time. Given phenomenon\nwas observed in carbon nanotubes, produced by high en-\nergy ion beam modification of diamond single crystals\nin/angbracketleft100/angbracketrightdirection. The fact itself of observation of un-\ncompensated antiferromagnetic spin wave resonance isdirect proof of the formation of antiferromagnetic order-\ning [uncompensated], which is found rather strong. It is\ncomparable with magnetic ordering in classical magnetic\nsubstances, elementary units of which contains the ele-\nments with unfilled inner atomic shells. Given property\nof carbon is established also for the first time. Spin wave\nresonance observed has two main peculiarities.\n1.The opposite deviation of the asymmetry extent ra-\ntio A/B from 1 of resonance modes in comparison with\nmain AFM mode, at that it increases with mode num-\nber increase. It is explained qualitatively by existence of\nnodes like to explanation of the asymmetry extent of the\nresonance lines in a dx2−y2superconductors.\n2.The substantial increase of the intensity of AFM-\nSWR modes with mode number increase. It is explained\nby taking into account the presence of the magnetic fluc-\ntuationspectrumconsistingofthecontinuumoftheAFM\nspin fluctuations peaked at AFM vector /vectorQ. For AFM-\nSWR modes wave vector |/vector q| /negationslash= 0 and |/vector q|is increasing\nwith mode number increase, coming near to the value\nof/vectorQ. Then the dynamical magnetization will be deter-\nmined by Fourier component of the magnetic fluctuation\nfield with the frequency, coinciding with the operating\nmicrowave frequency of the spectrometer. 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We consider the numerical approximation of a continuum model of anti-\nferromagnetic and ferrimagnetic materials. The state of the material is described in\nterms of two unit-length vector fields, which can be interpreted as the magnetizations\naveraging the spins of two sublattices. For the static setting, which requires the solution\nof a constrained energy minimization problem, we introduce a discretization based on\nfirst-order finite elements and prove its Γ-convergence. Then, we propose and analyze\ntwo iterative algorithms for the computation of low-energy stationary points. The al-\ngorithms are obtained from (semi-)implicit time discretizations of gradient flows of the\nenergy. Finally, we extend the algorithms to the dynamic setting, which consists of a\nnonlinear system of two Landau–Lifshitz–Gilbert equations solved by the two fields, and\nwe prove unconditional stability and convergence of the finite element approximations\ntoward a weak solution of the problem. Numerical experiments assess the performance\nof the algorithms and demonstrate their applicability for the simulation of physical pro-\ncesses involving antiferromagnetic and ferrimagnetic materials.\n1.Introduction\nAntiferromagnetic(AFM)andferrimagnetic(FiM)materials, materialsinwhichneigh-\nboring magnetic moments tend to align antiparallel to each other (see Figure 1), have\nbeenknownformanyyears. However, theyhaverecentlygainedrenewedinterest, because\nseveral theoretical and experimental studies have shown that AFM and FiM materials\nhave features that could lead to strong improvements of the functionality of spintronics\ndevices, compared to those based on ferromagnetic (FM) materials [8, 19].\n(a) FM (b) AFM (c) FiM\nFigure 1. Classes of magnetic materials.\nIn this work, we consider a continuum model that is the state of the art for micro-\nmagnetic simulations of devices based on magnetic processes involving of AFM and FiM\nmaterials; see, e.g., the works [25, 26, 29, 32, 28] on AFM materials and [23, 24, 14]\non FiM materials. The main elements of the model, an extension of the classical mi-\ncromagnetic model of FM materials [12], are an order parameter, which consists of two\nDate: December 11, 2023.\n2010Mathematics Subject Classification. 35K61, 65M12, 65M60, 65Z05.\nKey words and phrases. antiferromagnetism; ferrimagnetism; finite element method; Γ-convergence;\nLandau–Lifshitz–Gilbert equation.\n1arXiv:2312.04939v1 [math.NA] 8 Dec 2023unit-length vector fields that can be interpreted as the normalized magnetizations averag-\ning the magnetic moments of two sublattices, and an energy functional, which consists of\nseveral contributions, each of them representing a specific physical effect. A key feature\nof the model, that is necessary to describe the antiparallel alignment of the spins in AFM\nand FiM materials, is a more complex expression of the exchange energy than in FM ma-\nterials, which involves not only the classical Heisenberg exchange interaction penalizing\nnonuniform configurations, but also the interaction of the two fields with each other (see\nthe last two terms in the energy functional (1) below). Similarly to the classical micro-\nmagnetic theory, the static problem consists of minimizing the energy functional over all\npairs of unit-length vector fields, whereas the dynamics of each field out of equilibrium\nis governed by the Landau–Lifshitz–Gilbert (LLG) equation (see (9) below), with the\neffective field being the Gateaux derivative of the energy with respect to the respective\nfield. However, due to the energy contributions involving the interaction between the\ntwo fields, both the Euler–Lagrange equations associated with the minimization problem\nand the system of LLG equations are nonlinearly coupled systems of nonlinear partial\ndifferential equations.\nBuildingonpreviousworkontheapproximationof(theheatflowof)harmonicmaps[10]\nand of the classical model of FM materials [3, 1], we propose fully discrete numerical\nschemes for the approximation of both the static and the dynamic problems.\nFor the static problem, we propose a discretization based on first-order finite elements\nand prove that the discrete energy functional converges to the continuous one in the sense\nofΓ-convergence [11]. Moreover, we propose two iterative algorithms for the computation\nof low energy stationary points based on time discretizations of the gradient flow of the\nenergy functional (see Algorithm 4.4 and Algorithm 4.5 below). These two algorithms\ndiffer from each other in the time discretization (fully implicit for Algorithm 4.4, semi-\nimplicit for Algorithm 4.5). For both algorithms, we prove well-posedness of the iteration,\nan energy-decreasing property, termination of the iterative loop, an upper bound for the\nerror in the unit-length constraint, and (under a restrictive assumption on the coefficients\nappearing in the energy functional) convergence toward a stationary point. Moreover,\nwe perform numerical experiments to compare the two algorithms and to assess their\nperformance.\nThen, we extend the best performing algorithm (and its analysis) to the dynamic\nproblem and show that the resulting integrator (Algorithm 5.1) is well-posed, stable, and\ngenerates approximations that are unconditionally convergent toward a weak solution of\nthe coupled system of LLG equations. A by-product of our constructive convergence\nproof is the first proof of existence of weak solutions for this problem.\nIn general, the mathematical literature on AFM and FiM materials is much less devel-\noped than that of FM materials. We refer, e.g., to [6, 7] for works discussing discrete-to-\ncontinuum variational limits of a two-dimensional atomistic model of AFM materials. As\nfar as the continuum model considered in this work is concerned, to the authors’ knowl-\nedge, the only other work addressing it is [21], where extensions of the Gauss–Seidel\nprojection method [33, 22] have been proposed for its numerical approximation (but no\nconvergence analysis is discussed).\nTo sum up, the novel contributions of the present work are the following:\n•We provide the first mathematically rigorous formulation of a state-of-the-art model\ncurrently used by applied scientists to simulate processes and devices involving AFM and\nFiM materials.\n2•Extending the techniques that have been developed for the approximation of the clas-\nsical model of FM materials, we introduce and analyze the first convergent numerical\nschemes for this model of AFM and FiM materials.\nThe remainder of the paper is organized as follows: In Section 2, we present the\nmathematical model of AFM and FiM materials. In Section 3, we introduce the main\ningredient of our discretization. The algorithms for the static problem, their properties,\nand two numerical experiments are discussed in Section 4. In Section 5, we extend one of\nthe algorithm and its analysis to the dynamic case, and use it to simulate the dynamics of\nmagnetic skyrmions in antiferromagnets. In Section 6, we collect the proofs of all results\nof the work. Finally, in Appendix A, we show how to pass from the formulation of the\nmodel in physical units to the dimensionless setting considered in this work.\n2.Mathematical model\nLetΩ⊂R3be a bounded Lipschitz domain representing the volume occupied by an\nAFM or a FiM material. The magnetic state of the material is described in terms of two\nunit-length vector fields, m1andm2. The total magnetization of the sample is given by\nm=ηs,1m1+ηs,2m2, where ηs,1, ηs,2>0are dimensionless constants.\nIn what follows, for Lebesgue, Sobolev, and Bochner spaces and norms, we will use the\nstandard notation [16]. To denote (spaces of) vector-valued or matrix-valued functions,\nwe use bold letters, e.g., for any domain U, we denote both L2(U;R3)andL2(U;R3×3)\nbyL2(U). Moreover, we will denote by ⟨·,·⟩the inner product of L2(Ω)and by ∥·∥\nthe corresponding norm (any other inner product or norm will be denoted by the same\nnotation, but supplemented with a suitable subscript). We will denote by ⟨·,·⟩also the\nduality pairing between H1(Ω)and its dual, and note that it coincides with the inner\nproduct of L2(Ω)if the arguments are in L2(Ω).\n2.1. Static problem. Stable magnetic configurations of the sample are described by\nminimizers m1,m2: Ω→S2of the energy functional\nE[m1,m2] =1\n22X\nℓ=1aℓℓ∥∇mℓ∥2+a12⟨∇m1,∇m2⟩ −a0⟨m1,m2⟩, (1)\nwhere the material constants a11, a22, a12, a0∈Rsatisfy the inequalities\na11+a22>0and a11a22> a2\n12. (2)\nThe three contributions in (1) are called inhomogeneous intralattice exchange ,inhomo-\ngeneous interlattice exchange , andhomogeneous interlattice exchange , respectively [29].\nMinimizers are sought in the set of admissible pairs of vector fields\nX: =H1(Ω;S2)×H1(Ω;S2)\n={(m1,m2)∈H1(Ω)×H1(Ω) :|m1|=|m2|= 1a.e. in Ω}.(3)\nNote that (2) guarantees that the energy is bounded from below in X, as there holds the\ninequality E[m1,m2]≥ −| a0||Ω|for all (m1,m2)∈ X, and that the energy functional is\nweakly sequentially lower semicontinuous in H1(Ω)×H1(Ω)(see Proposition 6.1 below).\nHence, existence of minimizers follows from the direct method of calculus of variations.\nStationary points of the energy are admissible pairs (m1,m2)∈ Xwhich, for all\nℓ= 1,2, solve\n−⟨heff,ℓ[m1,m2],φ−(mℓ·φ)mℓ⟩= 0for all φ∈H1(Ω)∩L∞(Ω),(4)\n3where the effective field heff,ℓ[m1,m2]is the (negative) Gateaux derivative of the energy\nwith respect to mℓ, i.e.,\n⟨heff,ℓ[m1,m2],ϕ⟩: =\u001c\n−δE[m1,m2]\nδmℓ,ϕ\u001d\n(1)=−aℓℓ⟨∇mℓ,∇ϕ⟩ −a12⟨∇m3−ℓ,∇ϕ⟩+a0⟨m3−ℓ,ϕ⟩.(5)\nEquivalently, a stationary point (m1,m2)∈ Xcan be seen as the solution of\n−⟨heff,ℓ[m1,m2],ϕ⟩= 0for all ϕ∈K[mℓ], (6)\nwhere\nK[mℓ] ={ψ∈H1(Ω) :mℓ·ψ= 0a.e. in Ω}. (7)\nNote that (4) and (6) can be interpreted as variational formulations of the boundary\nvalue problem\n−mℓ×heff,ℓ[m1,m2] =0inΩ,\n∂νmℓ=0on∂Ω,(8)\nwhere ν:∂Ω→S2denotes the outward-pointing unit normal vector to ∂Ω.\n2.2. Dynamic problem. Out of equilibrium, the dynamics of the time-dependent vec-\ntor fields m1,m2: Ω×(0,∞)→S2is governed by a coupled system of two Landau–\nLifshitz–Gilbert (LLG) equations, one for each vector field:\n∂tmℓ=−ηℓmℓ×heff,ℓ[m1,m2] +αℓmℓ×∂tmℓfor all ℓ= 1,2, (9)\nwhere ηℓ, αℓ>0are dimensionless constants. Note that the two LLG equations are cou-\npled to each other via their effective fields. To complete the setting, (9) is supplemented\nwith a suitable initial condition and the same boundary conditions as in (8), i.e.,\nmℓ(0) = m0\nℓinΩand ∂νmℓ=0on∂Ω×(0,∞)for all ℓ= 1,2,(10)\nfor some admissible pair (m0\n1,m0\n2)∈ X.\nIn the following definition, we state the notion of a weak solution to the initial bound-\nary value problem (9)–(10), which naturally extends to the present setting the notion\nintroduced in [5] for the standard LLG equation.\nDefinition 2.1 (weak solution) .Let(m0\n1,m0\n2)∈ X. A global weak solution of (9)–(10)\nis(m1,m2)∈L∞(0,∞;X)such that, for all T >0, the following properties are satisfied:\n(i)mℓ|ΩT∈H1(ΩT)for all ℓ= 1,2, where ΩT:= Ω×(0, T);\n(ii)mℓ(0) = m0\nℓin the sense of traces for all ℓ= 1,2;\n(iii)for all ℓ= 1,2, for all φ∈H1(ΩT), it holds that\nZT\n0⟨∂tmℓ(t),φ(t)⟩dt\n=−ηℓZT\n0⟨heff,ℓ[m1(t),m2(t)],φ(t)×mℓ(t)⟩dt+αℓZT\n0⟨mℓ(t)×∂tmℓ(t),φ(t)⟩dt;\n(11)\n(iv)it holds that\nE[m1(T),m2(T)] +2X\nℓ=1αℓ\nηℓZT\n0∥∂tmℓ(t)∥2dt≤ E[m0\n1,m0\n2]. (12)\n4The variational formulations in (11) are weak formulations of the LLG equations in (9)\nin the space-time cylinder ΩT, while (12) is a weak counterpart of the energy law\nd\ndtE[m1(t),m2(t)] =−2X\nℓ=1αℓ\nη���∥∂tmℓ(t)∥2≤0for all t >0,\nsatisfied by sufficiently smooth solutions of (9).\nRemark 2.2. For ease of presentation, we consider a dimensionless form of the energy\nfunctional. We refer to Appendix A.1 for its derivation (starting from the equations in\nphysical units usually encountered in the physical literature). Moreover, we restrict our-\nselves to the case in which the energy comprises only the exchange contribution. This is\nsufficient to capture the main mathematical features of the model: First, the analytical\nand numerical treatment of standard lower-order energy contributions (e.g., magnetocrys-\ntalline anisotropy, Zeeman energy, magnetostatic energy, Dzyaloshinskii–Moriya interac-\ntion) is well understood (see, e.g., [13, 18]). Second, lower-order terms do not entail the\ncoupling of the fields (see Appendix A.2 for more details). Hence, even in the presence of\nlower-order terms, the Euler–Lagrange equations (4)and the system of LLG equations (9)\nare exchange-coupled only.\n3.Preliminaries\nIn this section, we collect the notation and the definitions that are necessary to intro-\nduce our numerical schemes.\nFor the time discretization, we consider uniform partitions of the positive real axis with\nconstant time-step size τ >0, i.e., ti:=iτfor all i∈N0. Given a sequence {ϕi}i∈N0, for\nalli∈N0we define dtϕi+1:= (ϕi+1−ϕi)/τ. We consider the time reconstructions ϕτ,\nϕ−\nτ,ϕ+\nτdefined, for all i∈N0andt∈[ti, ti+1), as\nϕτ(t) :=t−ti\nτϕi+1+ti+1−t\nτϕi, ϕ−\nτ(t) :=ϕi,and ϕ+\nτ(t) :=ϕi+1.(13)\nNote that ∂tϕτ(t) =dtϕi+1for all i∈N0andt∈[ti, ti+1).\nThe spatial discretization is based on first-order finite elements. We assume Ωto be\na polyhedral domain and consider a family {Th}h>0of shape-regular tetrahedral meshes\nofΩparametrized by the mesh size h= max K∈Thdiam( K). We denote by Nhthe set\nof vertices of Th. For any K∈ T h, letP1(K)be the space of polynomials of degree at\nmost 1onK. We denote by S1(Th)the space of piecewise affine and globally continuous\nfunctions from ΩtoR, i.e.\nS1(Th) =\b\nvh∈C0(Ω) :vh|K∈ P1(K)for all K∈ Th\t\n.\nIt is well known that S1(Th)is a finite-dimensional subspace of H1(Ω)with dimS1(Th) =\nNh:= #Nh. LetIh:C0(Ω)→ S1(Th)denotethenodalinterpolationoperator, i.e., forall\nv∈C0(Ω),Ih[v]is the unique element of S1(Th)satisfying Ih[v](z) =v(z)for all z∈ N h.\nWe use the same notation to denote its vector-valued counterpart Ih:C0(Ω)→ S1(Th)3,\nwherethescalar-valuedoperatorisappliedtoeachcomponentofavector-valuedfunction.\nWe consider the mass-lumped L2-product ⟨·,·⟩hdefined by\n⟨ψ,ϕ⟩h=Z\nΩIh[ψ·ϕ]for all ψ,ϕ∈C0(Ω). (14)\nWe recall that this defines an inner product on S1(Th)3and that the induced norm ∥·∥h\nsatisfies the norm equivalence\n∥ϕh∥ ≤ ∥ϕh∥h≤√\n5∥ϕh∥for all ϕh∈ S1(Th)3, (15)\n5see [9, Lemma 3.9]. Moreover, we have that\n|⟨ϕh,ψh⟩ − ⟨ϕh,ψh⟩h| ≤Ch2∥∇ϕh∥L2(Ω)∥∇ψh∥L2(Ω)for all ϕh,ψh∈ S1(Th)3,(16)\nwhere C >0depends only on the shape-regularity of Th; see again [9, Lemma 3.9].\nWe conclude this section with a notational remark: In what follows, we will always\ndenoteby C >0agenericconstant, whichwillbealwaysindependentofthediscretization\nparameters, but not necessarily the same at each occurrence.\n4.Numerical energy minimization\nIn this section, we introduce a finite element discretization of the energy minimization\nproblem and show its convergence in the sense of Γ-convergence. Then, we introduce two\nfully discrete algorithms to approximate stationary points of the energy functional (1).\nWe state our results regarding well-posedness, stability, and convergence of the algo-\nrithms and underpin our theoretical results with numerical experiments. To make the\npresentation of the results concise, all proofs are postponed to Section 6.1.\n4.1. Finite element discretization. To discretize the set of admissible pairs in (3),\ngiven a mesh Thand a parameter δ >0, we consider the set\nXh,δ:={(m1,h,δ,m2,h,δ)∈ S1(Th)3× S1(Th)3:for all ℓ= 1,2,\n|mℓ,h,δ(z)| ≥1for all z∈ N hand∥Ih[|mℓ,h,δ|2]−1∥L1(Ω)≤δ}.\nNote that, at the discrete level, the unit-length constraint is relaxed [10, 1] and a mild\ncontrol of the error is enforced by the inequality involving the parameter δ.\nThe discrete static problem consists of seeking a minimizer of the energy functional (1)\nin the set of discrete admissible pairs in Xh,δ. In the following theorem, we show that the\ndiscrete energy functional Eh,δ[·,·] :=E|Xh,δ[·,·]converges toward the continuous one in\nthe sense of Γ-convergence. We note that our discretization is consistent, i.e., we do not\nmodify the energy functional, but we restrict the set in which minimizers are sought.\nTheorem 4.1 (Γ-convergence) .The following two properties hold:\n(i)Lim-inf inequality: For every sequence {(m1,h,δ,m2,h,δ)}with (m1,h,δ,m2,h,δ)∈ X h,δ\nfor all h, δ > 0such that, for some (m1,m2)∈ X,mℓ,h,δ⇀mℓinH1(Ω)ash, δ→0\nfor all ℓ= 1,2, we have that E[m1,m2]≤lim inf h,δ→0Eh,δ[m1,h,δ,m2,h,δ].\n(ii)Existence of a recovery sequence: For every (m1,m2)∈ X, there exists a sequence\n{(m1,h,δ,m2,h,δ)}with (m1,h,δ,m2,h,δ)∈ X h,δfor all h, δ > 0such that (m1,h,δ,m2,h,δ)→\n(m1,m2)inH1(Ω)×H1(Ω)andEh,δ[m1,h,δ,m2,h,δ]→ E[m1,m2]ash, δ→0.\nA well-known consequence of Γ-convergence is the convergence of discrete global min-\nimizers.\nCorollary 4.2. Let{(m1,h,δ,m2,h,δ)}be a sequence such that (m1,h,δ,m2,h,δ)∈ X h,δis\na global minimizer of the discrete energy functional Eh,δ[·,·]for all h, δ > 0. Then, every\naccumulation point (m1,m2)of the sequence belongs to Xand is a global minimizer of\nthe continuous energy functional E[·,·].\nWe omit the proof of Corollary 4.2 as it is based on standard Γ-convergence arguments;\nsee, e.g., [11, Section 1.5]. Moreover, we recall that Γ-convergence does not imply the\nconvergence of local minimizers.\n64.2. Computation of low energy stationary points. LetHbe a Hilbert space with\ninner product ⟨·,·⟩Hsuch that Xis continuously embedded in H×H. Furthermore, we\nsuppose that there exists a constant cH≥1such that\nc−1\nH∥ϕ∥ ≤ ∥ϕ∥H≤cH∥ϕ∥H1(Ω)for all ϕ∈H1(Ω). (17)\nTo find stationary points with low energy, we propose two iterative algorithms that are\nbased on two discretizations of the dissipative dynamics governed by the H-gradient flow\nof the energy\n⟨∂tmℓ,ϕ⟩H+\u001cδE[m1,m2]\nδmℓ,ϕ\u001d\n=0for all ϕ∈K[mℓ] (ℓ= 1,2).(18)\nThe spatial discretization of both methods is based on first-order finite elements as de-\nscribed in Section 3. As a discrete counterpart of the space of pointwise orthogonal\nvector fields in (7), for mh∈ S1(Th)3withmh(z)̸=0for all z∈ N h, we consider the\nfinite-dimensional space\nKh[mh] :=\b\nϕh∈ S1(Th)3:mh(z)·ϕh(z) = 0for all z∈ N h\t\n. (19)\nFor discrete functions, the pointwise orthogonality of (7) is required to hold only at the\nvertices of the mesh. Note that Kh[mh]is a subspace of S1(Th)3with dimension 2Nh.\nThe time discretization is based on two different time-stepping methods.\nRemark 4.3. In this section, we refer to the variable tastime(accordingly, we refer to\nτbelow as the time-step size ). However, note that we are considering the static setting,\nwith the time variable tplaying the role of a pseudo-time , introduced only for numerical\npurposes.\nThe first method is proposed in the following algorithm.\nAlgorithm 4.4 (coupled discrete gradient flow) .Discretization parameters: Mesh\nsizeh >0, time-step size τ >0, tolerance ε >0.\nInput: Initial guess (m0\n1,h,m0\n2,h)∈ S1(Th)3× S1(Th)3such that, for all ℓ= 1,2,\f\fm0\nℓ,h(z)\f\f= 1for all z∈ N h.\nLoop:For all i∈N0, iterate (i)–(ii)until the stopping criterion (stop)is met:\n(i)Given (mi\n1,h,mi\n2,h)∈ S1(Th)3×S1(Th)3, compute (vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]\nsuch that, for all (ϕ1,h,ϕ2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]andℓ= 1,2, it holds that\n⟨vi\nℓ,h,ϕℓ,h⟩H+aℓℓτ⟨∇vi\nℓ,h,∇ϕℓ,h⟩+a12\n2τ⟨∇vi\n3−ℓ,h,∇ϕℓ,h⟩ −a0\n2τ⟨vi\n3−ℓ,h,ϕℓ,h⟩\n=−aℓℓ⟨∇mi\nℓ,h,∇ϕℓ,h⟩ −a12⟨∇mi\n3−ℓ,h,∇ϕℓ,h⟩+a0⟨mi\n3−ℓ,h,ϕℓ,h⟩.(20)\n(ii)Define\nmi+1\nℓ,h:=mi\nℓ,h+τvi\nℓ,hfor all ℓ= 1,2. (21)\n(stop)Stop iterating (i)–(ii)if(vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]satisfies\nmax\nℓ=1,2\u0000\n∥vi\nℓ,h∥2\nH+τ∥∇vi\nℓ,h∥2\u0001\n≤ε2|Ω|. (22)\nOutput: Ifi∗∈N0denotes the smallest integer satisfying the stopping criterion (22),\ndefine the approximate stationary point (m1,h,m2,h) := (mi∗\n1,h,mi∗\n2,h).\nThe second method is proposed in the following algorithm.\n7Algorithm 4.5 (decoupled discrete gradient flow) .Discretization parameters: Mesh\nsizeh >0, time-step size τ >0, tolerance ε >0.\nInput: Initial guess (m0\n1,h,m0\n2,h)∈ S1(Th)3× S1(Th)3such that, for all ℓ= 1,2,\f\fm0\nℓ,h(z)\f\f= 1for all z∈ N h.\nLoop:For all i∈N0, iterate (i)–(ii)until the stopping criterion (stop)is met:\n(i)Given (mi\n1,h,mi\n2,h)∈ S1(Th)3×S1(Th)3, compute (vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]\nsuch that, for all (ϕ1,h,ϕ2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]andℓ= 1,2, it holds that\n⟨vi\nℓ,h,ϕℓ,h⟩H+aℓℓτ⟨∇vi\nℓ,h,∇ϕℓ,h⟩\n=−aℓℓ⟨∇mi\nℓ,h,∇ϕℓ,h⟩ −a12⟨∇mi\n3−ℓ,h,∇ϕℓ,h⟩+a0⟨mi\n3−ℓ,h,ϕℓ,h⟩.(23)\n(ii)Define\nmi+1\nℓ,h:=mi\nℓ,h+τvi\nℓ,hfor all ℓ= 1,2. (24)\n(stop)Stop iterating (i)–(ii)if(vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]satisfies\nmax\nℓ=1,2\u0000\n∥vi\nℓ,h∥2\nH+τ∥∇vi\nℓ,h∥2\u0001\n≤ε2|Ω|. (25)\nOutput: Ifi∗∈N0denotes the smallest integer satisfying the stopping criterion (25),\ndefine the approximate stationary point (m1,h,m2,h) := (mi∗\n1,h,mi∗\n2,h).\nIn both Algorithm 4.4 and Algorithm 4.5 the iteration stops when the size of the\nupdatesissufficientlysmall(accordingto(22)and(25), respectively). Thealgorithmsare\ncharacterizedbyadifferenttreatmentoftheinhomogeneousandhomogeneousinterlattice\nexchange contributions, which are treated implicitly in Algorithm 4.4 and explicitly in\nAlgorithm 4.5. One immediate consequence is that in Algorithm 4.4 the two equations\nare coupled (as they are in the continuous problem) and one iteration of the algorithm\nrequires the solution of one4Nh-by-4Nhlinear system, whereas in Algorithm 4.5 the\ntwo equations are decoupled and one iteration of the algorithm requires the solution of\ntwo2Nh-by-2Nhlinear systems (that are independent of each other and thus can be\nsolved in parallel). This difference will affect the solvability and energetic behavior of the\nalgorithms, which will be the subject of the following propositions.\nIn the following proposition, we establish the properties of Algorithm 4.4.\nProposition 4.6 (properties of Algorithm 4.4) .There hold the following statements:\n(i)Suppose that τsatisfies c2\nH|a0|τ <2, where a0is one of the coefficients in (1)andcH\nis the constant in (17). Then, for all i∈N0,(20)admits a unique solution (vi\n1,h,vi\n2,h)∈\nKh[mi\n1,h]×Kh[mi\n2,h].\n(ii)Under the assumption of part (i), suppose that τadditionally satisfies cHcT|a0|τ <1,\nwhere cT>0is a constant which depends only on the shape-regularity of the family\nof meshes. Then, Algorithm 4.4 terminates within a finite number of iterations. In\nparticular, the approximate stationary point (m1,h,m2,h)is well defined.\n(iii)Under the assumption of part (i), for all i∈N0, the iterates of Algorithm 4.4 satisfy\nE[mi+1\n1,h,mi+1\n2,h]− E[mi\n1,h,mi\n2,h] =−τ2X\nℓ=1∥vi\nℓ,h∥2\nH−τ2\n22X\nℓ=1aℓℓ∥∇vi\nℓ,h∥2.(26)\nIn particular, the sequence of energies generated by the algorithm is monotonically de-\ncreasing, i.e., it holds that E[mi+1\n1,h,mi+1\n2,h]≤ E[mi\n1,h,mi\n2,h].\n(iv)Under the assumptions of part (ii), there exists C > 0such that the approximate\n8stationary point (m1,h,m2,h)satisfies\n∥Ih[|mℓ,h|2]−1∥L1(Ω)≤Cτ \n1 +2X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω)!\nfor all ℓ= 1,2.\nThe constant C >0depends only on a11,a12,a22,a0,cH, and the shape-regularity of the\nfamily of meshes.\nCorresponding results for Algorithm 4.5 are the subject of the following proposition.\nProposition 4.7 (properties of Algorithm 4.5) .There hold the following statements:\n(i)For all i∈N0,(23)admits a unique solution (vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h].\n(ii)Suppose that τsatisfies cH(cH/2 +cT)|a0|τ <1, where a0is one of the coefficients\nin(1),cHis the constant in (17), and cT>0is a constant which depends only on the\nshape-regularity of the family of meshes. Then, Algorithm 4.5 terminates within a finite\nnumber of iterations. In particular, the approximate stationary point (m1,h,m2,h)is well\ndefined.\n(iii)For all i∈N0, the iterates of Algorithm 4.5 satisfy\nE[mi+1\n1,h,mi+1\n2,h]− E[mi\n1,h,mi\n2,h] =−τ2X\nℓ=1∥vi\nℓ,h∥2\nH−τ2\n22X\nℓ=1aℓℓ∥∇vi\nℓ,h∥2\n+a12τ2⟨∇vi\n1,h,∇vi\n2,h⟩ −a0τ2⟨vi\n1,h,vi\n2,h⟩.(27)\nMoreover, if τsatisfies c2\nH|a0|τ≤2, then the sequence of energies generated by the\nalgorithm is monotonically decreasing, i.e., it holds that E[mi+1\n1,h,mi+1\n2,h]≤ E[mi\n1,h,mi\n2,h].\n(iv)Under the assumptions of part (ii), there exists C > 0such that the approximate\nstationary point (m1,h,m2,h)satisfies\n∥Ih[|mℓ,h|2]−1∥L1(Ω)≤Cτ \n1 +2X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω)!\nfor all ℓ= 1,2.\nThe constant C >0depends only on a11,a12,a22,a0,cH, and the shape-regularity of the\nfamily of meshes.\nEach iteration of Algorithm 4.4 is well defined if the time-step size is sufficiently small.\nMoreover, the algorithm unconditionally generates a monotonically decreasing sequence\nof energies. Conversely, each iteration of Algorithm 4.5 is unconditionally well defined,\nbut the sequence of energies it generates is monotonically decreasing only if the time-step\nsize is sufficiently small. Furthermore, we note that the inequalities in point (iv) of both\nProposition 4.6 and Proposition 4.7 show that if the initial guesses are uniformly bounded\ninH1(Ω)(in the sense of (28) below), then the approximate stationary points generated\nby the algorithms belong to the set of admissible pairs Xh,δwith δof the form δ=Cτ.\nIn the following theorem, we show that the sequence of approximate stationary points\ncomputed with both algorithms converges toward an admissible pair in Xas the dis-\ncretization parameters go to zero. If we neglect the inhomogeneous interlattice exchange\ncontribution, wecanidentifythelimitwithastationarypointoftheenergyfunctional(1).\nTheorem 4.8 (convergence of Algorithm 4.4 and Algorithm 4.5) .Suppose that there\nexists c0>0, independent of the discretization parameters h,τ, and ε, such that\nsup\nh>0 2X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω)!\n≤c0. (28)\n9Suppose that τ→0andε→0ash→0. Then, as h→0, the sequence of approximate\nstationary points {(m1,h,m2,h)}h>0generated by either Algorithm 4.4 or Algorithm 4.5,\nupon extraction of a subsequence, converges weakly in H1(Ω)×H1(Ω)toward a point\n(m1,m2)∈ X. Ifa12= 0, the limit (m1,m2)∈ Xis stationary point of the energy\nfunctional (1).\nA byproduct of Theorem 4.8 is the existence of weak solutions to the Euler–Lagrange\nequations (4) for the case a12= 0.\nRemark 4.9. In our analysis, we can identify the limit of the sequence of approximate\nstationary points with a stationary point of the energy only if we assume that a12= 0,\ni.e., if we neglect the inhomogeneous interlattice exchange contribution from the energy.\nThis restriction is related to the fact that, if a12̸= 0, the weak formulation of the approxi-\nmate Euler–Lagrange equations satisfied by (m1,h,m2,h)contains a term that involves the\nL2-product of ∇m1,hand∇m2,h. Since (m1,h,m2,h)converges to (m1,m2)only weakly\ninH1(Ω)×H1(Ω), we are not allowed to pass this term to the limit. We believe that this\nissue comes from the fact that our algorithms do not use any regularization, so that the\nstability analysis does not yield any additional regularity (and thus no stronger conver-\ngence properties) that would allow us to use arguments based on compensated compactness\n(see, e.g., [15, Chapter 5] or[31, Section I.3] ). However, we note that our numerical ex-\nperiments suggest that the algorithms behave well even if a12̸= 0. Moreover, in many\nsituations (see, e.g., [26, 28]), the inhomogeneous interlattice exchange contribution is\nof limited physical value and is omitted, so that the current theory already covers many\napplications.\n4.3. Numerical experiments. Before moving to the dynamic case, we show the ef-\nfectivity of the proposed algorithms with two numerical experiments. The computations\npresented in this section (and in Section 5.2 below) have been performed with an im-\nplementation based on the open-source finite element library Netgen/NGSolve [30] (ver-\nsion 6.2.2302). Lower-order energy contributions such as magnetocrystalline anisotropy,\nDzyaloshinskii–Moriya interaction, and Zeeman energy (cf. Section A.2), omitted in our\nanalysis, are treated explicitly (and thus contribute only to the right-hand-sides of (20)\nand (23)); see [13, 1]. The orthogonality constraint in (20) and (23) is enforced using the\nnull-space method discussed in [27, 20]. The resulting linear systems are solved using the\ngeneralized minimal residual method (GMRES) with an incomplete LU decomposition\npreconditioner. We note that in the static case, the use of the conjugate gradient method\nis possible due to symmetry, but we use GMRES in these tests to maintain consistency\nwith the dynamic case (see Section 5.2 below). All computations have been made on an\ni5-9500CPUwith 16 GBofinstalledmemory. Magnetizationconfigurationsarevisualized\nwith ParaView [2].\n4.3.1.Comparison of the algorithms. In this experiment, we aim to compare to each\nother Algorithm 4.4 and Algorithm 4.5, and to evaluate the impact on their performance\nof the choice of the gradient flow metric, i.e., the inner product ⟨·,·⟩Hin (18).\nFor the dimensionless setting discussed in Section 2, we consider a toy problem on\nthe unit cube Ω = ( −1/2,1/2)3. The total energy consists of exchange and uniaxial\n10anisotropy, i.e.,\nE[m1,m2] =1\n22X\nℓ=1aℓℓZ\nΩ|∇mℓ|2+a12Z\nΩ∇m1:∇m2−a0Z\nΩm1·m2\n+q2\n1\n2Z\nΩ[1−(a·m1)2] +q2\n2\n2Z\nΩ[1−(a·m2)2],\nwith exchange constants a11= 2,a22= 1,a12=−1/2, and a0=−100, anisotropy\nconstants q1= 5andq2= 10, and easy axis a= (1,1,1)/√\n3. It is easy to see that for\nthis setup the energy minimization problem admits two global minimizers (m±\n1,m±\n2)≡\n±(a,−a)and that the energy value at the minimizers is E[m±\n1,m±\n2] =−100.\nFor the discretization, we consider a tetrahedral mesh generated by Netgen with mesh\nsizeh≈0.209(1433vertices and 6201elements), and we set τ= 10−3andε= 10−4.\nStarting from the constant initial guess m0\n1,h≡(1,0,0)andm0\n2,h≡(0,1,0), we run\nAlgorithm 4.4 and Algorithm 4.5 for three different choices for the gradient flow metric:\ntheL2-metric ⟨·,·⟩H=⟨·,·⟩, the mass-lumped L2-metric ⟨·,·⟩H=⟨·,·⟩h(see (14)), and\ntheH1-metric ⟨·,·⟩H=⟨·,·⟩+⟨∇(·),∇(·)⟩. Forallsixruns(twoalgorithms, threemetrics\neach), the iterative algorithm returns as approximate stationary point an approximation\nof the minimizer (m−\n1,m−\n2)≡(−a,a).\nIn Table 1, we compare the performance of each combination in terms of\n•the final energy E[m1,h,m2,h]of the approximate stationary point ( energy);\n•the difference E[ˆm1,h,ˆm2,h] + 100between the final energy E[ˆm1,h,ˆm2,h]of the\nnormalized approximate stationary point ( proj. energy err. ) and the expected\nenergy −100, where ˆmℓ,h=Ih[mℓ,h/|mℓ,h|]for all ℓ= 1,2;\n•the number of iterations necessary to meet the stopping criterion ( num. iter. );\n•the average solve time per iteration ( solve time ), measured in s, where the solve\ntime is defined as the time needed to solve the linear system (20) for Algorithm 4.4\nand as the sum of the times needed to solve the two linear systems (23) (one for\nℓ= 1and one for ℓ= 2) for Algorithm 4.5;\n•the error in the unit-length constraint measured in the L1-norm, i.e., errL1:=\nmax ℓ=1,2\r\rIh\u0002\n|mℓ,h|2\u0003\n−1∥L1(Ω);\n•the error in the unit-length constraint measured in the L∞-norm, i.e., errL∞:=\nmax ℓ=1,2∥mℓ,h∥L∞(Ω)−1.\nMoreover, in Figure 2, for all six combinations of algorithms and metrics, we plot the\nevolution of the energy during the iteration.\nAlgorithm 4.4 (coupled) Algorithm 4.5 (decoupled)\nH (L2(Ω),∥·∥)(L2(Ω),∥·∥h)H1(Ω) (L2(Ω),∥·∥)(L2(Ω),∥·∥h)H1(Ω)\nenergy −111.59 −111.59 −111.59 −111.38 −111.38 −111.38\nproj. energy err. 8.56·10−118.56·10−118.56·10−119.90·10−119.90·10−119.90·10−11\nnum. iter. 249 249 249 275 275 275\nsolve time 0.223 0.232 0.376 0.095 0.094 0.209\nerrL∞ 0.049 0.049 0.049 0.047 0.047 0.047\nerrL1 0.100 0.100 0.100 9.61·10−29.61·10−29.61·10−2\nTable 1. Experiment of Section 4.3.1: Comparison of algorithms and gradient flow\nmetrics (constant initial guess).\nIn the very first part of the gradient flow dynamics (corresponding roughly to the first\n15 iterations), the constant initial guess with m1andm2perpendicular to each other\nevolves to reach a constant state with an antiparallel alignment of the fields. This yields\n110 50 100 150 200 250−100−500\niterationenergyAlg. 4.4 Alg. 4.5\n(L2(Ω),∥·∥)\n(L2(Ω),∥·∥h)\nH1(Ω)\nFigure 2. Experiment of Section 4.3.1: Evolution of the energy with different al-\ngorithms and gradient flow metrics (constant initial guess).\na strong reduction of the a0-modulated homogeneous interlattice exchange contribution\n(with the total energy abruptly dropping from an initial value of about 41 to -42). The\nrest of the dynamics is slower and consists in a rotation of the pair of constant fields\nwhich make them align to the direction of the easy axis as prescribed by the anisotropy\nenergy contribution.\nLooking at the results, we observe that Algorithm 4.4 and Algorithm 4.5 require ap-\nproximately the same number of iterations to fulfill the stopping criterion (those of Algo-\nrithm4.4areslightlylessthanthoseofAlgorithm4.5). TheenergydecayofAlgorithm4.4\nis faster than the one of Algorithm 4.5, but this does not lead to a significantly smaller\nnumber of iterations. On the other hand, the average solve time of Algorithm 4.5 is\nabout half of the one of Algorithm 4.4, which makes the simulations performed with the\ndecoupled algorithm significantly faster. The different metrics are practically identical\n(except for a minimal difference in the average solve time). For the L2- and mass-lumped\nL2-metric, this is unsurprising, since they are equivalent to each other; see (15). We be-\nlieve that the equivalence to the H1-metric in this example (with constant initial guess)\nis due to the fact that the updates vi\nℓ,hare essentially uniform, and hence their gradients\n∇vi\nℓ,hare essentially zero. It follows that in the numerical scheme, the gradient part of\ntheH1-metric is small, reducing to the L2-metric.\nThere is a significant discrepancy between the value of the energy at the minimizer\n(E[m+\n1,m+\n2] =−100) and the one of its approximation ( E[m1,h,m2,h]≈-111). However,\nif we remove the error in the unit-length constraint by normalizing the fields at the\nvertices of the mesh, we obtain the desired value up to the tenth digit. This shows that\nour projection-free algorithms are perfectly able to identify the minimizers. However,\nfor a quantitative match of the energy values, the error in the constraint needs to be\nremoved or reduced (applying a nodal projection to the final configuration or decreasing\nthe time-step size).\nNext, we repeat the experiment, but this time we start from a random initial guess\n(the same for all simulations). For all six runs, the iterative algorithm again returns as\napproximate stationary point an approximation of the minimizer (m−\n1,m−\n2)≡(−a,a).\nThe results are displayed in Table 2. The faster average solve time of Algorithm 4.5\nobserved for the case of a constant initial guess is confirmed. However, in this case, we\nobserve a clear difference between the L2-metrics (with and without mass lumping) and\ntheH1-metric, with the latter requiring a significantly larger number of iterations (ca\n13700 against 200–300), which results in longer computational times. However, as far as\n12Algorithm 4.4 (coupled) Algorithm 4.5 (decoupled)\nH (L2(Ω),∥·∥)(L2(Ω),∥·∥h)H1(Ω) (L2(Ω),∥·∥)(L2(Ω),∥·∥h)H1(Ω)\nenergy -182.36 -158.04 -100.38 -182.70 -158.03 -100.38\nproj. energy err. 5.70·10−116.33·10−116.02·10−96.54·10−117.70·10−116.02·10−9\nnum. iter. 231 231 13713 252 254 13718\nsolve time 0.220 0.227 0.347 0.093 0.092 0.170\nerrL∞ 1.185 0.923 0.004 1.196 0.905 0.004\nerrL1 0.668 0.433 2.51·10−30.670 0.428 2.51·10−3\nTable 2. Experiment of Section 4.3.1: Comparison of algorithms and gradient flow\nmetrics (random initial guess).\nthe unit-length constraint is concerned, the H1-metric is characterized by a much better\naccuracy.\nOverall, our experiments show that the decoupled approach of Algorithm 4.5, due to its\ncomputationalefficiency, ispreferableoverthecoupledoneofAlgorithm4.4. Ontheother\nhand, the choice of the gradient flow metric is more delicate. While for a constant initial\nguess (with low exchange energy) the metrics are essentially equivalent, for a random\ninitial guess (with large exchange energy) the H1-metric guarantees a significantly smaller\nviolation of the unit-length constraint at the discrete level (which, however, is obtained\nat the price of higher computational costs).\n4.3.2.Skyrmion formation. In this experiment, we aim to highlight the capability of our\nalgorithms to compute stable magnetization configurations in AFM materials.\nThe domain is an AFM nanodisk of thickness 1 nm(aligned with x3-axis) and diam-\neter60 nm(aligned with the x1x2-plane). The energy consists of exchange, out-of-plane\nuniaxial anisotropy, and interfacial Dzyaloshinskii–Moriya interaction, and reads as\nE[m1,m2] =1\n22X\nℓ=1aℓℓZ\nΩ|∇mℓ|2−a0Z\nΩm1·m2+q2\n2Z\nΩ[1−(a·m1)2]\n+q2\n2Z\nΩ[1−(a·m2)2] +Z\nΩbD: (∇m1×m1) +Z\nΩbD: (∇m2×m2),\nwhere the dimensionless parameters a11,a22,a0,qandbDare obtained from the material\nparameters collected in Table 3 as explained in Appendix A.\nParameter Value\nMs,1,Ms,2 376 kA /m\nA11,A22 6.59 pJ /m\nA12 0\nA0 −6.59 pJ /m\na 1 nm\nK 0.15 MJ /m3\na e3\nD D(−e1⊗e2+e2⊗e1)\nD 3 mJ/m2\nTable 3. Experiment of Section 4.3.2: Material parameters. All values are taken\nfrom [28], except those of aandD. Here, we denote by {e1,e2,e3}the canonical\nbasis of R3.\nFor the discretization, we consider a tetrahedral mesh Thgenerated by Netgen with\nmesh size 3.36 nm(1660vertices and 4694elements), i.e., well below the exchange length\n13ofℓex=q\n2A11/(µ0M2\ns,1) = 8 .61 nm, Here, µ0>0denotes the vacuum permeability (in\nN/A2).\nFigure 3. Experiment of Section 4.3.2: Initial guess for Algorithm 4.5. The initial\nmagnetisation for m0\nh,1(resp., m0\nh,2) is shown in red (resp., blue), with the internal\nregion facing in the e3(resp., −e3) direction.\nAs an initial guess, we consider a perturbed skyrmion-like AFM state; see Figure 3.\nMore precisely, we consider the auxiliary function\nfinit(x, y, z ) :=1\n1 + exp\u0010\n20\u0010p\nx2+y2−10\u0011\u0011−1\n2\nand start from the initial condition m1,2= (0,0,±finit). This is then interpolated using\nthe built-in Oswald-type interpolation of NGSolve before undergoing a nodal projection,\nrandom perturbation (up to 0.3in each component) and another nodal projection. The\nvalue 10intheexpressionof finitcorrespondsto 10 nmandreferstotheradiusoftheinner\ncircle. The decay constant 20makes the transition reasonably sharp before projecting.\nStarting from this configuration, we run Algorithm 4.5 (in our opinion, the best per-\nforming one in Section 4.3.1) with dimensionless time-step size τand stopping tolerance\nεboth equal to 1·10−3.\n(a)mh,1\n (b)mh,2\nFigure 4. Experiment of Section 4.3.2: Stable AFM configurations computed using\nAlgorithm 4.5 with H1-metric. In the pictures, the color scale refers to the third\ncomponent of the fields, which attains values between -1 (blue) and 1 (red).\nIn Figure 4, we show the stable configurations obtained running Algorithm 4.5 with\nH1-metric. We see that both fields are Néel-type skyrmions [17] (with the cores pointing\nup for mh,1and down for mh,2, in line with the orientation of the field in the internal\nregion for the corresponding initial condition), which is typical for magnetic systems\n14characterized by interfacial Dzyaloshinskii–Moriya interaction. Moreover, as expected\nfor an AFM material, we have that mh,1≈ −mh,2.\nAlgorithm 4.5\nH (L2(Ω),∥·∥)(L2(Ω),∥·∥h)H1(Ω)\nenergy −4.362·105−4.287·105−4.256·105\nnum. iter. 17 552 17 704 17 784\nsolve time 0.060 0.050 0.058\nerrL∞ 0.165 0.100 0.061\nerrL1 95.134 44.173 23.411\nTable 4. Experiment of Section 4.3.2: Comparison of gradient flow metrics for\nAlgorithm 4.5.\nIn Table 4, we compare the performance of the three gradient flow metrics considered\nin Section 4.3.1. We see that, in terms of final energy value, number of iterations, and\naverage solve time, the performance of the three metrics is comparable. On the other\nhand, as far as the violation of the unit-length constraint is concerned, the H1-metric\nexhibits the best performance.\n5.Numerical approximation of the LLG system\nIn this section, starting from Algorithm 4.5, we introduce a fully discrete algorithm\nto approximate solutions of the initial boundary value problem (9)–(10) for the coupled\nsystem of LLG equations modeling the dynamics of AFM and FiM materials. We state\nwell-posedness, stability, and unconditional convergence of the approximations toward a\nweak solution of the problem, and present numerical experiments to show its applicability\nfor the simulation of the dynamics of magnetic skyrmions in AFM materials. To make\nthe presentation of the results concise, all proofs are postponed to Section 6.2.\n5.1. Numerical algorithm and main results. The method we propose, stated in the\nfollowing algorithm, is based on the projection-free tangent plane scheme from [1, 10, 18]\nand employs the decoupled approach of Algorithm 4.5. Like in the static case, the spatial\ndiscretization is based on first-order finite elements (see Section 3).\nAlgorithm5.1 (tangentplanescheme) .Discretization parameters: Mesh size h >0,\ntime-step size τ >0.\nInput:Approximate initial condition (m0\n1,h,m0\n2,h)∈ S1(Th)3×S1(Th)3such that, for all\nℓ= 1,2,\f\fm0\nℓ,h(z)\f\f= 1for all z∈ N h.\nLoop:For all i∈N0, iterate (i)–(ii)until the stopping criterion (stop)is met:\n(i)Given (mi\n1,h,mi\n2,h)∈ S1(Th)3×S1(Th)3, compute (vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]\nsuch that, for all ℓ= 1,2, for all (ϕ1,h,ϕ2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h], it holds that\nαℓ⟨vi\nℓ,h,ϕℓ,h⟩h+⟨mi\nℓ,h×vi\nℓ,h,ϕℓ,h⟩h+ηℓaℓℓτ⟨∇vi\nℓ,h,∇ϕℓ,h⟩\n=−ηℓaℓℓ⟨∇mi\nℓ,h,∇ϕℓ,h⟩ −ηℓa12⟨∇mi\n3−ℓ,h,∇ϕℓ,h⟩+ηℓa0⟨mi\n3−ℓ,h,ϕℓ,h⟩.(29)\n(ii)Define\nmi+1\nℓ,h:=mi\nℓ,h+τvi\nℓ,hfor all ℓ= 1,2. (30)\nOutput: Sequence of approximations {(mi\n1,h,mi\n2,h)}i∈N0.\n15Starting from approximations m0\n1,h,m0\n2,h∈ S1(Th)3of the initial conditions, in each\nstep of Algorithm 5.1, the new approximations are computed updating the current ones\nusing a predictor-corrector approach. In the predictor step, (29) are discretizations of\nαℓ∂tmℓ+mℓ×∂tmℓ=ηℓheff,ℓ[m1,m2]−ηℓ(heff,ℓ[m1,m2]·mℓ)mℓfor all ℓ= 1,2,\nan equivalent reformulation of (9) that can be obtained using standard vector identities\nas well as the relations |mℓ|= 1andmℓ·∂tmℓ= 0[4]. The discrete problems are\nposed in the discrete tangent space (19), which yields a natural linearization. Like in\nAlgorithm 4.5, the inhomogeneous intralattice exchange contribution is treated implicitly,\nwhereas the interlattice contributions are treated explicitly. By doing this, the system of\nLLG equations is decoupled and one has to solve two, independent of each other, 2Nh-\nby-2Nhlinear systems per time-step. The corrector step (30) is a simple projection-free\nfirst-order time-stepping.\nIn the following proposition, we show that Algorithm 5.1 is well-defined.\nProposition 5.2 (well-posedness of Algorithm 5.1) .For all i∈N0,(29)admits a unique\nsolution (vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]. In particular, each iteration of Algorithm 5.1\nis well-defined.\nIn the following proposition, we characterize the energy behavior of Algorithm 5.1.\nProposition 5.3 (discrete energy law and stability of Algorithm 5.1) .There hold the\nfollowing statements:\n(i)For all i∈N0, the approximations generated by Algorithm 5.1 satisfy the identity\nE[mi+1\n1,h,mi+1\n2,h]− E[mi\n1,h,mi\n2,h] =−τ2X\nℓ=1αℓ\nηℓ∥vi\nℓ,h∥2\nh−τ2\n22X\nℓ=1aℓℓ∥∇vi\nℓ,h∥2\n+a12τ2⟨∇vi\n1,h,∇vi\n2,h⟩ −a0τ2⟨vi\n1,h,vi\n2,h⟩.(31)\n(ii)Ifτ < 2 max{α1, α2}/|a0|, for all j∈N, the approximations generated by Algo-\nrithm 5.1 satisfy the inequality\n2X\nℓ=1∥mj\nℓ,h∥2\nH1(Ω)+τj−1X\ni=02X\nℓ=1∥vi\nℓ,h∥2\nh+τ2j−1X\ni=02X\nℓ=1∥∇vi\nℓ,h∥2≤C. (32)\nThe constant C > 0depends only on the problem data and the shape-regularity of the\nfamily of meshes.\nThe discrete energy law of Algorithm 5.1 is an approximation of the one satisfied\nby weak solutions (see (12)). The LLG-inherent energy dissipation, modulated by the\ndamping parameters α1andα2, is enhanced by the dissipation coming from the second\nterm on the right-hand side, which is due to the implicit treatment of the homogeneous\nintralattice exchange contribution. The last two terms on the right-hand side of (31), in\ngeneral unsigned, are perturbations arising from the explicit treatment of the interlattice\nexchange contributions.\nWith the sequence of approximations delivered by Algorithm 5.1, for ℓ= 1,2, we define\nthe piecewise affine time reconstruction mℓ,hτ: [0,∞)→ S1(Th)3as\nmℓ,hτ(t) :=t−ti\nτmi+1\nℓ,h+ti+1−t\nτmi\nℓ,hfor all i∈N0andt∈[ti, ti+1]\n(see (13)). In the following theorem, we state the convergence of the finite element\napproximations toward a weak solution of (9) in the sense of Definition 2.1.\n16Theorem5.4 (convergenceofAlgorithm5.1) .Suppose that m0\n1,h→m0\n1andm0\n2,h→m0\n2\ninH1(Ω)ash→0. Then, there exist (m1,m2)∈L∞(0,∞;X)and a (nonrelabeled)\nsubsequence of {(m1,hτ,m2,hτ)}which converges toward (m1,m2)ash, τ→0. In par-\nticular, as h, τ→0, for all ℓ= 1,2it holds that mℓ,hτ∗⇀mℓinL∞(0,∞;H1(Ω)). If\na12= 0, the limit (m1,m2)is a weak solution of (9)in the sense of Definition 2.1.\nLike in the stationary case, we need to assume that a12= 0to be able to show that the\nlimit toward which the finite element approximations are converging satisfies the vari-\national formulation (11) (cf. Remark 4.9). Under this assumption, Theorem 5.4 shows\nexistence of a weak solution to (9) and convergence (without rates) of the time recon-\nstructions generated using the snapshots computed using Algorithm 5.1 toward it.\n5.2. Numerical experiments. In this section, we aim to show the capability of Algo-\nrithm 5.1 to simulate dynamic processes involving AFM materials.\n5.2.1.LLG-based energy minimization. Starting from the observation that the dynam-\nics of m1andm2governed by the system of LLG equations (9) is dissipative, with\nthe energy dissipation being modulated by the damping parameters α1andα2, we re-\npeat the experiment of Section 4.3.1, but to compute low-energy stationary points we\nuse Algorithm 5.1 (instead of the gradient flow-based approaches of Algorithm 4.4 and\nAlgorithm 4.5). More precisely, we consider the same setup and the same spatial dis-\ncretization of Section 4.3.1 and run Algorithm 5.1 with η1=η2= 1, different damping\nparameters α1=α2=α∈ {1,1/2,1/4,1/8,1/16}, and τ= 10−3, using the constant\nfieldsm0\n1,h≡(1,0,0)andm0\n2,h≡(0,1,0)as initial condition. The iteration is stopped\nwhen the α-independent stopping criterion (22) with ∥·∥2\nH=∥·∥2\nhandε= 10−4is satis-\nfied.\nAlg. 4.5 Algorithm 5.1\nα 1 1 1/2 1/4 1/8 1/16\nenergy −111.38 −112.26 −126.64 −160.60 −214.79 −998.44\nproj. energy err. 9.90·10−111.03·10−104.81·10−118.27·10−116.13·10−113.64·10−11\nnum. iter. 275 310 334 600 2954 46698\nsolve time 0.094 0.093 0.095 0.098 0.097 0.094\nerrL∞ 0.047 3.932·10−29.497·10−20.219 0.408 6.705\nerrL1 9.61·10−28.019·10−20.199 0.486 0.982 3.995\nTable 5. Experiment of Section 5.2.1: Comparison of Algorithm 4.5 (with mass-\nlumped L2-metric and α= 1) with Algorithm 5.1 (with α= 1,1/2,1/4,1/8,1/16).\nWe display the results of our computations in Table 5. Noting that Algorithm 5.1\ncoincides with Algorithm 4.5 with mass-lumped L2-metric if η1=η2=α1=α2= 1\nand the precession term ⟨mi\nℓ,h×vi\nℓ,h,ϕℓ,h⟩his omitted from (29), in the first column of\nthe table we include the results from Section 4.3.1 of this instance of Algorithm 4.5 for\nthe sake of comparison. We see that as αis lowered, the final energy is further from\nthe expected value of −100, which is due to the slower dissipation resulting in lengthier\ndynamics(largernumberofiterations)andmorerotations(astheprecessiontermismade\nstronger in a relative sense) before reaching the minimizing state, thereby increasing the\naverage length of mh,1andmh,2(as seen in the error rows). Similarly to Table 1 we\nsee that after applying a nodal projection, the energy is within 10 decimal places of\n−100, indicating that a minimizer is still identified. As expected the average solve time\nis independent of α. For α= 1/16we see that the violation of the unit-length constraint\nand the number of iterations are significantly larger. We suppose that this is related to a\n17possible instability of Algorithm 5.1, since, as shown Proposition 5.3(ii), stability requires\nτto be sufficiently small, with the threshold for the time-step size being proportional to\nthe damping parameter. Indeed, for α= 1/16, we observe that it is sufficient to reduce\nτto regain a good performance of the algorithm.\n0 100 200 300−0.500.51\niteration\n(a) Alg. 4.5, α= 1.0 100 200 300−0.500.51\niteration\n(b) Alg. 5.1, α= 1.0 100 200 300−0.500.51\niteration\n(c) Alg. 5.1, α= 1/2.\n0 200 400 600−101\niteration\n(d) Alg. 5.1, α= 1/4.0 1,000 2,000 3,000−101\niteration\n(e) Alg. 5.1, α= 1/8.0 2 4\n·104−1012\niteration\n(f) Alg. 5.1, α= 1/16.\nFigure 5. Experiment of Section 5.2: Evolution of ⟨m1(t)·e1⟩(red) and ⟨m2(t)·\ne1⟩(blue). (a) Algorithm 4.5 with mass-lumped L2-metric and α= 1. (b)–(f)\nAlgorithm 5.1 with α= 1,1/2,1/4,1/8,1/16.\nThe fact that lowering the value of αresults in less dissipation can also be seen in\nFigure 5, where we display the evolution of the average first component of both fields, i.e.,\n⟨mℓ(t)·e1⟩=|Ω|−1R\nΩmℓ(t)·e1for all ℓ= 1,2, for all cases. Interestingly, we see that the\ngradient flow dynamics and the LLG dynamics for α= 1/8,1/16return as approximate\nstationary point an approximation of the minimizer (m−\n1,m−\n2)≡(−a,a), whereas the\nLLG dynamics for α= 1,1/2,1/4return an approximation of (m+\n1,m+\n2)≡(a,−a). This\nis not surprising, since different dynamics can result in convergence to different stationary\npoints, even with the same initial condition. As far as the LLG dynamics is concerned,\nwe see that the oscillations of the average first components increase as αis lowered, which\ncan be explained by the greater relative weight of the precessional term on the right-hand\nside of the LLG equation for smaller values of α.\n5.2.2.Skyrmion dynamics. Inspired the experiment in [18, Section 4.3], we simulate the\ndynamics of isolated magnetic skyrmions in an AFM nanodisk in response to an applied\nfield pulse.\nThe setup (domain, energy, and material parameters) is the same as in Section 4.3.2,\nwhich we complete with the additional parameters needed for the dynamic case, i.e., the\nrescaled gyromagnetic ratios γ1=γ2=γ0≈2.21·105m/(A s)and the Gilbert damping\nparameters α1=α2= 5·10−3(see (51) below). Given the same spatial discretization\n(mesh) as in Section 4.3.2, as initial conditions m0\n1,handm0\n2,hfor Algorithm 5.1, we\n18consider the nodal projections of the Néel-type skyrmions shown in Figure 4. Moreover,\nfor the time discretization, we use of a constant time-step size of 2 fs.\nStarting from this configuration, we perturb the system from its equilibrium by ap-\nplying an in-plane pulse field of the form Hext(t) = ( H(t),0,0)of maximum intensity\nµ0Hmax= 100 mT for150 ps; see Figure 6(a). Then, we turn off the applied external field,\ni.e.,Hext(t)≡(0,0,0), and let the system relax to equilibrium. The overall simulation\ntime is 1 ns.\n00.150 10Hmax\nt(ns)H(t)\n(a) Applied pulse field.0 0.2 0.4 0.6 0.8 100.51·10−3\nt(ns)\n(b)⟨m(t)·e1⟩.\nFigure 6. Experiment of Section 5.2.2: (a) Structure of the applied pulse field. (b)\nTime evolution of ⟨m(t)·e1⟩.\nIn Figure 6(b), we show the time evolution of the average first component of the total\nmagnetization m=m1+m2. We see a perfect match between the applied pulse field\nand the total magnetization. When the field is turned off, the state immediately comes\nback to the initial configuration, which confirms its stability.\n6.Proofs\nIn this section, we collect the proof of all results presented in the paper.\n6.1. Static problem. We start with showing the weak sequential lower semicontinuity\nof the energy functional.\nProposition 6.1. The energy functional (1)is weakly sequentially lower semicontinuous\ninH1(Ω)×H1(Ω), i.e., if {(m1,k,m2,k)}k∈N⊂H1(Ω)×H1(Ω)and(m1,m2)∈H1(Ω)×\nH1(Ω)are such that (m1,k,m2,k)⇀(m1,m2)inH1(Ω)×H1(Ω)ask→ ∞, then\nE[m1,m2]≤lim inf k→∞E[m1,k,m2,k].\nThe result is a special case of the following lemma.\nLemma 6.2. LetVandHtwo Hilbert spaces such that V⊂Hwith compact inclusion.\nLeta:V×V→Rbe a continuous bilinear form satisfying a so-called Gårding inequality,\ni.e., there exists C1>0andC2∈Rsuch that\na(v, v)≥C1∥v∥2\nV−C2∥v∥2\nHfor all v∈V. (33)\nThen, the quadratic functional J:V→Rdefined by J[v] :=a(v, v)for all v∈Vis\nweakly sequentially lower semicontinuous in V, i.e., if {vk}k∈N⊂Vandv∈Vare such\nthatvk⇀ vinVask→ ∞, then J[v]≤lim inf k→∞J[vk].\n19Proof.Let{vk}k∈N⊂Vandv∈Vbe such that vk⇀ vinVask→ ∞. From the\ncompact inclusion V⊂H, it follows that vk→vinH. Using (33), we see that\nC1∥v−vk∥2\nV−C2∥v−vk∥2\nH≤a(v−vk, v−vk) =a(v, v)−a(vk, v)−a(v, vk) +a(vk, vk).\nWe now take the liminf as k→ ∞of this inequality. For the left-hand side we have that\nlim inf\nk→∞\u0000\nC1∥v−vk∥2\nV−C2∥v−vk∥2\nH\u0001\n≥0.\nFor the right-hand side, noting that vk⇀ vinVimplies that a(vk, v)→a(v, v)as\nk→ ∞, we have that\nlim inf\nk→∞[a(v, v)−a(vk, v)−a(v, vk) +a(vk, vk)] =−a(v, v) + lim inf\nk→∞a(vk, vk)\n=−J[v] + lim inf\nk→∞J[vk].\nThis shows that J[v]≤lim inf k→∞J[vk]and thus concludes the proof. □\nWe now prove Theorem 4.1 establishing the Γ-convergence of our finite element dis-\ncretization.\nProof of Theorem 4.1. Part (i) of the theorem immediately follows from the weak sequen-\ntial lower semicontinuity of the energy functional established in Proposition 6.1.\nToshowpart(ii),let (m1,m2)∈ Xbearbitrary. Since C∞(Ω;S2)isdensein H1(Ω;S2)\n(see [31, Theorem III.6.2]), for all k∈Nthere exists (m1,k,m2,k)∈C∞(Ω;S2)×\nC∞(Ω;S2)such that ∥mℓ−mℓ,k∥H1(Ω)≤1/kfor all ℓ= 1,2.\nLetε > 0. The above convergence guarantees the existence of k∈Nsuch that\n∥mℓ−mℓ,k∥H1(Ω)≤ε/2. Define mℓ,k,h:=Ih[mℓ,k]for all ℓ= 1,2. By construction, for\nallℓ= 1,2,|mℓ,k,h(z)|= 1for all z∈ N hand0 =∥Ih[|mℓ,k,h|2]−1∥L1(Ω)≤δfor all δ >0.\nHence, (m1,k,h,m2,k,h)belongs to Xh,δfor all δ >0. Moreover, a classical interpolation\nestimate yields that ∥mℓ,k−mℓ,k,h∥H1(Ω)≤Ch∥D2mℓ,k∥. Therefore, we have that\n∥mℓ,k−mℓ,k,h∥H1(Ω)≤ε/2ifhis chosen sufficiently small. Using the triangle inequality,\nwe thus obtain that ∥mℓ−mℓ,k,h∥H1(Ω)≤ε. Since ε >0was arbitrary, this shows that\nthe sequence {(m1,h,δ,m2,h,δ)}defined by (m1,h,δ,m2,h,δ) := (( m1,k,h,m2,k,h))∈ X h,δ\nsatisfies the desired convergence property toward (m1,m2)ash, δ→0(note that our\nconstruction is independent of δ, so the limit δ→0is trivial). This implies also that\nEh,δ[m1,h,δ,m2,h,δ]→ E[m1,m2]ash, δ→0and concludes the proof. □\nIn view of the analysis of the discrete gradient flows presented in Section 4, we now\nintroduce the following algorithm.\nAlgorithm 6.3 (general discrete gradient flow) .Discretization parameters: Mesh\nsizeh >0, time-step size τ >0, tolerance ε >0, parameters 0≤θ1, θ2, θ3≤1.\nInput: Initial guess (m0\n1,h,m0\n2,h)∈ S1(Th)3× S1(Th)3such that, for all ℓ= 1,2,\f\fm0\nℓ,h(z)\f\f= 1for all z∈ N h.\nLoop:For all i∈N0, iterate (i)–(ii)until the stopping criterion (stop)is met:\n(i)Given (mi\n1,h,mi\n2,h)∈ S1(Th)3×S1(Th)3, compute (vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]\nsuch that, for all (ϕ1,h,ϕ2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]andℓ= 1,2, it holds that\n⟨vi\nℓ,h,ϕℓ,h⟩H+aℓℓθ1τ⟨∇vi\nℓ,h,∇ϕℓ,h⟩+a12θ2τ⟨∇vi\n3−ℓ,h,∇ϕℓ,h⟩ −a0θ3τ⟨vi\n3−ℓ,h,ϕℓ,h⟩\n=−aℓℓ⟨∇mi\nℓ,h,∇ϕℓ,h⟩ −a12⟨∇mi\n3−ℓ,h,∇ϕℓ,h⟩+a0⟨mi\n3−ℓ,h,ϕℓ,h⟩.(34)\n(ii)Define\nmi+1\nℓ,h:=mi\nℓ,h+τvi\nℓ,hfor all ℓ= 1,2. (35)\n20(stop)Stop iterating (i)–(ii)if(vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]satisfies\n2X\nℓ=1\u0000\n∥vi\nℓ,h∥2\nH+τ∥∇vi\nℓ,h∥2\u0001\n≤ε2|Ω|. (36)\nOutput: Ifi∗∈N0denotes the smallest integer satisfying the stopping criterion (36),\ndefine the approximate stationary point (m1,h,m2,h) := (mi∗\n1,h,mi∗\n2,h).\nThe parameters 0≤θ1, θ2, θ3≤1modulates the ‘degree of implicitness’ in the treat-\nment of the three contributions of the energy. It is easy to see that Algorithm 4.4 and\nAlgorithm 4.5 are special instances of Algorithm 6.3, where θ1= 1(backward Euler) and\nθ2=θ3= 1/2(Crank–Nicolson) in Algorithm 4.4, whereas θ1= 1(backward Euler) and\nθ2=θ3= 0(forward Euler) in Algorithm 4.5.\nFor ease of presentation, in Section 4, we have decided not to present Algorithm 6.3 in\nits full generality, but we have restricted ourselves to two of its instances. This has been\nmotivated by the following two reasons: First, we believe that the two proposed cases\nare the most relevant in practical computations. Second, the properties and the analysis\nof the algorithm for general θ1, θ2, θ3resemble the ones of the two presented prototypical\ncases (excluding the combinations involving values θ1, θ2<1/2, which require severe\nrestrictions of the form τ=O(h2)for stability and therefore have been ignored).\nIn the following proposition, we show well-posedness of each iteration of Algorithm 6.3.\nProposition 6.4. Suppose that θ1,θ2,θ3, and τsatisfy the following conditions:\nθ1>0, a 11a22θ2\n1> a2\n12θ2\n2,and c2\nH|a0|θ3τ <1, (37)\nwhere a11,a22,a12, and a0are the coefficients in (1), whereas cHis the constant in (17).\nThen, for all i∈N0,(34)admits a unique solution (vi\n1,h,vi\n2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h].\nProof.Leti∈N0. The sum of the left-hand sides of (34) for ℓ= 1,2yields a bilinear\nform bi: (Kh[mi\n1,h]×Kh[mi\n2,h])×(Kh[mi\n1,h]×Kh[mi\n2,h])→R, which is defined by\nbi((ψ1,h,ψ2,h),(ϕ1,h,ϕ2,h)) =⟨ψ1,h,ϕ1,h⟩H+⟨ψ2,h,ϕ2,h⟩H\n+a11θ1τ⟨∇ψ1,h,∇ϕ1,h⟩+a22θ1τ⟨∇ψ2,h,∇ϕ2,h⟩\n+a12θ2τ⟨∇ψ2,h,∇ϕ1,h⟩+a12θ2τ⟨∇ψ1,h,∇ϕ2,h⟩\n−a0θ3τ⟨ψ2,h,ϕ1,h⟩ −a0θ3τ⟨ψ1,h,ϕ2,h⟩\nfor all (ψ1,h,ψ2,h),(ϕ1,h,ϕ2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h]. Owing to the second inequality\nin (17), the bilinear form is bounded with respect to the H1-norm. To show coercivity,\nfor an arbitrary (ϕ1,h,ϕ2,h)∈Kh[mi\n1,h]×Kh[mi\n2,h], we first compute\nbi((ϕ1,h,ϕ2,h),(ϕ1,h,ϕ2,h)) =∥ϕ1,h∥2\nH+∥ϕ2,h∥2\nH−2a0θ3τ⟨ϕ2,h,ϕ1,h⟩\n+a11θ1τ∥∇ϕ1,h∥2+a22θ1τ∥∇ϕ2,h∥2\n+ 2a12θ2τ⟨∇ϕ2,h,∇ϕ1,h⟩.\nThe terms involving the gradients of (ϕ1,h,ϕ2,h)make up a quadratic form, which is\npositive definite if and only if the underlying 2-by-2 matrix is positive definite, which is\ntrue if and only if the first two inequalities in (37) hold.\nThanks to (17), it holds that\n∥ϕ1,h∥2\nH+∥ϕ2,h∥2\nH−2a0θ3τ⟨ϕ2,h,ϕ1,h⟩ ≥(c−2\nH− |a0|θ3τ)\u0000\n∥ϕ1,h∥2+∥ϕ2,h∥2\u0001\n.\nThis shows that the L2-part of the bilinear form is coercive if the third inequality in (37)\nholds.\n21Hence, weconcludethatthebilinearform bi(·,·)iscoercivewithrespecttothe H1-norm\nif (37) is satisfied. Observing that the sum over ℓ= 1,2of the right-hand sides of (34)\ndefines a bounded linear form, well-posedness of (34) then follows from the Lax–Milgram\ntheorem. □\nIn the following proposition, we establish the discrete energy law satisfied by the ap-\nproximations generated by Algorithm 6.3.\nProposition 6.5. Letθ1,θ2,θ3, and τsatisfy the assumptions of Proposition 6.4. For\nalli∈N0, the iterates of Algorithm 6.3 satisfy\nE[mi+1\n1,h,mi+1\n2,h]− E[mi\n1,h,mi\n2,h] =−τ2X\nℓ=1∥vi\nℓ,h∥2\nH−(2θ1−1)\n2τ22X\nℓ=1aℓℓ∥∇vi\nℓ,h∥2\n−a12(2θ2−1)τ2⟨∇vi\n1,h,∇vi\n2,h⟩+a0(2θ3−1)τ2⟨vi\n1,h,vi\n2,h⟩.(38)\nSuppose that θ1,θ2,θ3, and τsatisfy also the following conditions:\nθ1≥1/2, a 11a22(2θ1−1)2≥a2\n12(2θ2−1)2,andc2\nH|a0||2θ3−1|τ≤2.(39)\nThen, the sequence of energies generated by Algorithm 6.3 is monotonically decreasing,\ni.e., it holds that E[mi+1\n1,h,mi+1\n2,h]≤ E[mi\n1,h,mi\n2,h]for all i∈N0.\nProof.Leti∈N0. Testing (34) with ϕℓ,h=vi\nℓ,h∈Kh[mi\nℓ,h]forℓ= 1,2and summing\nthe resulting equations, we obtain that\n2X\nℓ=1\u0000\n∥vi\nℓ,h∥2\nH+aℓℓθ1τ∥∇vi\nℓ,h∥2\u0001\n+ 2a12θ2τ⟨∇vi\n1,h,∇vi\n2,h⟩ −2a0θ3τ⟨vi\n1,h,vi\n2,h⟩\n=2X\nℓ=1\u0000\n−aℓℓ⟨∇mi\nℓ,h,∇vi\nℓ,h⟩ −a12⟨∇mi\n3−ℓ,h,∇vi\nℓ,h⟩+a0⟨mi\n3−ℓ,h,vi\nℓ,h⟩\u0001\n.(40)\nIt follows that\nE[mi+1\n1,h,mi+1\n2,h]\n(35)=E[mi\n1,h,mi\n2,h] +1\n2τ2X\nℓ=1aℓℓ\u0000\n2⟨∇mi\nℓ,h,∇vi\nℓ,h⟩+τ∥∇vi\nℓ,h∥2\u0001\n+a12τ\u0010\n⟨∇mi\n1,h,∇vi\n2,h⟩+⟨∇mi\n2,h,∇vi\n1,h⟩+τ⟨∇vi\n1,h,∇vi\n2,h⟩\u0001\n−a0τ\u0000\n⟨mi\n1,h,vi\n2,h⟩+⟨mi\n2,h,vi\n1,h⟩+τ⟨vi\n1,h,vi\n2,h⟩\u0011\n(40)=E[mi\n1,h,mi\n2,h]−τ2X\nℓ=1∥vi\nℓ,h∥2\nH−(2θ1−1)\n2τ22X\nℓ=1aℓℓ∥∇vi\nℓ,h∥2\n−a12(2θ2−1)τ2⟨∇vi\n1,h,∇vi\n2,h⟩+a0(2θ3−1)τ2⟨vi\n1,h,vi\n2,h⟩,\nwhich is (38). Arguing as in the proof of Proposition 6.4, it is easy to see the right-hand\nside of (38) is nonpositive if the inequalities in (39) are satisfied. This shows that the\nsequenceofenergiesgeneratedbythealgorithmismonotonicallydecreasingandconcludes\nthe proof. □\nIn the following lemma, we prove two auxiliary estimates, which will be useful in the\nproof of convergence of Algorithm 6.3.\n22Lemma 6.6. For all ℓ= 1,2, for all j∈N, the iterates of Algorithm 6.3 satisfy\nc−1\nT∥Ih[\f\fmj\nℓ,h\f\f2]−1∥L1(Ω)≤τ2j−1X\ni=0∥vi\nℓ,h∥2(41)\nc−1\nT∥mj\nℓ,h∥2≤ |Ω|+τ2j−1X\ni=0∥vi\nℓ,h∥2, (42)\nwhere cT>0depends only on the shape-regularity of the family of meshes.\nProof.We follow [10]. Let ℓ= 1,2andj∈N. For all i= 0, . . . , j −1, from (35), since\nvi\nℓ,h∈Kh[mi\nℓ,h], we deduce that\f\fmi+1\nℓ,h(z)\f\f2=\f\fmi\nℓ,h(z)\f\f2+τ2\f\fvi\nℓ,h(z)\f\f2for all z∈ N h.\nIterating in iand using that\f\fm0\nℓ,h(z)\f\f= 1for all z∈ N h, we obtain that\n\f\fmj\nℓ,h(z)\f\f2= 1 + τ2j−1X\ni=0\f\fvi\nℓ,h(z)\f\f2.\nThen, both (41) and (42) follow from the equivalence of the Lp-norm of discrete functions\nwith the weighted ℓp-norm of the vector collecting their nodal values (with equivalence\nconstants depending only on the shape-regularity of the family of meshes); see, e.g., [9,\nLemma 3.4]. □\nIn the following lemma, we prove stability of Algorithm 6.3.\nLemma 6.7. Letθ1,θ2,θ3, and τsatisfy the assumptions of Proposition 6.4 as well as\nthe inequalities\nθ1>1/2, a 11a22(2θ1−1)2> a2\n12(2θ2−1)2,andc2\nH|a0||2θ3−1|τ <2.(43)\nThen, there exists a threshold τ0>0such that, if τ < τ 0, the iterates of Algorithm 6.3\nsatisfy, for all j∈N, the stability estimate\n2X\nℓ=1∥mj\nℓ,h∥2\nH1(Ω)+τj−1X\ni=02X\nℓ=1∥vi\nℓ,h∥2\nH+τ2j−1X\ni=02X\nℓ=1∥∇vi\nℓ,h∥2≤C \n1 +2X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω)!\n.\n(44)\nThe threshold τ0depends on a0,θ3,cH, and the shape-regularity of the family of meshes,\nwhereas the constant C >0depends only on |Ω|,a11,a12,a22,a0,θ1,θ2,θ3,cH, and the\nshape-regularity of the family of meshes.\nProof.Letj∈N. For all i= 0, . . . , j −1, we apply Proposition 6.5, which yields (38).\nSumming (38) over i= 0, . . . , j −1, we obtain that\nE[mj\n1,h,mj\n2,h] +τj−1X\ni=02X\nℓ=1∥vi\nℓ,h∥2\nH+(2θ1−1)\n2τ2j−1X\ni=02X\nℓ=1aℓℓ∥∇vi\nℓ,h∥2\n+a12(2θ2−1)τ2j−1X\ni=0⟨∇vi\n1,h,∇vi\n2,h⟩ −a0(2θ3−1)τ2j−1X\ni=0⟨vi\n1,h,vi\n2,h⟩=E[m0\n1,h,m0\n2,h].\nUsing (43) and arguing as in the proof of Proposition 6.4, one can show that\nE[mj\n1,h,mj\n2,h] +λ1τj−1X\ni=02X\nℓ=1∥vi\nℓ,h∥2\nH+λ2τ2j−1X\ni=02X\nℓ=1∥∇vi\nℓ,h∥2≤ E[m0\n1,h,m0\n2,h]\n23for some positive values λ1=λ1(a0, θ3)andλ2=λ2(a11, a12, a22, θ1, θ2). From (2) and\nYoung’s inequality, it follows that\nE[mj\n1,h,mj\n2,h]≥λ32X\nℓ=1∥∇mj\nℓ,h∥2−|a0|\n22X\nℓ=1∥mj\nℓ,h∥2\nfor some λ3=λ3(a11, a12, a22)>0. Moreover, it holds that\nE[m0\n1,h,m0\n2,h]≤λ42X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω)\nfor some λ4=λ3(a11, a12, a22, a0)>0. Altogether, we thus obtain that\nλ32X\nℓ=1∥∇mj\nℓ,h∥2−|a0|\n22X\nℓ=1∥mj\nℓ,h∥2+λ1τj−1X\ni=02X\nℓ=1∥vi\nℓ,h∥2\nH+λ2τ2j−1X\ni=02X\nℓ=1∥∇vi\nℓ,h∥2\n≤λ42X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω).(45)\nFrom Lemma 6.6 and (17), we deduce that\n|a0|2X\nℓ=1∥mj\nℓ,h∥2≤2cT|a0||Ω|+cT|a0|cHτ2j−1X\ni=02X\nℓ=1∥vi\nℓ,h∥2\nH, (46)\nwhere cT>0is the constant appearing in (42) (which depends only on the shape-\nregularity of the family of meshes). Combining (45) and (46), we thus obtain that\nλ32X\nℓ=1∥∇mj\nℓ,h∥2+|a0|\n22X\nℓ=1∥mj\nℓ,h∥2+ (λ1−cT|a0|cHτ)τj−1X\ni=02X\nℓ=1∥vi\nℓ,h∥2\nH\n+λ2τ2j−1X\ni=02X\nℓ=1∥∇vi\nℓ,h∥2≤2cT|a0||Ω|+λ42X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω).\nHence, if τ < τ 0:=λ1/(cT|a0|cH), all terms on the left-hand side are nonnegative and\nwe obtain (44), where the (explicitly computable) constant C > 0depends only on |Ω|,\na11,a12,a22,a0,θ1,θ2,θ3,cH, and cT. □\nIn the following proposition, combining the results we have proved so far, we establish\nthe main properties of Algorithm 6.3\nProposition 6.8. Letθ1,θ2,θ3, and τsatisfy the assumptions of Lemma 6.7. If the\ntime-step size τis sufficiently small, then Algorithm 6.3 is well defined: Each iteration\nis well defined and the stopping criterion (36)is met in a finite number of iterations. In\nparticular, the approximate stationary point (m1,h,m2,h)is well defined. Moreover, for\nallℓ= 1,2, it holds that\n∥Ih[|mℓ,h|2]−1∥L1(Ω)≤Cτ \n1 +2X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω)!\n, (47)\nwhere the constant C > 0depends only on |Ω|,a11,a12,a22,a0,θ1,θ2,θ3,cH, and the\nshape-regularity of the family of meshes.\n24Proof.The well-posedness of each iteration of the algorithm is a consequence of Propo-\nsition 6.4. Now, let τ0>0be the threshold guaranteed by Lemma 6.7. If τ < τ 0,\nthen (44) holds. Since the left-hand side of (44) is nonnegative and the right-hand side\nis independent of j, we deduce that the series\n∞X\ni=02X\nℓ=1\u0000\n∥vi\nℓ,h∥2\nH+τ∥∇vi\nℓ,h∥2\u0001\nisconvergent. ItfollowsthatP\nℓ=1,2∥vi\nℓ,h∥2\nH+τ∥∇vi\nℓ,h∥2→0asi→ ∞, whichguarantees\nthat the stopping criterion (36) is satisfied if iis sufficiently large. Estimate (47) is a\nconsequence of (44) and (41) from Lemma 6.6. This concludes the proof. □\nIn the following theorem, we show the convergence of the sequence generated by Algo-\nrithm 6.3.\nTheorem 6.9. Letθ1andθ2satisfy the inequalities\nθ1>1/2, a 11a22θ2\n1> a2\n12θ2\n2,and a11a22(2θ1−1)2> a2\n12(2θ2−1)2.\nSuppose that there exists c0>0, independent of the discretization parameters h,τ, and\nε, such that\nsup\nh>0 2X\nℓ=1∥m0\nℓ,h∥2\nH1(Ω)!\n≤c0. (48)\nSuppose that τ→0andε→0ash→0. Then, as h→0, the sequence of approximate\nstationary points {(m1,h,m2,h)}h>0generated by Algorithm 6.3, upon extraction of a\nsubsequence, converges weakly in H1(Ω)×H1(Ω)toward a point (m1,m2)∈ X. If\na12= 0, the limit (m1,m2)is a stationary point of the energy functional (1).\nProof.Since τ→0, we can assume that it is sufficiently small such that the algo-\nrithm is well defined (cf. Proposition 6.8) and that the stability estimate (44) holds (cf.\nLemma 6.7). Together with (48), it thus follows that the sequence {(m1,h,m2,h)}h>0is\nuniformly bounded in H1(Ω)×H1(Ω). Hence, there exists (m1,m2)∈H1(Ω)×H1(Ω)\nand a (nonrelabeled) weakly convergence subsequence of {(m1,h,m2,h)}h>0such that\n(m1,h,m2,h)⇀(m1,m2)inH1(Ω)×H1(Ω)and(m1,h,m2,h)→(m1,m2)inL2(Ω)×\nL2(Ω). Combining (48) with (47), we see that, for all ℓ= 1,2,∥Ih[|mℓ,h|2]−1∥L1(Ω)→0\nash→0. Hence, applying [9, Lemma 7.2], we obtain that (m1,m2)∈ X.\nTo conclude the proof, it remains to show that, if a12= 0,(m1,m2)∈ Xsatisfies (4).\nWe start with observing that each approximate stationary point (m1,h,m2,h)generated\nby Algorithm 6.3 satisfies the variational formulation\n−aℓℓ⟨∇mℓ,h,∇ϕℓ,h⟩+a0⟨m3−ℓ,h,ϕℓ,h⟩=Rℓ,h(ϕℓ,h)\nfor all ϕℓ,h∈Kh[mℓ,h]andℓ= 1,2, where the reminder terms on the right-hand side are\ngiven by\nRℓ,h(ϕℓ,h) =⟨vi∗\nℓ,h,ϕℓ,h⟩H+aℓℓθ1τ⟨∇vi∗\nℓ,h,∇ϕℓ,h⟩ −a0θ3τ⟨vi∗\n3−ℓ,h,ϕℓ,h⟩\nand satisfy\f\fRh(ϕℓ,h)\f\f≤Cε∥ϕℓ,h∥H1(Ω)for all ϕℓ,h∈H1(Ω); see (34) and (36). Here,\nC > 0depends only on a11,a22,a0, and |Ω|. Note that, since ε→0ash→0, we\nhave that Rh→0inH1(Ω)∗ash→0. Let ψ∈C∞(Ω). Choosing the test function\nϕℓ,h=Ih[mℓ,h×ψ]∈Kh[mℓ,h]in (34) and passing to the limit as h→0(using the\navailable convergence results as in the proof of [9, Theorem 7.6]), we obtain that\n−aℓℓ⟨∇mℓ,mℓ×∇ψ⟩+a0⟨m3−ℓ,mℓ×ψ⟩= 0 (49)\n25for all ℓ= 1,2. Since ψwas arbitrary, by density we have that this identity holds for\nallψ∈H1(Ω). Finally, let φ∈H1(Ω)∩L∞(Ω)be arbitrary. Choosing ψ=mℓ×φ\nin(49)andperformingsimplealgebraicmanipulationsbasedontheidentities a×(b×c) =\n(a·c)b−(a·b)c(forall a,b,c∈R3),|mℓ|= 1(a.e.in Ω, forall ℓ= 1,2)and ∂imℓ·mℓ= 0\n(a.e. in Ω, for all i= 1,2,3andℓ= 1,2), we obtain that (m1,m2)∈ Xsolves (4) for the\ncasea12= 0. This shows that (m1,m2)is a stationary point of the energy and concludes\nthe proof. □\n6.2. Dynamic problem. In this section, we aim to present the proofs of the results\nconcerning Algorithm 5.1 discussed in Section 5. However, for the sake of brevity, we\nomit those of Proposition 5.2 and Proposition 5.3, because they can be obtained following\nline by line those of Proposition 6.4, Proposition 6.5, and Lemma 6.7. We focus on the\nproof of the main convergence result.\nProof of Theorem 5.4. We follow the argument of the seminal paper on the tangent plane\nscheme [3], which we adapt in order to take the projection-free update (30) (see also [1,\n18]) and the different expression of the energy into account. For the sake of clarity, we\nsplit the proof into three steps:\n•Step 1:Existence of the limit (m1,m2)∈L∞(0,∞;X).\nLetT >0be arbitrary. From the stability estimate (32) (cf. Proposition 5.3), which holds\nuniformlyin handτ(ifτissufficientlysmall), itfollowsthat, forall ℓ= 1,2, thepiecewise\naffine time reconstruction mℓ,hτand the piecewise constant time reconstructions m±\nℓ,hτ\n(defined according to (13)) are both uniformly bounded in L∞(0,∞;H1(Ω)). Moreover,\nmℓ,hτ|ΩTis uniformly bounded in H1(ΩT). By compactness, successive extractions of\n(nonrelabeled) subsequences and standard Sobolev embeddings yield the existence of\nm1,m2∈L∞(0,∞;H1(Ω))∩H1(ΩT)such that, for all ℓ= 1,2, ash, τ→0we have\nthe convergences mℓ,hτ|ΩT⇀m|ΩTinH1(ΩT),mℓ,hτ|ΩT→m|ΩTinHs(ΩT)for all s∈\n(0,1),mℓ,hτ,m±\nℓ,hτ∗⇀mℓinL∞(0,∞;H1(Ω)),mℓ,hτ,m±\nℓ,hτ⇀mℓinL2(0,∞;H1(Ω)),\nmℓ,hτ|ΩT,m±\nℓ,hτ|ΩT→mℓinL2(0, T;Hs(Ω))for all s∈(0,1),mℓ,hτ|ΩT,m±\nℓ,hτ|ΩT→mℓ\ninL2(ΩT)andpointwisealmosteverywherein ΩT. Fromtheprojection-freeupdates(30),\narguing as in the proof of Lemma 6.6, we obtain that (41) holds for all ℓ= 1,2and for\nallj∈N, from which it follows (see Step 3 of the proof of [18, Proposition 6]) that\n|m1|=|m2|= 1a.e. in Ω×(0,∞). This shows that (m1,m2)∈L∞(0,∞;X). Finally,\nfrom the stability estimate, it also follows that, for all ℓ= 1,2,τ∇(∂tmℓ,hτ)|ΩT→0in\nL2(ΩT)ash, τ→0.\n•Step 2:Ifa12= 0,(m1,m2)satisfies the variational formulation (11).\nLetφ∈C∞(ΩT)be an arbitrary smooth test function. We consider the smallest integer\nj∈Nsatisfying T≤jτand extend φby zero in (T, tj). Let ℓ= 1,2. For all i=\n0, . . . , j −1, we choose ϕℓ,h=Ih[mi\nℓ,h×φ(ti)]∈Kh[mi\nh]in (29), we obtain\nαℓ⟨vi\nℓ,h,Ih[mi\nℓ,h×φ(ti)]⟩h+⟨mi\nℓ,h×vi\nℓ,h,Ih[mi\nℓ,h×φ(ti)]⟩h\n+ηℓaℓℓτ⟨∇vi\nℓ,h,∇Ih[mi\nℓ,h×φ(ti)]⟩\n=−ηℓaℓℓ⟨∇mi\nℓ,h,∇Ih[mi\nℓ,h×φ(ti)]⟩+ηℓa0⟨mi\n3−ℓ,h,Ih[mi\nℓ,h×φ(ti)]⟩,\nDue to the properties of the mass-lumped scalar product, we can remove the nodal\ninterpolant from the first two terms on the left-hand side without altering the value of\n26the integrals. Multiplication by τand summation over i= 0, . . . , j −1then yield\nαℓZtj\n0⟨∂tmℓ,hτ(t),m−\nℓ,hτ(t)×φ−\nτ(t)⟩hdt\n+Ztj\n0⟨m−\nℓ,hτ(t)×∂tmℓ,hτ(t),m−\nℓ,hτ(t)×φ−\nτ(t)]⟩hdt\n+ηℓaℓℓτZtj\n0⟨∇∂tmℓ,hτ(t),∇Ih[m−\nℓ,hτ(t)×φ−\nτ(t)]⟩dt\n=−ηℓaℓℓZtj\n0⟨∇φ−\nτ(t),∇Ih[m−\nℓ,hτ(t)×φ−\nτ(t)]⟩dt\n+ηℓa0Ztj\n0⟨m−\n3−ℓ,hτ(t),Ih[m−\nℓ,hτ(t)×φ−\nτ(t)]⟩dt,\nwhere we note that we have rewritten the equation in terms of the time reconstruc-\ntions (13). Using (16) and the approximation properties of the nodal interpolant, in all\nintegrals we substitute the mass-lumped inner products by L2-products and remove the\nnodal interpolant (see [3]). Moreover, exploiting the fact that the integrands are all uni-\nformly bounded, we modify the domain in integration in time from (0, tj)to(0, T). All\nthese actions generate an error which goes to zero in the limit as h, τ→0. In particular,\nwe obtain\nαℓZT\n0⟨∂tmℓ,hτ(t),m−\nℓ,hτ(t)×φ−\nτ(t)⟩dt\n+ZT\n0⟨m−\nℓ,hτ(t)×∂tmℓ,hτ(t),m−\nℓ,hτ(t)×φ−\nτ(t)]⟩dt\n+ηℓaℓℓτZT\n0⟨∇∂tmℓ,hτ(t),∇[(m−\nℓ,hτ(t)×φ−\nτ(t)]⟩dt\n=−ηℓaℓℓZT\n0⟨∇φ−\nτ(t),∇[m−\nℓ,hτ(t)×φ−\nτ(t)]⟩dt\n+ηℓa0ZT\n0⟨m−\n3−ℓ,hτ(t),m−\nℓ,hτ(t)×φ−\nτ(t)⟩dt+o(1).\nUsing the convergence results available from Step 1, we can pass this formulation to the\nlimit as h, τ→0and obtain that the last term on the left-hand side goes to zero, whereas\nall other terms converge toward the corresponding ones in (11). For the details of the\nargument, we refer to [3] for all terms but the second one on the left-hand side, which, due\nto the omission of the nodal projection from (30), requires a more careful treatment (see\nStep 2 of the proof of [18, Theorem 1]). This shows that, for all ℓ= 1,2,mℓsatisfies (11)\nfor all φ∈C∞(ΩT). By density, the result then holds for all φ∈H1(ΩT).\n•Step 3: (m1,m2)satisfies the energy inequality (12).\nWe start from the discrete energy law (31) established in Proposition 5.3. Using (2) and\na combination of Cauchy–Schwarz’ an Young’s inequalities, we obtain that\nE[mj\n1,h,mj\n2,h]+τj−1X\ni=02X\nℓ=1\u0012αℓ\nηℓ−|a0|τ\n2\u0013\n∥vi\nℓ,h∥2\nh+λτ2j−1X\ni=02X\nℓ=1∥∇vi\nℓ,h∥2≤ E[m0\n1,h,m0\n2,h],\n27where λ >0is the minimum eigenvalue of the 2-by-2 matrix\u0012\na11a12\na12a22\u0013\n. The last term\non the left-hand side is nonnegative and can be omitted. Rewriting the inequality in\nterms of the time reconstructions (13), we get\nE[m+\n1,hτ(T),m+\n2,hτ(T)] +2X\nℓ=1\u0012αℓ\nηℓ−|a0|τ\n2\u0013ZT\n0∥∂tmℓ,hτ(t)∥2\nhdt≤ E[m−\n1,hτ(0),m−\n2,hτ(0)].\nPassing to the limit as h, τ→0, using the convergence results available from Step 1,\nstandard lower semicontinuity arguments yield (12). This concludes the proof. □\n7.Acknowledgment\nMR is a member of the ‘Gruppo Nazionale per il Calcolo Scientifico (GNCS)’ of the\nItalian ‘Istituto Nazionale di Alta Matematica (INdAM)’. Part of the work on this pa-\nper was undertaken when the authors were visiting the Hausdorff Research Institute for\nMathematics of the University of Bonn during the Trimester Program on Mathematics for\nComplex Materials , funded by the German Research Foundation (DFG) under Germany’s\nExcellence Strategy – EXC-2047/1– 390685813. The kind hospitality of the institute is\nthankfully acknowledged.\nReferences\n[1] C. Abert, G. Hrkac, M. Page, D. Praetorius, M. Ruggeri, and D. 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García-Cervera, and W. E. A Gauss-Seidel projection method for micromagnetics\nsimulations. J. Comput. Phys. , 171(1):357–372, 2001. doi:10.1006/jcph.2001.6793.\n29Appendix A.The equations in physical units\nIn this appendix, for the convenience of all interdisciplinary readers, we present the\nmodel in physical units (used for physical investigations, e.g., in [25, 23, 26, 24, 28, 29,\n32, 14]) and show how to obtain from it the dimensionless setting described in Section 2\nand analyzed in the paper. By doing this, we also justify the setup and the choice of the\nmaterial parameters in the numerical experiments presented in the work.\nA.1. Nondimensionalization. LetΩ⊂R3be the volume occupied by an AFM or\nFiM material. Let the vector field M: Ω→R3denote the total magnetization of the\nsample (in A/m). The total magnetization can be decomposed as M=M1+M2, where\nM1,M2: Ω→R3, the magnetization vectors of two sublattices (in A/m), satisfy the\nconstraints |M1|=Ms,1and|M2|=Ms,2. Theconstants Ms,1, Ms,2>0arethesublattice\nsaturation magnetizations (in A/m). Let m1,m2: Ω→S2be the dimensionless unit-\nlength vector fields m1=M1/Ms,1andm2=M2/Ms,2. The total Gibbs free energy (in\nJ) of the system (assumed, for simplicity, to include only exchange contributions in this\nsection) reads as\nE[m1,m2] =Eex[m1,m2]\n=2X\nℓ=1AℓℓZ\nΩ|∇mℓ|2+A12Z\nΩ∇m1:∇m2−4A0\na2Z\nΩm1·m2,(50)\nwhere the exchange constants A11, A22>0andA12, A0∈Rare in J/m, whereas a >0\nis the lattice constant (in m). The first contribution in (50) is called inhomogeneous in-\ntralattice exchange and models the classical ferromagnetic exchange for m1andm2. The\nsecond term is called inhomogeneous interlattice exchange , which arises from a nearest-\nneighbor approximation of the exchange interaction between spins. The last contribution\nis called homogeneous interlattice exchange and takes the local interaction between m1\nandm2into account.\nThe dynamics of m1andm2is governed by a coupled system of two LLG equations\n∂tmℓ=−γℓmℓ×Heff,ℓ[m1,m2] +αℓmℓ×∂tmℓforℓ= 1,2, (51)\nwhere γℓ>0(inm/(A s)) and αℓ>0(dimensionless) are the sublattice rescaled gy-\nromagnetic ratios and Gilbert damping parameters, respectively. In (51), the effective\nfieldsHeff,ℓ[m1,m2](inA/m) are equal, up to a negative multiplicative constant, to the\nfunctional derivatives of the total energy with respect to mℓ, i.e.,\nHeff,ℓ[m1,m2] =−1\nµ0Ms,ℓE[m1,m2]\nδmℓ,\nwhere µ0is the vacuum permeability (in N/A2). Assuming no flux boundary conditions,\nthe strong form of the resulting effective fields reads as\nHeff,ℓ[m1,m2] =2Aℓℓ\nµ0Ms,ℓ∆mℓ+A12\nµ0Ms,ℓ∆m3−ℓ+4A0\nµ0Ms,ℓa2m3−ℓ.\nWe now start the nondimensionalization. Let Ms>0andγ0>0be some reference satu-\nrationmagnetization(in A/m)andrescaledgyromagneticratio(in m/(A s)), respectively.\nFor all ℓ= 1,2, define the positive dimensionless parameters ηs,ℓ:=Ms,ℓ/Msandηγ,ℓ:=\nγℓ/γ0. The dimensionless total magnetization is given by m=M/Ms=ηs,1m1+ηs,2m2.\nLetL > 0is some intrinsic length of the problem. We rescale the space and time\nvariables to obtain the dimensionless variables x′=x/Landt′=γ0Mst. Accord-\ningly, we rescale also the domain Ω′= Ω/L. We consider the rescaled unit-length\n30vector fields m′\nℓ(x′, t′) =mℓ(Lx′, t′/(γ0Ms))(ℓ= 1,2) and the rescaled total magne-\ntization m′(x′, t′) =m(Lx′, t′/(γ0Ms)). Moreover, we rescale the energy as E′[m′\n1,m′\n2] =\nE[m1,m2]/(µ0M2\nsL3), which yields the expression\nE′[m′\n1,m′\n2] =E′\nex[m′\n1,m′\n2]\n=1\n22X\nℓ=12Aℓℓ\nµ0M2\nsL2Z\nΩ′|∇′m′\nℓ|2+A12\nµ0M2\nsL2Z\nΩ′∇′m′\n1:∇′m′\n2−4A0\nµ0M2\nsa2Z\nΩ′m′\n1·m′\n2.\nDefining the dimensionless coefficients aℓℓ= 2Aℓℓ/(µ0M2\nsL2)>0(ℓ= 1,2),a12=\nA12/(µ0M2\nsL2)∈R, and a0= 4A0/(µ0M2\nsa2)∈R, and omitting all ‘primes’ for sim-\nplicity, we obtain the dimensionless energy functional (1) of Section 2. By construction,\nthe dimensionless rescaled effective fields defined in (5) are related to the ones in physical\nunits according to the relation\nheff,ℓ[m′\n1,m′\n2](5)=−δE′[m′\n1,m′\n2]\nδm′\nℓ=η2\ns,ℓ\nMs,ℓHeff,ℓ[m1,m2]for all ℓ= 1,2.\nRescaling the LLG equations in (51) according to the above change of variables and\nintroducing all dimensionless quantities, we obtain\n∂t′m′\nℓ=−ηγ,ℓ\nηs,ℓm′\nℓ×heff,ℓ[m′\n1,m′\n2] +αℓm′\nℓ×∂t′m′\nℓfor all ℓ= 1,2,\nDefining the dimensionless parameter ηℓ:=ηγ,ℓ/ηs,ℓ>0and omitting all ‘primes’, we\nobtain the dimensionless system (9) of LLG equations of Section 2.\nA.2. Lower-order energy contributions. In practically relevant simulations, to be\nable to describe complex physical processes involving AFM and FiM materials, more\nenergy contributions (in addition to the exchange ones) need to be taken into account\nin (50):\n•Themagnetocrystalline anisotropy energy incorporates the existence of preferred\ndirections of alignment for the fields. In the uniaxial case, it reads as\nEani[m1,m2] =K1Z\nΩ[1−(a1·m1)2] +K2Z\nΩ[1−(a2·m2)2],\nwhere K1, K2>0are physical constants (in J/m3), whereas a1,a2∈S2are the\nso-called easy axes of the material (usually it holds that K1=K2anda1=a2).\n•TheDzyaloshinskii–Moriya interaction is used to incorporate chiral effects into\nthe model. Its general expression for AFM and FiM materials is given by\nEDMI[m1,m2] =Z\nΩD1: (∇m1×m1) +Z\nΩD2: (∇m2×m2),\nwhere D1,D2∈R3×3are the so-called spiralization tensors (with coefficients in\nJ/m2), whereas, for ℓ= 1,2,∇mℓ×mℓdenotesthematrixwithcolumns ∂jm×m\nforj= 1,2,3(again, usually it holds that D1=D2).\n•TheZeeman energy models the interaction of the total magnetization with an\napplied external field (assumed to be magnetization-independent) and reads as\nEext[m1,m2] =−µ0Z\nΩHext·(Ms,1m1+Ms,2m2),\nwhere Hext∈R3denotes an applied external field (in A/m).\n31•Themagnetostatic energy can be understood as the energy associated with the\ninteraction of the total magnetization with the stray field Hs∈R3, which solves\nthe magnetostatic Maxwell equations\n∇ ·Hs=−∇ · [χΩ(Ms,1m1+Ms,2m2)]and∇ ×Hs=0inR3.\nThe energy contribution is given by\nEext[m1,m2] =−µ0\n2Z\nΩHext·(Ms,1m1+Ms,2m2),\nwhere χΩ:R3→ {0,1}denotes the indicator function of the domain Ω.\nNote that in all the above energy contributions the two fields are decoupled (for the\nmagnetostatic energy, this is a consequence of the fact that the operator mapping the\ntotal magnetization to the solution of the magnetostatic Maxwell equations is linear).\nHence, even in the presence of the above contributions, the system of Euler–Lagrange\nequations associated with the minimization problem and the system of LLG equations\nare only exchange-coupled.\nIn the numerical experiments of the work (see Sections 4.3 and 5.2), we considered\ndimensionless forms of magnetocrystalline anisotropy energy, Dzyaloshinskii–Moriya in-\nteraction and Zeeman energy, namely\nEani[m1,m2] =q2\n1\n2Z\nΩ[1−(a1·m1)2] +q2\n2\n2Z\nΩ[1−(a2·m2)2],\nEDMI[m1,m2] =Z\nΩbD1: (∇m1×m1) +Z\nΩbD2: (∇m2×m2),\nEext[m1,m2] =−Z\nΩhext·(ηs,1m1+ηs,2m2) =−Z\nΩhext·m.\nIn these expressions, which can be obtained rescaling the energy contributions as de-\nscribed in the previous section, the dimensionless parameters are related to the physical\nones via the relationships qℓ=p\n2Kℓ/(µ0M2\ns),bDℓ=Dℓ/(µ0M2\nsL)(ℓ= 1,2), and\nhext=Hext/Ms.\nTo conclude, we note that for AFM and FiM materials, differently from what happens\nfor FM materials, the Zeeman and the magnetostatic energies are usually of limited\nphysical importance, because they depend on the total magnetization of the sample,\nwhich is in general very small.\nDepartment of Mathematics and Statistics, University of Strathclyde, 26 Richmond\nStreet, Glasgow G1 1XH, United Kingdom\nEmail address :hywel.normington@strath.ac.uk\nDepartment of Mathematics, University of Bologna, Piazza di Porta San Donato 5,\n40126 Bologna, Italy\nEmail address :m.ruggeri@unibo.it\n32" }, { "title": "2108.07180v1.Losses_of_Interface_Waves_in_Plasmonic_and_Gyrotropic_Structures.pdf", "content": "1\nLosses of Interface Waves in Plasmonic and Gyrotropic Structures\nA. Schuchinsky\nUniversity of Liverpool, L3 5TQ, Liverpool, UK, a.schuchinsky@liverpool.ac.uk\nAbstract – The loss mechanisms of slow interface waves in the layered resonant media are\nexamined and illustrated by the examples of (i) surface plasmon polaritons in an isotropic plasma\nlayer, (ii) magnetoplasmons in magnetised plasma and (iii) spin waves in ferrimagnetic layers. It\nis shown that losses of all these interface waves grow at the same rate of Im ~ Re3, whereis the\nwavenumber. These abnormal losses are caused by vortices of the power flow of the interface\nwaves near their resonance cut-off. The basic properties of the slow interface waves discussed in\nthe paper are inherent to the waves of hyperbolic type in the layered resonant media.\nI. INTRODUCTION\nSlow electromagnetic waves guided by the layers and interfaces of the resonant plasmonic and ferrimagnetic\nstructures represent a distinct class of the surface waves. These waves are of hyperbolic type and exist only in the\nfinite frequency bands, being resonantly absorbed at their upper frequency cut-offs. Their main properties are\ndiscussed in this paper by the examples of interface waves (IWs) such as surface plasmon polaritons [1]-[7],\nmagnetoplasmons [8]- [13] and spin waves [14]-[17]. The slow bulk waves (BWs) like magnetoplasmons and spin\nwaves exist when the magnetic bias has components directed along the wave propagation or normal to the guiding\ninterface. The mechanisms of the BW propagation, dissipation and power flow are somewhat similar to the IWs\nand they are not discussed in detail here.\nThe properties, functionality and applications of the slow IWs and BWs in the resonance media have been\nextensively studied in the literature, see e.g., [1]-[17] and references therein. In contrast to the conventional surface\nwaves guided by dielectric layers, IWs and BWs are slower than the plane waves in the constituent media of the\nguiding structure. Therefore, these waves of the hyperbolic type cannot be described by a basic superposition of\nplane waves. The waves in the hyperbolic metamaterials have recently attracted increased attention [18]-[26].\nTheir various applications have been proposed and explored in the literature [19], [20], [23]-[26], including the\npromise of enhancing the sub-wavelength resolution [19], [20] and realising “slow light” [27], [28]. However, the\npublished practical demonstrators exhibited high losses, which were notably higher than in the conventional\ndevices based on dielectric waveguides and resonators, and optical fibres. Therefore, the detailed analysis of the\nloss mechanisms of the slow IWs and BWs in imperfect hyperbolic media is essential for their practical use.\nThe effects of the medium losses on the properties of IWs and their Poynting vector are examined and quantified\nin this work. The properties of the IW propagation and dissipation are elucidated with the examples of the waves\nguided by the interfaces of dielectric layers with isotropic and magnetised plasma and ferrimagnetic layers. The\npaper scope includes the analysis of\n-the basic modes of the slow IWs,\n-the dispersion and attenuation characteristics of the IWs, and\n-the effect of the power flow vorticity on the IW propagation and resonance losses.\nThe main properties of the IWs are discussed by the three examples:\n(i) Surface plasmon polaritons (SPPs) in isotropic plasma layer,\n(ii) Magnetoplasmons (MPs) in tangentially magnetised plasma layer and\n(iii) Spin waves (SWs) in tangentially magnetised ferrimagnetic layer.\nThe results of this work demonstrate that the anomalous losses are inherent to the IWs. In contrast to the\nconventional surface waves, the IWs are of the hyperbolic type, and their attenuation constants are proportional to\nacube of their propagation constants. This is why the IWs exhibit strong attenuation in the proximities of their\nhigh frequency resonance cut-offs. It is shown that both the slow propagation and the high losses of the IWs are\nintrinsically linked to vorticity of the power flow of the IWs in the hyperbolic medium, and the examples of the\nthree types of IWs illustrate this effect.\nThe paper is organised as follows. The canonical 3-layer structure, used for the analysis of the SPPs, MPs and\nSWs, is described in Section II. SPPs in thin metallic layers are discussed in Section III. MPs in the tangentially\nmagnetised plasma layer are considered in Section IV, and spin waves in Section V. The main properties of the\nIWs, their dissipation and power flow are summarised in Conclusion.2\nII.CANONICAL STRUCTURE\nThe basic 3-layer planar structure shown in Fig. 1 is used for the\nstudy of IWs bound to the central layer of thickness a0=2ac. Two\nisotropic dielectric layers have relative permittivities 1,2and\nthicknesses a1,2. The whole structure is bounded by the perfect electric\nconductor1(PEC) walls located at y=(a1+ac), -(a2+ac). The central\nlayers are of the following types\n(i)an isotropic plasmonic layer with Drude scalar permittivity p(),\n(ii)a gyrotropic plasma slab, magnetised along the x-axis and\ndescribed by V oigt permittivity tensor m(),\n(iii)a ferrimagnetic layer, magnetised along the x-axis and described\nby a scalar permittivity gandPolder permeability tensor g().\nThese types of the central layers have negative effective permittivity or permeability in the finite frequency\nbands limited by the intrinsic resonances of the medium. They support propagation of the IWs of hyperbolic type,\nwhich have the high frequency resonance cut-offs. In the proximities of the cut-off frequencies, losses of the IWs\nrapidly grow in the resonant manner, and the attenuation constants of the IWs are proportional to a cube of the\npropagation constants. It is shown below that the high losses of the IWs is their inherent property related to vorticity\nof their Poynting vector. The main mechanisms of losses and power flow of the IWs are elucidated by the examples\nof SPPs, MPs and SWs in isotropic and gyrotropic (magnetised plasma and ferrimagnetic) layers.\nIII.SURFACE PLASMON POLARITONS IN PLASMONIC LAYER\nLet us consider the 3-layer structure shown in Fig. 1 where the central layer is the isotropic plasma with Drude\npermittivity p() defined as\n2\n1p\np L(1)\nwhere=-j,is angular frequency, pandare the plasma and collision frequencies, respectively, and L\nis the background permittivity.\nEigenwaves in a planar structure shown in Fig. 1 include TE and TM waves [29] .TE waves with the field\ncomponents Ex,HyandHzare the ordinary surface waves. They are the guided modes of the plasmonic layer only\nat Rep() > max(1,2) and layer thickness a0about a half wavelength . For thin layers, these conditions are\nfulfilled only at very high frequencies >>p. Therefore, TE waves are not considered here.\nTM waves with the field components Hx,EyandEzare the extraordinary modes of the plasmonic layer. They\nhave been extensively studied in the literature, see, e.g., [2]-[7], [10] and references therein. The dispersion\nequation (DE) of TM waves with the wave propagator exp{ j(t-z} is readily obtained by enforcing the boundary\nconditions of the tangential field continuity at the layer interfaces and at PEC enclosure. The DE can be presented\nas follows\n2\n1 2\n00\nsinhp\npK K\na\n \n \n (2)\nwhere 2 2 2 2\n0 0 0 coth , tanh , , 1, 2; ;m\nm p m p p m m m m m p p\nmK V a V a k m k \nandk0are longitudinal and free space wavenumbers, respectively. The main features of the TM wave dispersion\nand attenuation are illustrated in Fig. 2, obtained by numerical solution of (2).\nSpectrum of the fundamental TM modes in the plasmonic layer includes the conventional surface waves and\nSPPs. The surface waves are the bound modes only at >p, see Fig. 2, when the plasmonic layer acts as a\ndispersive dielectric waveguide with Re p() > max(1,2).The propagation constants of these surface waves vary\nin the range 1 2 0 max , Re Rep L k . At0 > 1, SPPs are forward waves, and their power flows in the dielectric layers is greater\nthan in the plasmonic layer, as evident in Fig. 3. When the dielectric layer at the guiding interface becomes thinner,\nSPP in the lossless plasmonic layer turns into a backward wave SPP' 3shown in Fig. 2. However, losses\nqualitatively alter the eigenwave properties in the proximity of SPP resonance. At the result, the backward wave\nSPP' 3turns into a complex wave SPP 3, which is very strongly attenuated at >r2due to its high losses in the\nplasmonic layer. This effect is illustrated by Fig. 2 , where the attenuation constant of backward SPP 3exceeds the\npropagation constant at >r2.\nAs frequency approaches the SPP resonances at r1andr2, the SPPs slow down. Their Re and Imgrow\nand vorticity of the power flow increases. At the cut-off frequencies, the oppositely directed power flows become\nequal in the plasmonic and dielectric layers, and vortices of Poynting vectors are trapped. Thus, vorticity the\nPoynting vector at the layer interfaces causes the anomalous losses of SPPs at their resonance cut-off.\nFig. 3. Normalized cross-sectional distributions of the SPP fields Hx,Ey,Ez, and Poynting vector Pz,Py\ncomponents in gold film located at | y| max(1,2) only. This\ncondition is satisfied at frequencies 1u<2u>qu, where 2 2\n1 2 2u p c c p ,\n2 2\nqu p c and 2 2\n2 2 2u p c c . At<1uandqu<<2u, Ree() is negative, and only\nMPs are guided by the magnetised plasma layer.\nNonreciprocity is the distinctive feature of MPs. In DE (9) it is described by the last term of Lm() dependent on\na. In the case of identical dielectric layers, V1=V2, wavenumbers of the oppositely directed waves are the same.\nBut the field distributions differ due to the nonreciprocal field displacement dependent on sign . MPs exist in the\ntwo frequency bands and have the resonance cut-offs at the upper bound of each band. In these two bands, MPs\ntravelling in the same direction are guided by the opposite interfaces of the magnetised plasma layer due to the\nopposite signs of Re{ at[12]. For example, if a MP is attached to the upper interface of the plasma layer\nshown in Fig. 1 at <1u, MP of the same direction is displaced to the lower interface in the frequency band\nqu<<2u.\nAs frequency approaches the MP resonances, 1uor2u, the MP wavenumbers grow similar to those of SPPs.\nAt 1,2 0 | | ma x ,L k , the asymptotic solution of DE (9) has the same form as (4), where pmust be replaced\nby 1m c\nm t a . Then the propagation and attenuation constants of MPs are approximated as follows\n\n\n2\n0\n0 2\n2\n30\n2 2\n0Re 12\n2Im\nIm Re\n1c\nm m\nm m c\nm m m\nc\nm\nm mc\nmm mkk O\nkO\nk \n\n\n \n \n\n (10)6\nwherec\nmdepends on frequency and2Re\n, 1,2c c\nm m m\nm c c\nm m mm\n\n\n \n. It is important to note that i n the proximity\nof the MP resonances\n2\n21 1\n1L\nm m\nm cO andmare small at<<±c, similar to SPPs.\nThe dispersion characteristics of MPs in the proximities of their resonances are shown in Fig. 4 . They illustrate\ntwo cases permittivities of dielectric layers are (a) the same, 1=2= 4.7, and (b) different, 1= 4.7,2= 2.4 . The\npropagation and attenuation of MPs in Fig. 4 demonstrate that in the proximities of the resonance cut-offs,\nattenuation constants Immgrow much faster than the propagation constants Remas predicted by (10). It is\nnoteworthy that permittivity of a dielectric layer at the guiding interface influences the cut-off frequencies of MPs,\nand they slightly increase at lower 1,2. At frequencies above the MP resonances, the dispersion curves of TM\nwaves are in the shaded areas at the left from the dotted lines where only the fast waveguide modes of the dielectric\nlayers exist. These waves are guided by the dielectric layers and are not the bound to the plasma layer.\nThe fact that MP losses in (10) grow at the same rate as SPPs in (4), i.e., Imm~ (Rem)3,suggests that dissipation\nof MPs is also the result of their power flow vorticity .ThePoynting vector distribution of MPs in (11) shows that\nit differs from that for SPPs by the dependence on a/t\nFig. 4. Asymptotic dispersion characteristics of magnetoplasmons in the three-layer structure\nof Fig. 1. Propagation constants (solid lines) and attenuation constants (dashed lines) in the\ncases of dielectric layers with the same permittivities, 1=2= 4.7 (red and blue lines) and\ndifferent permittivities with 1= 4.7,\n2= 2.4 (red and green lines). In the shaded areas to the\nleft from dotted lines, where no interface waves are guided by the\nplasmonic layer. Parameters of the plasmonic layer: L= 13.1,p= 2.17c,= 0.02c.\nRe/k0, Im/k0/c\n0 2 4 6\n3\n2\n11u/c2u/c\nqu/c\n1= 4.7\n2= 4.7\n*2= 2.47\n 2\n1\n2 0\n2 0cosh 1\n1 1 , , 1 , 1,2cosh\nRe\ncosh sinh1 , , , ,cosh sinhm\nm m c m ma\ne e c\nm t m m\nz\np p a a a\ne e e e c c\ne t t p c t p ca a y\nW y a ma\nP y Qk\ny yS y W W y aa a \n \n \n\n\n\n\n(11)\nwhere0,k0,Q,e() are defined in (5), (9), \ntanh tanh\n,\ntanh tanhp c p\np\np c pW a y\nS y W\na W y \n\n \n\n, and nonreciprocity of\nthe MP power flow is described by the terms dependent on We(e,a/t) defined in (6).\nPoynting vector distribution Pz(y) in (11) shows that at Re e() < 0, power flows of MPs inside and outside\nplasma layer are counter-directional, similar to SPPs in the isotropic plasma layer. Therefore, vorticity of the power\nflow is the main propagation mechanisms of the MPs responsible for their anomalous losses. It is necessary to note\nthat despite the similarities in the power flows of SPPs and MPs, Poynting vector distribution is more intricate due\nto gyrotropy of the magnetised plasma layer [31], [32].\nNonreciprocity is the distinctive property of MPs. It manifests itself not only in asymmetry of the field and\npower flow distributions but also in the nonreciprocity of the cut-off frequencies. Therefore, the propagation and\nattenuation constants of MPs, displaced to the opposite interfaces of the magnetised plasma layer, differ when the\ndielectric layers are not the same. It is necessary to note that vorticity of the power flow is stronger in MPs than in\nSPPs due to the effect of the nonreciprocal field displacement. As the result, losses of MPs are higher than SPPs,\ndespite the cubic relation between their attenuation and propagation constants remains the same.\nV.SPINWAVES IN FERRIMAGNETIC LAYERS\nLet us consider a ferrimagnetic layer located in the middle of a planar structure shown in Fig. 1. It is magnetised\nto saturation along the x-axis (V oigt configuration) and characterised by a scalar relative permittivity gand Polder\ntensor of relative permeability g[33]\ng t a j μxxIxx xI (12)\nwhere2 2\n2 2 2 2,M\nt a\nH H ,0 , 4 , ,H M s B H M H B H j H M ,H0\nis internal DC magnetic bias, His the ferrimagnetic resonance linewidth, 4 Msis the saturation magnetization,\nand the gyromagnetic ratio = 2.8 MHz/Oe.\nSpectrum of eigenwaves in the ferrimagnetic layer with the tensor permeability gincludes TE and TM waves.\nTM waves with the field components Hx,Ey,Ezare the ordinary waves, which are not affected by the layer\ngyrotropy. Therefore, they are not considered here. TE waves with the field components Ex,Hy,Hzare the\nextraordinary waves, which strongly interact with the ferrimagnetic medium described by the tensor permeability\ng. The spectrum of TE waves includes the dynamic waves and surface spin waves (SSWs). Properties of the TE\nwaves have been extensively explored in the literature and are used in the nonreciprocal devices [14]-[17], [34],\n[35]. This is particularly concerned of the dynamic waves, which are the workhorse of the passive ferrite devices\nsuch as circulators, isolators and phase shifters, etc., see e.g., [36]-[40]. Applications of spin waves has been limited\nby their high losses. Despite significant efforts in mitigating SW losses, they remain high. It is shown below that\nthe attenuation constants of SSWs are proportional to a cube of the propagation constant due to vorticity of the\npower flow, similar to SPPs and MPs.\nLet us consider the TE waves with the propagator exp{j(t-z)}. Their DE can be presented in the form similar\nto the DE for SPPs and MPs\n2\n1 2\n00\nsinhg\ngM M\na\n\n \n \n (13)\nwhere 2 2\n0 0 coth 1 , coth , , 1,2;m a\nm e m g g m m m m m m\ntM U a U a k m \n2 2 2\n0 , ;g g e e t a tk k0andare the free space and the longitudinal wavenumbers.\nThe solutions of DE (13) include the dynamic waves and SSWs. The dynamic waves are the conventional\nsurface waves, which exist only at t·e> 0, i.e., at frequencies B, whereH= ReHand8\nB=ReBare the ferrimagnetic resonance and plasma frequencies [33]. In contrast to the dynamic waves, SSWs\nare the IWs. They exist in the finite frequency band H<0 experience the high frequency\ncut-off at the SSW resonance sw=(H+B)/2. Similar to MPs, SSWs are the nonreciprocal waves. Their\nnonreciprocity is described by the last term in Mm() proportional to a/t.In the case of identical dielectric\nlayers, U1=U2and wavenumbers of the oppositely directed SSWs are the same . But their field distributions differ\ndue to nonreciprocity of their field displacement, which depends on the propagation direction, i.e., the sign .\nExamples of the dispersion characteristics and field patterns of SSWs in the planar structure of Fig. 1 are shown\ninFig. 5 at several thicknesses of the dielectric layers.\nThe properties of SSWs in the presence of magnetic losses were studied in detail in [15]-[17], and it was found\nthat a t frequencies close tosw, the wavenumbers of the SSWs grow similar to those of SPPs and MPs. The\nasymptotic solutions of DE (13) at0g k can be also represented in the form of (10) wherec\nmandmare\nreplaced bycr t a and2Re\n1cr cr\ncr cr, respectively. Then the asymptotic expansions of the SSW\npropagation and attenuation constants are approximated as follows\n\n\n2\n0\n0 2\n2\n3 0\n2 2\n0Re 11\nImIm Re\n2 1g cr\ncr\ng cr\ncrkk O\nkO\nk \n \n \n \n\n m=1, 2 (14)\nwherecrandare frequency dependent. Then at the SSW resonance frequency,2\n241H H\nM MO and\n<< 1 atH< τc\n100 μm\nFigure 3: Critical pulse-duration threshold \u001ccmeasured as a function of composition for Gd x(FeCo) 100\u0000xalloys. HI-AOS\nis achieved if the pulse duration \u001csatis\fes the condition \u001c < \u001c c, but HI-AOS fails if \u001c > \u001c c. The maximum achievable \u001c\nwas 6 ps in these experiments, implying that \u001cc>6 ps forx\u001526. Insets: typical background-corrected magneto-optical\nimages obtained for Gd 23(FeCo) 77indicative of HI-AOS (bottom-right inset, \u001c= 1:4 ps) and demagnetization (top-left inset,\n\u001c= 1:5 ps). Adapted with permission from Ref. [42].\nThe variation of alloy composition in Gd x(FeCo) 100\u0000xis qualitatively similar in impact to considering\na \fxed alloy composition at di\u000berent equilibrium temperatures T0[44]. At a starting temperature T=T0,\nthe constituent sublattices of GdFeCo have angular momenta SGdjT0andSFejT0, hereafter labelled SGd,0\nandSFe,0respectively. Varying either xorT0leads to a change in SGd,0 andSFe,0, naturally leading to\nthe question of how the starting temperature T0in\ruences HI-AOS. In the case of GdFeCo, Davies et\nal.showed that increasing T0results in a monotonic decrease of \u001cc[27]. Importantly, this measurement\nshowed that HI-AOS can be achieved at temperatures below and far above the compensation point Tcomp\ni.e. the temperature at which the net magnetization and angular momentum is zero. In a similar manner,\nthe threshold \u001ccfor HI-AOS displayed by two samples of MRG decreases with T0(Fig. 4) [27]. On the\ncontrary, the HI-AOS tested in a total of \ffteen MRG samples appears to always be constrained to starting\ntemperatures below Tcomp [26, 27].\nThe energy required for activating HI-AOS represents another important consideration. Laser-delivered\n40 100\nT0 - Tcomp (K)012345\n-100 20 40 -60 80 -20 -40 60 -80τc (ps)Gd25(FeCo)75\nMn2Ru0.80GaMn2Ru0.75GaFigure 4: Threshold pulse duration \u001ccfor one GdFeCo and two MRG \flms measured as a function of the di\u000berence be-\ntween the compensation temperature Tcomp and the measurement base temperature T0. The compensation temperatures of\nGd25(FeCo) 75, Mn 2Ru0:8Ga, and Mn 2Ru0:75Ga are 320 K, 345 K, and 370 K, respectively. Adapted with permission from\nRef. [27].\noptical pulses are almost always Gaussian in spatial distribution of energy, leading to a Gaussian distribution\nof heating at the focus. The non-uniform heating of the illuminated spot results in substantially more energy\nbeing deposited at the center of the spot compared to that deposited at the outer perimeter. This result\nis very useful for distinguishing the e\u000bects of pulse duration and energy. If the pulse duration is su\u000ecient\nfor switching but the absorbed \ruence is excessive, the pulse generates a ring of switched magnetization\nencircling a spot of demagnetization [27, 35, 41]. At the same time, the Gaussian distribution conveniently\nallows absorbed-\ruence thresholds associated with HI-AOS to be identi\fed, with a minimum absorbed\n\ruenceFcbeing necessary for HI-AOS. Measurements of Fcfor GdFeCo range from 0.75 to 3 :14 mJ=cm2[40],\nwithFc= 0:82 mJ=cm2[41] for a Gd 24(FeCo) 76sample using \u001c= 55 fs-long pulses at room temperature.\nAs expected, the starting temperature, sample composition and pulse duration all in\ruence Fc, but it is\nquite remarkable that increasing the pulse duration up to 15 ps for Gd 26(FeCo) 74only increases Fcto\n\u00191:5 mJ=cm2[41]. Measurements for HI-AOS in MRG alloys have shown that Fccan be minimized by\nreducing\u001cand bringing T0below but increasingly closer to Tcomp [27].\nThe \fnal parameter of the stimulus we consider is the type of excitation. Yang et al. were the \frst\nto show experimentally that laser pulses with high photon energy are not compulsory for switching, with\nelectronic pulses generated by an Auston switch being equally su\u000ecient [45]. This represented a technological\nbreakthrough since it can be argued that picosecond-long electrical pulses can already be excited in integrated\ncircuits [12]. Further evidence of the indi\u000berence of HI-AOS to the type of excitation was provided by Davies\net al. , who showed that single optical pulses with photon energies ranging from 1 :55 eV to as low as 50 meV\nare similarly capable of driving HI-AOS in both GdFeCo and MRG [27, 42]. While HI-AOS clearly only\nrequires a gentle excitation of the electronic population just above the Fermi level, the latter measurements\ncon\frmed that such an excitation must still be shorter in duration than \u001cc.\n3. Phenomenological description of AOS\nIn this section, we present a general theoretical framework describing ultrafast spin dynamics in multi-\nsublattice magnets [10, 46, 47]. This framework, containing longitudinal relaxation terms of both exchange\nand relativistic origin, accounts for how angular momentum \rows from one sublattice to another with con-\nservation of the total angular momentum, and with accelerated and independent \row of angular momentum\nfrom each sublattice to an external bath (i.e. the lattice). We argue here that this model, while being\nphenomenological and somewhat simple in approach, can be used to qualitatively understand how HI-AOS\ncan both be achieved and constrained by the conditions described in Section 2.3.\n53.1. Equations of motion for longitudinal spin dynamics\nThe basis of our theoretical framework is the description of spin dynamics for a two-sublattice magnetic\nstructure provided by Onsager's relations [48]. Iwata [49, 50] and Baryakhtar [51, 52, 53] independently\ndeveloped this approach in the mid-1980s, showing that Onsager's relations naturally yields dynamics of\nthe macroscopic length of the magnetization. Moreover, when taking into account the symmetry of the\nexchange interaction in multisublattice systems, dynamics of the length of the magnetization belonging to\neach sublattice is possible, even when the total angular momentum is conserved. These dynamics evolve\nacross the timescale relevant to the exchange interaction, where conventional transverse (precessional) dy-\nnamics of the angular momentum can be considered to be negligible. In the limit of considering longitudinal\ndynamics of the macroscopic angular momentum S\u0017belonging to sublattice \u0017, the equations of motion for\ntwo non-equivalent collinear sublattices can be written as\ndSa=dt =\u0015aHa+\u0015e(Ha\u0000Hb); (1)\ndSb=dt =\u0015bHb+\u0015e(Hb\u0000Ha): (2)\nThe angular momentum S\u0017, which can be either positive or negative in sign, is related to the magnetization\nM\u0017by the gyromagnetic ratio \r\u0017viaS\u0017=M\u0017=\r\u0017. The\u0015terms indicate the relative strength of the\nrelaxation pathways. The relativistic transfer of angular momentum between sublattice \u0017=a;band the\nenvironment is described by \u0015\u0017, whereas\u0015eis of exchange origin and stems from spin-spin interactions,\nconserving the total angular momentum by allowing angular momentum to transfer between the sublattices.\nThe e\u000bective magnetic \felds H\u0017acting on each sublattice will be discussed extensively in the following\nSection.\n3.2. Non-equilibrium free energy and e\u000bective magnetic \felds\nThe equations of motion given by Eqs. (1)-(2) assume that the time-dependent perturbation is induced\nby an external magnetic \feld. In the phenomenological theory presented here, the external \feld is replaced\nby an e\u000bective magnetic \feld, with the latter being derived from a free energy taking into account internal\ninteractions. Such an approach is very useful since it does not alter the structure of the equations of motion\nthemselves.\nLandau and Lifshitz were the \frst to introduce the concept of an e\u000bective magnetic \feld [54], de\fning\nit as the functional derivative of the macroscopic free energy H\u0017=\u0000\u000eF=\u000eS\u0017, whereFis the free energy\nandS\u0017=hP\nis\u0017\nii=Vis the operator of the total angular momentum per unit volume Vof sublattice \u0017.\nAiming to derive the macroscopic free energy from a microscopic Hamiltonian H, we use for simplicity the\nHeisenberg spin model, which can be written for 2 sublattices ( \u0017=a;b) in the form\nH=\u0000X\ni2a\ni02a0\nJaa\nii0sa\nisa\ni0\u0000X\ni2b\ni02b0\nJbb\nii0sb\nisb\ni0\u0000X\ni2a\ni02b0\nJab\nii0sa\nisb\ni0\u0000X\ni2b\ni02a0\nJab\nii0sb\nisa\ni0; (3)\nHere,J\u0017\u00170\nii0are exchange parameters andP0\ni;i0=P\niP\ni06=iindicates a double summation where each sum\nruns over a whole sublattice. In the following, and similar in approach to atomistic spin dynamics, we treat\nthe spinss\u0017\nias classical, but we note that calculations for quantum spins also proceed in a similar way [55].\nTo calculate the relationship between Hand the free energy F, it is possible in principle to use the\nfundamental relation F(H) =hHi\u0000TSwhere Sis the entropy and Tthe temperature of the medium in\nwhich the spin system is embedded. While this relationship can be used both in and out of equilibrium, actual\ncalculation is di\u000ecult, and so we limit ourselves to the mean \feld approximation S\u0017=N\u0017hs\u0017\niiwhereN\u0017is\nthe number of spins of sublattice \u0017per unit volume. We can ensure that the the mean-\feld approximation\nofFis still valid when considering non-equilibrium scenarios by performing statistical perturbation theory.\nWritingH=H0+ (H\u0000H0), we obtain (to \frst order in H\u0000H0)\nF\u0014F(H0) +hH\u0000H0i0\u0011\b; (4)\n6wherehxi0=hxexp(\u0000\fH0)i=hexp(\u0000\fH0)iindicates averaging over the equilibrium distribution function of\nthe trial Hamiltonian H0. In the classical and quantum case, this inequality is named Gibbs and Bogoliubov\nrespectively [56].\nTo arrive at the mean-\feld approximation, we choose the simple form\nH0=\u0000X\ni2ahasa\ni\u0000X\ni2bhbsb\ni; (5)\nwhereh\u0017are variational parameters. Direct calculation gives\nF(H0) =\u0000\f\u00001[NalnZa+NblnZb]; (6)\nhH0i0=\u0000@\n@\f(NalnZa+NblnZb) =\u0000Nahasa\u0000Nahbsb; (7)\nand\nhHi0=\u0000NazaaJaas2\na\u0000NbzbbJbbs2\nb\u00002NpJabsasb; (8)\nwith\ns\u0017=hs\u0017\nii0=@lnZ\u0017\n@\fh\u0017=\u001b\u0017L(\fh\u0017\u001b\u0017): (9)\nHere, we use the Langevin function L(x) = coth(x)\u00001=xand the single-spin partition function Z\u0017=\n4\u0019sinh(\fh\u0017\u001b\u0017)=(\fh\u0017\u001b\u0017), with\u001b\u0017=js\u0017\nijbeing the length of the local spin moment. Np=Nazab=Nbzba\nrepresents the number of pairs and z\u0017\u00170the number of neighbors in sublattice \u00170of a spin in sublattice\n\u0017. Eq. (9) is a one-to-one relation between the mean-\feld spin moment per site s\u0017and the variational\nparameterh\u0017. Hence, we can consider h\u0017to be an explicit function of s\u0017, and thereby write the mean-\feld\nfree energy both in and out of equilibrium in the form\n\b(sa;sb) =Nafa(sa) +Nbfb(sb)\u00002NpJabsasb; (10)\nwith\nf\u0017(s\u0017) =\u0000\u0002\n\f\u00001(lnZ\u0017\u0000\u0011\u0017s\u0017=\u001b\u0017) +z\u0017\u0017J\u0017\u0017s2\n\u0017\u0003\n; (11)\nwhere\u0011\u0017=\fh\u0017\u001b\u0017=L\u00001(s\u0017=\u001b\u0017). Equations (10)-(11) provide the link between the microscopic spin\nHamiltonian and the macroscopic non-equilibrium free energy in the mean-\feld approximation.\nWe are now in a position to explicitly derive the e\u000bective \felds from the free energy. From Eq. (9), we\n\fnd that@\n@s\u0017(lnZ\u0017\u0000\u0011\u0017s\u0017=\u001b\u0017) =@lnZ\u0017\n@\u0011\u0017@\u0011\u0017\n@s\u0017\u0000@\u0011\u0017\n@s\u0017s\u0017\n\u001b\u0017\u0000\u0011\u0017=\u001b\u0017=\u0000\u0011\u0017=\u001b\u0017; (12)\nsuch that the e\u000bective \felds H\u0017=\u00001\nN\u0017@\b\n@s\u0017become\nHa=\u0000\f\u00001\u0011a=\u001ba+ 2zaaJaasa+ 2zabJabsb; (13)\nHb=\u0000\f\u00001\u0011b=\u001bb+ 2zbbJbbsb+ 2zbaJabsa; (14)\nIn equilibrium, and with the assumption that no external \felds are present, the e\u000bective \felds vanish.\nWe thus obtain the special values h\u0017=\f\u00001\u0011\u0017=\u001b\u0017given by\nha= 2zaaJaasa+ 2zabJabsb; (15)\nhb= 2zbbJbbsb+ 2zbaJabsa; (16)\nwhere the equilibrium values s\u0017can be determined by the self-consistent solution of the coupled set of\nequations\ns\u0017=\u001b\u0017L(\f\u001b\u0017h\u0017): (17)\nThe same result is obtained in the usual equilibrium mean-\feld theory with the quantities h\u0017being inter-\npretable as Weiss \felds. Such equilibrium mean-\feld theory is often used in the derivation of e\u000bective \felds\nentering precessional dynamics. However, as long as the magnetic sublattices are not yet in equilibrium\nwith each other, h\u00176=h\u0017. It is therefore mandatory to use h\u0017in the e\u000bective \felds to obtain the correct\nrelaxation to equilibrium.\n73.3. Classi\fcation of dynamics\nEquations (13)-(14), in combination with Eqs. (1)-(2) form a closed set of equations which provide the\nbasis for phenomenologically describing the non-equilibrium dynamics of the longitudinal angular momenta\nof multi-sublattice magnets. Generally, the exchange energies f\u0017andJabinFare parametrically dependent\non the temperature of the environment. Hence, the aforementioned set of equations can be used to simulate\nthe dynamics in response to heat pulses. Such numerical simulations were employed in Ref. [42] and it was\nfound that the model encompasses the counterintuitive switching of magnetic sublattices in various regimes,\nfeaturing all known results from much more computationally-demanding atomistic simulations [9, 57]. Here,\nwe use this result to classify the dynamics in two distinct regimes [10, 42]. For this purpose, it is convenient\nto analyze the dynamics by expanding the free energy to leading order in s2\n\u0017\u0000s2\n\u0017, which recovers the familiar\nphenomenological Landau expressions [46].\nThe \frst regime is de\fned for T\u001dTC, which we call the temperature-dominated regime. Such a scenario\ncan be realized by suddenly heating the electron system using a fs laser pulse. On the timescale of 10-100 fs,\nthe electronic temperature, and hence the value of the \\temperature\" that enters the expression for the\nnonequilibrium free energy, far exceeds the equilibrium Curie temperature [7, 39]. The system thus behaves\nacross this timescale like a paramagnet, and so we can write f\u0017=S2\n\u0017=(2\u001f\u0017) where\u001f\u0017\u00181=Tdenotes the\nlongitudinal susceptibility of sublattice \u0017. Since, in the paramagnetic regime, Jab\u001ckBT, the inter-sublattice\ninteraction can be neglected, and the independent transfer of angular momentum from each sublattice to\nthe environment dominates the dynamics. Consequently, the sublattices exhibit decoupled Bloch relaxation\nwith a characteristic longitudinal relaxation time given by \u001c\u0017=\u001f\u0017=\u0015\u0017. Microscopic calculations [58, 59, 60]\nshow that this is equivalent to\n\u001c\u0017=\u001b\u0017=(2\u000b\u0017kBT); (18)\nwhere\u000b\u0017is a microscopic parameter of relativistic origin determining the coupling to the heat bath. Im-\nportantly, Eq. (18) shows that the atomic spin moment \u001b\u0017controls the speed of the longitudinal relaxation,\nwith smaller magnetic moments relaxing faster. This \fnding is in accordance with the result shown in Fig. 1,\nsince the magnetic moment of iron is roughly four times smaller than that of gadolinium [44].\nThe second regime, hereafter-referred to as the exchange-dominated regime, is relevant in non-equilibrium\nscenarios where T < T C, typically appearing on the picosecond timescale. Exchange interactions in an\nordered system are generally stronger than relativistic interactions and so the exchange-dominated regime is\ncharacterized by \u0015\u0017\u001c\u0015e. In this regime, the transfer of angular momentum from one sublattice to the other\ndominates, with the conservation of total angular momentum Sa+Sb=constant in the limit of dynamics\ndriven entirely by exchange-relaxation. As a consequence, for any form of the free energy F, the changes of\nthe sublattice-speci\fc angular momentum sum up (approximately) to zero, leading to dSa=dt=\u0000dSb=dt.\nThis yields highly counter-intuitive dynamics when the spin of one of the sublattices is close to zero, whereby\ndS\u0017=dtremains \fnite even when S\u0017= 0. This represents the only pathway by which the coupled spins can\ncross zero and reverse sign.\nTo illustrate the e\u000bect of exchange-dominated dynamics schematically in the simplest possible way i.e.,\nwithout considering speci\fc pulse and system parameters, we can solve Eqns. (1)-(2) at \fxed temperature\nof the heat bath. For a typical rare-earth ( \u0017=a) transition-metal ( \u0017=b) ferrimagnet, Jaais substantially\nsmaller than Jbb, and so one can model the sublattices's free energies in the form of fa=AS2\na=2 and\nfb=B(S2\nb\u0000S2\nb)2=4. Using exemplary values A=B = 0:4,B= 1 =Sb,Jab=B=\u00000:15,\u0015a=\u0015b= 0:15\nand\u0015e= 1, Mentink et al. calculated the results shown in Fig. 5 [10]. This phase diagram conveniently\nvisualizes the dynamics of Saas a function of Sbfor various initial conditions in one graph. In this phase\nplane, a pure form of exchange-relaxation appears as trajectories inclined at -45\u000erelative to the horizontal\naxis (parallel to the dashed-dotted purple line), ful\flling dSa=dt=\u0000dSb=dt.\nIn general, the phase diagram shown in Fig. 5 shows the trajectory of longitudinal angular-momenta\ndynamics one would obtain for a ferrimagnetic system that is brought out of equilibrium. The starting\nsituation of the ferrimagnet in stable equilibrium is indicated by the green spheres in the upper-left or\nlower-right quadrants i.e., the quadrants in which SaandSbhave opposite sign. Following excitation, the\nferrimagnet is brought out of equilibrium, bringing the system to a di\u000berent \\starting\" coordinate ( Sb,Sa) on\nthe phase diagram. The solid lines with arrows show the ferrimagnet's direction of subsequent relaxation to\n8Sb / Sb,00 0.5 1 -0.5 -100.51\n-0.5\n-1\nSa / Sb,0\nFigure 5: Numerical solution of the longitudinal equations of motion in the exchange-dominated regime. The evolution of Sais\nshown as function of Sbfor various initial conditions, where both are normalized to the equilibrium value of angular momentum\nSb;0. The green spheres indicate stable equilibrium points and the arrows indicate the direction of relaxation with increasing\ntime. The origin is a saddle point, and the dashed blue line indicates a stable manifold of solutions whereas the dotted-dashed\npurple line corresponds to the condition Sa+Sb= 0. The blue shaded area encompasses the initial conditions from which the\nlongitudinal relaxation will proceed to reversal via temporal ferromagnetic alignment. Adapted with permission from Ref. [10].\nequilibrium. For a system starting in the top-left quadrant, the unshaded majority of the phase space leads\nto a return back to its original state. If, however, the excitation substantially and su\u000eciently demagnetizes\nSbwithjSbj0,2@fa\n@S2a>\u0000Jab(1 +\u0015b=\u0015e)>0: (19)\nSubstitution of the Landau form of fagiven above in Eq. (19) yields \u0015b<\u0015e(A=jJabj\u00001). Since by de\fnition\n\u0015b\u00150, we \fnd that reversal is thus only possible when exchange relaxation is included ( \u0015e>0).\nThe phase diagram shown in Fig. 5 represents a powerful tool for understanding how exactly HI-AOS can\narise. In principle, any excitation which brings the angular momenta of the two sublattices from the stable\nequilibrium point to the blue shaded manifold will result in HI-AOS. We emphasize here that this result\nshows that the transient ferromagnetic state is notthe critical prerequisite for the switching, but rather\nit is the strongly demagnetized state indicated by the blue shaded manifold, in which jSbj< SaandSbis\nsubstantially demagnetized. By considering di\u000berent system parameters, such as inter- and intra-sublattice\nexchange couplings, starting temperature and relaxation constants, the blue manifold shown in Fig. 5 can\nwiden, shrink and even rotate. We refer the reader to Ref. [46] for examples of how such behavior can be\nrealized.\n3.4. Phenomenological framework of HI-AOS in GdFeCo and MRG\nThe main result of Section 3.3 is that HI-AOS can only be achieved by propelling the ferrimagnetic system\nin to the strongly non-equilibrium state shaded in blue in Fig. 5. The question that we aim to answer, in\nthis Section, relates to how we can drive entry in to this non-equilibrium state. We emphasize that we do\n9not aim to use the phenomenological model to quantitatively reproduce experimental measurements, since\nthis would involve \ftting many parameters that would not bring much further insight. A phenomenological\napproach rather o\u000bers substantial predictive powers due to its lack of material-speci\fc assumptions and\ncomputational overheads. Indeed, we \fnd that our phenomenological approach qualitatively explains the\nunique kinetics of HI-AOS and the origins of the experimentally-observed conditions described in Section 2.3.\nTo model GdFeCo alloys in equilibrium, we can solve either Eqs. (15)-(16) self-consistently or Eqs. (13)-\n(14) with Eqs. (1)-(2) in equilibrium. These both yield the equilibrium temperature dependence of the\nangular momentum for each constituent sublattice within the ferrimagnet [61], shown in the inset of Fig. 6.\nIn the spirit of using phase diagrams to explain the process of HI-AOS, we recast in the main panel of\nFig. 6 the equilibrium thermal dependence as a dotted black line. This represents the size of the angular\nmomentum reservoirs SFe(T0) andSGd(T0) at equilibrium. At a given starting temperature, the angular\nmomentum of the ferrimagnet is con\fned to a point on this line (exemplary green spheres labelled [i] and\n[ii]). \\Slow\" variation of the temperature in equilibrium results in the angular momentum moving along\nthis line only, with both sublattices experiencing the same temperature T0.\nLet us now consider the HI-AOS displayed by GdFeCo. As discussed in Section 3.3, the longitudi-\nnal dynamics of angular momentum can be classi\fed in two distinct regimes, either being temperature-\nor exchange-dominated. The \frst scenario corresponds to ultrafast femtosecond heating, which essentially\ndecouples the two sublattices with loss of angular momentum to the environment at a rate inversely propor-\ntional to the sublattice's magnetic moment [Eq. (18)]. Because of the weakness of JGd\u0000Feand the magnetic\nmoment of Fe being four times smaller than that of Gd, the femtosecond pulse initially demagnetizes Fe\nroughly four times faster than Gd. This trajectory is sketched in Fig. 6 as the dashed black line, traversing\nthe phase diagram with a gradient of \u00194:-1. With the iron sublattice demagnetizing more rapidly, SFe\napproaches close to the axis SFe= 0. The trajectory then bends towards the origin ( SFe= 0;SGd= 0)\nsince this relativistic relaxation can only remove angular momentum from the sublattices. With the elec-\ntrons and lattice equilibrating across the timescale \u001ce\u0000l\u00192 ps [62], exchange relaxation begins to dominate\n(solid black line) [63]. The exchange-dominated dynamics, conserving the total angular momentum of the\ntwo sublattices, drives the ferrimagnet in to the ferromagnetic-like state with the Fe sublattice receiving\nangular momentum from the Gd sublattice i.e., dSFe=dt=\u0000dSGd=dt. In this case, the trajectory is given\nbySa+Sb=constant , parallel to the dashed-dotted purple line. The exchange relaxation then continues to\npushSGdacross zero as well, resulting in SGdandSFenow having switched signs. The spins then equilibrate\nwith the lattice across distinct timescales, and subsequent cooling eventually \fnalizes the HI-AOS with the\nferrimagnet's angular momentum residing in the bottom-right quadrant.\nA similar description can be used to explain the HI-AOS achieved by a substantially-stretched excitation,\nwith duration on the order of several picoseconds. In this scenario, the spins do not experience an intense\nsharp change in \\temperature\" that brings about decoupled Bloch relaxation. Instead, the sublattices\nprimarily demagnetize via exchange-relaxation, transferring angular momentum from the Gd sublattice to\nthe Fe one. Conserving the total angular momentum of the system, the trajectory of demagnetization only\nfollows the solid black line across Fig. 6. Provided that the ferrimagnet's starting point is above the dashed-\ndotted purple line e.g. at position [i], such exchange-relaxation is still capable of driving the system in to\nthe strongly non-equilibrium state (blue manifold in Fig. 6) from which HI-AOS emerges. For HI-AOS to\noccur,SFemust cross zero before SGd[32, 42, 64, 57]. This condition becomes evident when considering the\nstate in which both sublattices are considerably demagnetized and are starting to recover. The sublattice\nwith the stronger intra-sublattice exchange interaction will recover faster, and so HI-AOS demands SFeto\ncross zero \frst. It is also important during this process that both SFeandSGddo not simultaneously fall\nto too low a value where thermal spin correlations can dominate. This scenario leads to loss of magnetic\nmemory and subsequent demagnetization, as encountered if the absorbed \ruence is excessive.\nThe two pathways described above indicate the routes through which HI-AOS can be achieved. Equipped\nwith this understanding, we can comprehend how HI-AOS depends on the sample composition x, as shown\nexperimentally and numerically in Fig. 3. Increasing the percentage of Gd in the Gd x(FeCo) 100\u0000xalloy\nrapidly increases the size of the equilibrium angular momentum reservoir SGd, pushing the initial equilibrium\nstate upwards in the top-left quadrant in Fig. 6. Thus, increasingly longer pulses become capable of driving\nthe switching since the system can tolerate increasingly larger deviations from exchange-driven dynamics\n100SFeincreasing x SGd\n0\n600 400 200 0\nT0 (K)SGd,0\n|SFe,0|\nSFe,0increasing T 0\n\"slow\"\nheatingτ > ps's\"Ferromagnetic-like\" stateτ∼fs\nSGd + S\nFe = 0[i]\n[ii]\n234\n-11S0 (ħ)Figure 6: Conceptual phase diagram showing the di\u000berent pathways for thermally-induced relaxation of the sublattice-resolved\nangular momentum SFeandSGdof the ferrimagnetic alloy Gd xFe100\u0000x. The green spheres in the top-left and bottom-right\nquadrants indicate example positions of equilibrium (labelled [i] and [ii]). By varying xor the starting temperature T0, the\nequilibrium states move across the phase diagram, with the dotted line corresponding to the scenario of \\slow\" heating in\nequilibrium. Excitation of the ferrimagnet by a thermal pulse of duration \u001cleads to di\u000berent trajectories of demagnetization,\nwith a femtosecond pulse activating decoupled Bloch relaxation (dashed black line) followed by exchange-relaxation (solid black\nline line), and with a longer pulse activating exchange-relaxation only (solid black line). After both SFeandSGdchange sign,\nHI-AOS \fnalizes via \\slow\" relaxation to the equilibrium state in the bottom-right quadrant. The inset shows the thermal\ndependence of the equilibrium angular momentum SGd,0 andSFe,0(solid lines). To facilitate comparison, we also show jSFe,0j\n(dashed line). Adapted with permission from Ref. [42].\nintroduced by leakage of angular momentum to the lattice. Quantitative modelling of this scenario, using\nEqs. (1)-(2) in combination with Eqs. (13)-(14), was presented by Davies et al. in Ref. [42].\nSimilarly, we can also understand why the compensation temperature is rather insigni\fcant in the process\nof HI-AOS found in GdFeCo. At elevated temperatures T0> T comp, the ferrimagnet's starting point\n(SFe,0,SGd,0) sits below the dashed-dotted purple line (e.g. position [ii] in Fig. 6), and so exchange-driven\ndynamics becomes incapable of driving the angular momenta in to the blue shaded manifold. Decoupled\nBloch relaxation, however, can still meet this condition, thus explaining why only shorter pulses are capable\nof achieving HI-AOS at higher starting temperatures [42] or in alloys of Gd x(FeCo) 100\u0000xwith lower x\n(Fig. 3).\nWe now turn to the case of MRG, with the equilibrium thermal dependence of S4aandS4cgiven in\nthe inset of Fig. 7. In this alloy, the sublattice-speci\fc magnetic moments are similar (ratio \u00191:1.2) and\nare strongly coupled ( J4a\u00004c=\u000055 meV). Thus, decoupled Bloch relaxation at sublattice-speci\fc rates, as\nexpected from a femtosecond excitation, does not substantially dominate on the timescale before electron-\nlattice equilibration ( \u001ce\u0000l<1 ps [65, 66]). The system instead primarily relaxes under non-equilibrium\nvia exchange-driven demagnetization, conserving the total angular momentum [67]. This directly leads to\nthe implication that HI-AOS can only be achieved at starting temperatures below the compensation point,\nwhere the starting point ( S4a,0,S4c,0) is above the dashed-dotted purple line (case [i] in Fig. 7). As T0drops\nfurther below Tcomp,jS4c;0jgrows twice faster than jS4a;0j, permitting increasingly longer pulses to achieve\nHI-AOS since the system can tolerate some loss of angular momentum to the lattice alongside the exchange\n11S4aS4c\nT0 < T comp\n\"slow\"\nheating\"Ferromagnetic-like\" state\nT0 (K)600 400 200 000.511.5\n-0.5 S0 (ħ)\n-1S4c,0\n|S4a,0|\nS4a,0T0 > T comp\nS4a + S\n4c = 0Tcomp\nTcomp[i]\n[ii]Figure 7: Conceptual phase diagram showing the di\u000berent pathways for thermally-induced relaxation of the sublattice-resolved\nangular momenta S4aandS4cof the ferrimagnetic alloy Mn 2RuxGa. The green spheres in the top-left and bottom-right\nquadrants indicate example positions of equilibrium. By varying the starting temperature T0, the equilibrium states move\nacross the phase diagram, lying above or below the compensation point Tcomp whereS4a+S4c= 0. Non-equilibrated excitation\nof the ferrimagnet primarily activates exchange relaxation (solid black line). HI-AOS is activated if S4aswitches sign \frst,\nwhich can only be achieved when T0< Tcomp . If instead T0> Tcomp ,S4cswitches sign \frst and HI-AOS fails. The dotted\nline corresponds to the scenario of \\slow\" heating in equilibrium. The inset shows the thermal dependence of the equilibrium\nangular momentum S4c,0andS4a,0(solid lines). To facilitate comparison, we also show jS4a,0j(dashed line). Adapted with\npermission from Ref. [27].\nrelaxation. If the starting temperature instead shifts just below Tcomp, only a femtosecond pulse can realise\nHI-AOS since the system must conserve all angular momentum in order to access the appropriate non-\nequilibrium state (blue manifold). At starting temperatures above Tcomp (case [ii] in Fig. 7), however, pure\nexchange-driven dynamics becomes incapable of driving the system in to the prerequisite non-equilibrium\nstate required for HI-AOS. This qualitatively explains why MRG displays HI-AOS only when T0< T comp\n(Fig. 3).\n4. Outlook\nIn this review, we have concentrated on qualitatively explaining the kinetics of HI-AOS found in fer-\nrimagnetic GdFeCo and MRG alloys using a phenomenological framework. Since the \frst experimental\nidenti\fcation of HI-AOS, more than ten distinct models have been constructed to account for the pro-\ncess [9, 10, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. While these models all use di\u000berent methods, they produce\nsimilar temporal dynamics of HI-AOS, and are qualitatively consistent with the phenomenological model pre-\nsented here. Indeed, state-of-the-art atomistic simulations of HI-AOS also now embrace the same concepts\nof exchange and relativistic relaxations as used in the phenomenological approach [78]. Quantitative di\u000ber-\nences between the approaches naturally emerge, but one must always be strongly guided by experimental\nmeasurements when studying the counter-intuitive non-equilibrium dynamics of ultrafast demagnetization\nthat evidently underpin HI-AOS.\n12Despite the successes of the phenomenological model presented here, this model also exhibits limita-\ntions. For example, from the model it follows that the key ingredient to explain the counter-intuitive\ndynamics stems from exchange of angular momentum between di\u000berent magnetic sublattices in the exchange-\ndominated regime. It is natural to expect that such a regime indeed exists for magnetic sublattices which\nthemselves are weakly coupled to the environment. This situation may however not be realized in sys-\ntems featuring magnetic ions with sizable orbital moments, such as for Tb ions. These are generally more\nstrongly coupled to the lattice. It is possible to include single-ion anisotropy, even within the nonequilib-\nrium free energy [46], but this approach has so far not been investigated in the context of magnetization\nswitching. Moreover, in systems with strong orbital moments, non-collinear e\u000bects may become important.\nThe generalization of the current model to the non-collinear case has already been done for systems with\nweak coupling to the environment [79], disclosing an additional exchange-driven pathway for precessional\nswitching in ferrimagnets. In addition, the same equations were applied to describe the AFM-FM transition\nin FeRh systems [80, 81], which was found to yield profound di\u000berences between collinear and noncollinear\nphases. However, the combination of noncollinearity and sizeable orbital moments has not, however, been\ninvestigated so far. Moreover, being based on a macroscopic description of the sublattice magnetizations\nonly, the phenomenological model fails in describing possible noncollinear magnetic states that can emerge\nafter laser heating [82, 83, 84].\nAn open and outstanding question relates to whether the models referred to above, including the phe-\nnomenological model, can explain the HI-AOS experimentally identi\fed in the synthetic ferrimagnets Gd/Co\nand Tb/Co multilayers, and TbCo alloys doped with minute amounts of Gd. Beens et al. have successfully\ngeneralized the microscopic three-temperature model to account for the switching found in Gd/Co synthetic\nferrimagnets [32]. There, the switching is primarily identi\fed as an exchange-dominated e\u000bect with the\nreversal occurring close to the interface on the Co side, and subsequently propagating in to the bulk of the\nnanolayer. We anticipate that the phenomenological model presented here can reproduce this behavior, with\nappropriate division of the nanolayers in to even smaller thicknesses.\nAt the same time, it is not yet fully resolved why Gd facilitates HI-AOS whereas Tb does not. Early\nexperiments studying fs-laser-induced ultrafast demagnetization in TbFeCo alloys identi\fed the formation\nof a transient ferromagnetic-like state, with the magnetization of the Fe sublattice temporarily crossing\nzero [85]. Despite the ferromagnetic-like state persisting for more than 25 ps, the magnetization switched\nback. It was suggested that the strong spin-orbit coupling of the Tb sublattice provides an additional channel\nof angular momentum dissipation in competition with exchange-relaxation. Such a competing force is absent\nin Gd, which has an exactly half-\flled 4 fshell producing zero net orbital moment. This interpretation is\ncomplicated however by recent experimental works showing that TbCo alloys doped with just 1.5% of Gd\ndisplay HI-AOS [25]. Several research groups, using state-of-the-art atomistic models, currently argue that\nsublattice-speci\fc damping plays a crucial role in the kinetics of HI-AOS [24, 25].\nThe explanation of HI-AOS in synthetic ferrimagnets of Tb/Co nanostructures probably represents an\neven more di\u000ecult challenge. In the \frst experiments, Avil\u0013 es-F\u0013 elix et. al. tested HI-AOS as a function\nof nanolayer thickness tCo,Tb [33, 34, 86] and found that the switching can only be achieved for thickness\nratiostCo=tTb\u00141:2 withtCo>1 nm. As a point of comparison, the compensation point is at \u0019tCo=tTb\u0014\n1:1. Switching could be achieved using either 100 fs- or 5 ps-long pulses but, for some speci\fc combination\nof thicknesses, HI-AOS was only attainable using the 5 ps-long pulses, with the fs pulses only inducing\ndemagnetization. This behavior has not been identi\fed before in all prior demonstrations of HI-AOS,\nand indicates that the switching mechanism may be very di\u000berent from the process found in GdFeCo and\nMRG. Furthermore, single-shot pump-probe microscopy measurements indicate that the dynamics of HI-\nAOS found in Tb/Co multilayers initially involves the emergence of a ring of switched magnetization which\nsubsequently collapses to a single switched domain, stabilizing on a timescale of \u0019100 ps [87]. This behavior\nis not understood at the time of writing.\nA signi\fcant number of works have attempted already to formulate rules that give rise to HI-AOS in\nferrimagnetic alloys [42, 88, 89], but this generalization was severely impeded in the past by the fact that, for\nalmost a decade, only GdFeCo alloys were known to display HI-AOS. The recent discovery of HI-AOS in MRG\nhas o\u000bered a much better grounding for such proposals, with an empirical examination of common features\nbetween GdFeCo and MRG indicating the important role of exchange couplings within the ferrimagnet.\n13Speci\fcally, the sublattices must be strongly coupled antiferromagnetically to allow for exchange-relaxation,\nwhich is the only mechanism by which the constituent spins can cross zero and change sign. At the same\ntime, both GdFeCo and MRG have dissimilar intra-sublattice exchange couplings. This imbalance is vital\nfor the realization of exchange-driven switching since the angular momentum of one sublattice must grow at\nthe expense of the other [27, 42]. Beyond this, however, any further generalization must still fully explain,\nfor example, the role of spin-orbit coupling.\nTo address the aforementioned gaps in our understanding, further experimental campaigns studying the\ntemporal dynamics of HI-AOS are essential. While technically very challenging, it might be expected that\nXMCD measurements of the HI-AOS displayed by Tb/Co multilayers, using x-rays tuned to probe the\nsublattice-speci\fc response of Tb and Co, could provide the crucial insight required to understand why the\nswitching occurs or not. This approach would further require spatial resolution, in order to overcome the\napparent spatial non-uniformity of the magnetization dynamics [87].\nDespite the many outstanding questions relating to the general physics and kinetics of HI-AOS, tremen-\ndous progress has been made in rendering the process compatible with data storage and information pro-\ncessing technologies. HI-AOS can now be achieved, for example, using ultrafast electrical pulses that can\narguably be delivered in contemporary integrated circuitry [13]. Time-resolved measurements of HI-AOS\nin GdCo dots has revealed that reducing the dot's size towards the nanoscale actually enhances the speed\nof switching [23], with 75% reversal being achieved within \u00192 ps. Moreover, by growing GdCo on a silicon\nsubstrate, it is possible to switch magnetization repeatedly using two appropriately-tuned 250 fs-long pulses\nseparated in time by just 7 ps, corresponding to a write/erase frequency of 140 GHz [90]. Finally, large strides\nare being made in developing all-optically-switchable perpendicular magnetic tunnel junctions [91, 92], al-\nready yielding tunneling magnetoresistance signals in excess of 40% [34]. Data-recording technologies beyond\nthe state-of-the-art will undoubtedly bene\ft not only from the rapid progress made in understanding HI-\nAOS but also from the emerging progress in \fnding alternative means to all-optically switch magnetization,\nparticularly with reference to non-thermal ultrafast and minimally-dissipative methods based on the selective\nexcitation of electronic or phononic resonances [18, 21].\nAcknowledgements\nThis review has only been made possible by the dedicated work of our co-workers D. V. Afanasiev,\nC. Banerjee, J. Besbas, G. Bon\fglio, J. M. D. Coey, O. Eriksson, J. Hellsvik, B. A. Ivanov, T. Janssen,\nM. I. Katsnelson, A. F. G. van der Meer, K. Rode, P. Stamenov, A. Stupakiewicz and A. Tsukamoto. We are\nalso grateful to the skillful technical support provided by C. Berkhout, S. Semin, A. J. Toonen and the tech-\nnical sta\u000b at FELIX. We are grateful to J.-Y. Bigot, J. Bokor, U. Bovensiepen, R. Chantrell, O. Chubykalo-\nFesenko, H. D urr, V. N. 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Li, All-optical switching of magnetic tunnel junctions with single subpicosecond laser\npulses, Physical Review Applied 7 (2) (2017) 021001.\n[92] K. Borisov, D. Betto, Y.-C. Lau, C. Fowley, A. Titova, N. Thiyagarajah, G. Atcheson, J. Lindner, A. M. Deac, J. M. D.\nCoey, et al., Tunnelling magnetoresistance of the half-metallic compensated ferrimagnet Mn 2RuxGa, Applied Physics\nLetters 108 (19) (2016) 192407.\n18" }, { "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 ,\nhhc2(∆). Thegapvanishesattwocriticalpoints, h=hc1(∆)and\nh=hc2(∆). The isotropic case (∆ = 1) remains gapless in the intermediate re gionhc1(∆)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(∆)\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|\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 ≤h0. 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. 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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": "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": "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": "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": "1901.03286v2.Spin_wave_Confinement_and_Coupling_in_Organic_Based_Magnetic_Nanostructures.pdf", "content": " 1 Spin-wave Confinement and Coupling in Organic -\nbased Magnetic Nanostructures \nMichael Chilcote1, Megan Harberts1, Bodo Fuhrmann2, Katrin Lehmann3, Yu Lu4, \nAndrew Franson1, Howard Yu1, Na Zhu5, Hong Tang5, Georg Schmidt2,3, and Ezekiel Johnston -\nHalperin1,a) \n1Department of Physics, The Ohio State University, Columbus, OH 43210 -1117, USA \n2IZM, Martin -Luther -Universit ät Halle -Wittenberg, Halle, 06120, Germany \n3Institute für Physik, Martin -Luther -Universität Halle -Wittenberg, Halle, 01620, Germany \n4Department of Chemistry, The Ohio State University, Columbus, OH 43210 -1173, USA \n5Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA \na)Author to whom correspondence should be addressed. Electronic address : \njohnston -halperin.1@osu.edu \n 2 Abstract : Vanadium tetracyanoethylene (V[TCNE] x) is an organic -based ferrimagnet that \nexhibits robust magnetic ordering (T C of over 600 K), high quality -factor ( high-Q) microwave \nresonance ( Q up to 3,500), and compatibility with a wide variety of substrates and encapsulation \ntechnologies. Here, we substantially expand the potential scope and impact of this emerging \nmaterial by demonstrating the ability to produce engineer ed nanostructures with tailored magnetic \nanisotropy that serve as a platform for the exploration of cavity magnonics , revealing strongly \ncoupled quantum confined standing wave modes that can be tuned into and out of resonance with \nan applied magnetic field. Specifically, t ime-domain m icromagnetic simulations of the se \nnanostructures faithfully reproduce the experimentally measured spectra, including the quasi -\nuniform mode and higher -order spin-wave (magnon) modes. Finally, when the two domina nt \nmagnon modes present in the spectra are brought into resonance by varying the orientation of the \nin-plane magnetic field , we observe anti -crossing behavior indicating strong coherent coupling \nbetween these two magnon modes at room temperature . These resul ts position V[TCNE] x as a \nleading candidate for the development of coherent magnonics, with potential applications ranging \nfrom microwave electronics to quantum information. \n \n \n \n 3 The recent success of organic -based thin films in th e areas of optoelectronics and electronics \npromise s a new materials basis for these applications that is mechanically flexible, facile to \nsynthesize, and low cost when compared to traditional inorganic materials .1–3 This success should \nin principle extend to magnetic and spintronic functionality , and to some extent this promise has \nbeen realized in the observation of spin -dependent phenomenology including organic \nmagnetoresistance4–7 (OMAR), organic magneto -electroluminescence8,9 (OMEL), spin-pumping \nand spin transport ,10 and related phenomena .11–13 However , this phenomenology is constrained by \nthe fact that spins in these materials exhibit only diamagnetic, or at best paramagnetic, ordering \nand therefore miss the rich phenomenology found in extended magnetic order (such as ferro - and \nferrimagnetism). In particular, applications in the emerging field of coherent magnonics rely \nimplicitly on the a bility to excite and exploit long lived spin wave excitations in a magnetic \nmaterial. That requirement has led to the nearly universal reliance on yittrium iron garnet (YIG), \nwhich requires epitaxial synthesis on lattice matched substrates at temperatures above 800 °C to \nachieve high materials quality14–16 and has reigned for half a century as the unchallenged leader in \nlow loss magnetic resonance despite extensive efforts to identify alternative materials. \nSurprisingly, organic -based ferrimagnet s of the form M[Acceptor] x (M = transition metal ; x ≈ 2) \nprovide one of the most promising routes to realizing this goal , with the room -temperature , low -\nloss ferrimagnet vanadium tetracyanoethylene (V[TCNE] x) emerging as a compelling alternative \nto YIG . Manifestations of th e potential of this material system can be found in the demonstration \nof contro l of magnetic properties via ligand -tuning17–19 and metal -substitution ,20–25 optimized \nsynthesis26 (TC > 600 K) , extremely sharp (typically 1 Oe at 9.86 GHz ) ferromagnetic resonance \n(FMR) features ,27,28 the demonstration of FMR -driven spin -pumping,29 and encapsulation \nstrategies that stabilize the magnetic properties for weeks to months under ambient conditions .30 4 Here we build on this recent progress to demonstrate the ability to control the morphology of \nV[TC NE] x magnetic structures by creating arrays of templated nanowires, yielding control of \nmagnetic anisotropy and resulting spin-wave mode coupling and quantum confinement with no \nsubstantial increase in damping . This control is achieved through growth on SiO 2 substrates \npatterned with nanoscale grooves using ultraviolet interference lithography . After growth, t hese \nnanowire structures exhibit a high quality -factor (high -Q) quasi-uniform ferromagnetic resonance \n(FMR ) mode with uniaxial crystal -field driven magnetic anisotropy of 23.527 Oe ± 0.083 Oe, \noriented perpendicular to the nanowire axis. We perform time-domain micromagnetic simulations \nof the se nanostructures to provide additional insight into the mode structure present i n the \nexperimentally measured spectra. With the results , we identify the two dominant magnon modes \npresent in the spectra : one mode result s from the resonant excitation of the magnetic material in \nthe nanowire itself , while the other stems from the resonan t excitation of the nanostructured \nmagnetic material found within the trenches that lie between the wires . When the se two magnon \nmodes are brought into resonance by varying the orientation of an in -plane magnetic field , we \nobserve anti -crossing behavior consistent with stron g, coherent coupling between the two modes . \nThis study positions V[TCNE] x as a leading candidate for the development of coherent magnonics , \nwith functionality that directly challenges the best inorgan ic thin films demonstrated to date .14–16 \nV[TCNE] x samples are synthesized using a previously reported chemical vapor deposition \n(CVD ) growth process .26 Figure 1a shows a schematic view of a custom -built CVD reactor that is \nhoused within an argon glovebox . During the deposition, a rgon gas carries the two precursors, \nTCNE and V(CO) 6, into the reaction zone (shaded green in Fig. 1a) where V[TCNE] x is deposited \nonto one or more substrates . The system employs three independently temperature -controlled \nregions for the TCNE, V(CO) 6, and reaction zone with typical setpoints of 70 °C, 10 °C, and 50 °C, 5 respectively. For this experiment, patterned SiO 2 substrates are prepared using ultraviolet \ninterference lithography and reactive ion etching to produce an alternating pattern of trenches and \nridges . A variety of patterned substrates were prepared, with pitch es varying from 200 nm to 350 \nnm and ridge width s varying from 63 nm to 180 nm. All growth runs consist of deposition on to \none or more patterned substrates as well as a control sample consisting of either a flat SiO 2 or \nsapphire wafer die to account for any growth -to-growth v ariation in V[TCNE] x thin film \nproperties . \nThe result of a typical growth on the SiO 2 templates described abo ve can be seen in Fig. 1b, \nwhere cross -sectional scanning electron microscopy (SEM) shows the silicon wafer (dark grey), \nthe patterned SiO 2 layer (light grey), and the CVD deposited V[TCNE] x layer (dark grey) from the \nbottom to the top of the image, respectively. This microscopy reveals that the film growth around \nthe high aspect ratio fe ature s are governed by well -known growth dynamics controlled in part by \nthe differential arrival angles of gas -phase precursors around the features , which results in thicker \ncoverage on exterior angles than interior angles .31 Over the course of the deposition , V[TCNE] x \nforms into wire-like structures sitting atop the SiO 2 ridges and leaves closed -off voids within the \ntrenches (Fig. 1b). Studies of substrates oriented with trenches parallel and perpendicular to the \ngas flow direction in the CVD furnace reveal no significant changes in morphology or magnetic \ncharacteristics between the two orientations . \nAfter growth, samples are mount ed in the appropriate orientation and sealed in electron \nparamagnetic resonance (EPR) grade quartz tubes without exposure to air. When not being \nmeasured, the samples are stored in a -35 °C freezer within an argon glovebox and are found to be \nstable for weeks. The results of room -temperature DC magnetometry measurements as a function \nof applied magnetic field for a nanowire sample are shown in Fig. 1c. In contrast to the case for 6 uniform thin film s of V[TCNE] x, the magnetiz ation response to an in -plane field depends on \nwhether that field is applied parallel or perpendicular to the trench axis. This behavior is consistent \nwith the formation of an easy axis aligned perpendicular to the trenches (open triangles ), showing \nsaturation at a lower field than when the applied field is perpend icular to the wires (hard axis , filled \nsquares ). \nWhile these results are suggestive, a more complete study of t he magnetic anisotropy in the se \nV[TCNE] x nano structures can be found via FMR c haracterization of the anisotropy fields . Room -\ntemperature measurements are made using a Bruker electron paramagnetic resonance \nspectrometer , configured with an X -band bridge with 200 μW of applied microwave power and a \nmodulation field of 0.05 Oe. In standard operation, t he microwave frequency is tuned between 9 \nand 10 GHz for optimal microwave cavity performance before the measurement , and then the \nfrequency is fixed while the DC field is swe pt during the measurement . For consistency, the same \ntemplate material as that used for the SQUID measurement shown in Fig . 1c was used for the FMR \nstudies shown here . Figure 2a shows the FMR spectra of a V[TCNE] x nanowire array for the \nmagnetic field applie d in-plane ( = 90°; see inset to Fig. 2f) as the sample is rotated for values of \nϕ ranging from -90° to 270°. Figure 2b shows the integrated microwave absorption , as opposed to \nthe synchronously detected derivative spectra shown in Fig. 2a. These spectra show two sets of \nfeatures, each 90° phase shift ed from each other and covering a different field range. At angles \nwhere the external field is applied perpendicular to the trench (i.e. ϕ = -90°, 90°, 270 °), both high \nand low field features contain two peaks , suggesting the presence of higher order confined spin \nwave excitations supported by the nanostructured V[TCNE] x. This multimodal behavior persists \nthrough a subset of the full angular range shown in Figs. 2a and 2b. However, a t high symmetry \nangles where the external field is applied parallel to the trench (i.e. ϕ = 0°, 180 °), there is a n 7 isolated, single peak ed feature as well as a peak with much lower amplitude at lower field . Most \nstrikingly , at angles where these two sets of features would in principle cross ( approximately at \nϕ = -45°, 45°, 135 °, and 225 °) a gap appears in the spectra . This is most clearly ap parent in the \nintegrated spectra shown in Fig. 2b and in Fig. 2c, which shows the extracted center field values \nfor the higher intensity set of peaks . Both show an avoided crossing with a 14 Oe gap. \nThe peak with the highest intensity is ascribed to the quasi -uniform FMR mode of the nanowires \nas it involves the largest volume of magnetic material in the sample (this assumption will be \nvalidated by the analysis below) . The angular variation shown in Fig. 2c suggests a uniaxial \nanisotropy with an easy axis perpendicular to the nanowire/trench axis . Interestingly, this outcome \nis contrary to what one might expect from a simple magnetostatics argument using the shape \nanisotropy of a long thin rod (which would predict an anisotropy field of approximately 50 Oe \nwith an easy magnetization axis parallel to the wire axis) , and therefore further detailed analysis is \nnecessary to understand the origin of the magnetic anisotropy present in these nanostructures . In \norder to expand on the observations above , additional measurements are performed for rotations \nfrom in -plane to out of plane both for orientations parallel (ϕ = 0°) and perpendicular (ϕ = 90°) to \nthe nanowire axis with the results shown in Figs. 2d and 2e. The center fields for the quasi -uniform \nmode are extracted and all three data sets are simultaneously fit to the same set of equations (solid \nand dashed lines in Figs. 2c and 2f). \nDue to the complex geometry present in these samples many of the simplifying assumptions \ntypically employed in fitting thin magnetic films do not apply, as a result t he data are fit to \ndispersions obtained from the complete magnetic free energy of the nano wire array. The magnetic \nfree energy , including contributions from the Zeeman energy , an effective anisotropy energy , and \nan uniaxial anisotropy energy , is expressed as32,33 8 𝐹=−𝑴∙𝑯+ 1\n2𝑀[(𝒎∙𝑯eff)𝟐/𝐻eff−(𝒎∙𝑯∥)𝟐/𝐻∥], (1) \nwhere 𝑴 is the magnetization vector, 𝒎 is the magnetization unit vector given by 𝑴/𝑀, H is the \napplied bias field vector , 𝑯eff is an effective field resulting primarily from shape anisotropy (see \ndiscussion following Eq. 4), and 𝑯∥ is a uniaxial anisotropy field , which points in the direction of \nthe easy magnetization axis . The first term in Eq. 1 corresponds to the Zeeman energy while the \nsecond term includes contributions from the demagnetization energy and uniaxial anisotropy \nenergy. In general, the effective field 𝑯eff is not in the same direction as uniaxial anisotropy field \n𝑯∥. For the present case, the data presented in Fig. 2 indicate that 𝑯∥ is in-plane, perpendicular to \nthe nanowire axis (i.e. along 𝜃 = 90°, 𝜙 = 90°). The shape anisotropy of the nanowire array, given \nits high packing fraction, is similar to that of a thin film , and so 𝑯eff is oriented normal to the \nsurface of the sample (i.e. 𝜃 = 0°) as is typically the case for thin films . Note that 𝑯eff includes \ncontributions from any perpendicular uniaxial crystal -field anisotropy present in the sample since \nboth th is anisotropy and the demagnetization energy have an identical angular dependence , and \ntherefore are indistinguishable in FMR studies . For the analysis presented here these additional \ncontribution s to the anisotropy energy are not explicitly considered . \nThe resonance frequency , ω, can then be determined using the formalism provided by Smit, \nBeljers, and Suhl ,34,35 \n𝜔=𝛾\n𝑀sin𝜃(𝐹𝜃𝜃𝐹𝜙𝜙−𝐹𝜃𝜙2)1/2 (2) \nwhere γ is the gyromagnetic ratio and Fij is the second derivative of the free energy F with respect \nto the angles i and j. The FMR resonance fields (Fig. 2) are more than an order of magnitude larger \nthan the typical saturation field for V[TCNE] x, and therefore we assume that the magnetization is \neffectively parallel to the applied magnetic field (i.e. 𝜙≈𝜙𝐻 and 𝜃≈𝜃𝐻 where θ, 𝜙 and θH, 𝜙𝐻 \nare the polar and azimuthal angles of the magnetization M and the applied bias field H, 9 respectively ). With this framework , we may now separately obtain the dispersion relation for each \nsample orientation in Fig. 2 using Eqs. 1 and 2 : \n𝜔\n𝛾=√(𝐻−𝐻∥ cos2𝜙𝐻) (𝐻+𝐻eff+𝐻∥ sin2𝜙𝐻) , (𝜃𝐻 = 90°) (3a) \n𝜔\n𝛾=√(𝐻−𝐻∥−𝐻effcos2𝜃𝐻) (𝐻−𝐻effcos2𝜃𝐻) , (𝜙𝐻 = 0°) (3b) \n𝜔\n𝛾=√1\n2(𝐻−(𝐻∥+𝐻eff)cos2𝜃𝐻) (2𝐻+𝐻∥−𝐻eff−(𝐻∥+𝐻eff) cos2𝜃𝐻) , (𝜙𝐻 = 90°) (3c) \nThe first equation (Eq. 3a) is for an in-plane rotation (𝜃𝐻 = 90°) as the applied field is rotated \nthrough 𝜙𝐻. The second and third equation are for in -plane to out of plane rotations where the field \nis applied either along the nanowire axis (Eq. 3b; 𝜙𝐻 = 0°) or perpendicular to the nanowire axis \n(Eq. 3b; 𝜙𝐻 = 90°) as the applied field is rotated through 𝜃𝐻. Using an alternate form of the Smit -\nBeljers -Suhl formula ,36 this set of equations may be written as a single equation . This relation \napplies for arbitrary rotations of the applied field through both θH and 𝜙𝐻 and reduces to Eqs. 3a–\nc given the appropriate constraints . For rotations along the symmetry axes of the geometry \npresented here (i.e. for 𝜃𝐻 rotations along 𝜙𝐻 = 0°, 90°, 180 °, 270 ° and for 𝜙𝐻 rotations along \n𝜃𝐻 = 90°; or more fo rmally, when cos𝜃𝐻sin2𝜙𝐻=0), this relation reduces to: \n𝜔\n𝛾=√1\n2(𝐻−(𝐻eff+𝐻∥sin2𝜙𝐻)cos2𝜃𝐻) (2(𝐻−𝐻∥cos2𝜙𝐻)−(𝐻eff+𝐻∥sin2𝜙𝐻)−(𝐻eff+𝐻∥sin2𝜙𝐻) cos2𝜃𝐻) . (3d) \nAs noted above 𝜙≈𝜙𝐻 and 𝜃≈𝜃𝐻, and so from here forward we will drop the subscript, using \nsimply 𝜙 and 𝜃. \nWhen all three data sets are simultaneously fit (solid and dashed lines in Figs. 2c and 2f; the \nopen data points shown in Fig. 2c , which are affected by the presence of the avoided crossing, are \nexcluded from the fitting process ), this set of equations allows for the self -consistent extraction of \n𝐻eff and 𝐻∥, yielding values of 91.18 8 Oe ± 0.510 Oe and 23.527 Oe ± 0.083 Oe, respectively. \nThis value of 𝐻eff is consistent with previous reported values of 4𝜋𝑀S for uniform thin films ; for \nexample, 4𝜋𝑀S of 95 Oe is reported in Ref erence 27. In considering the origin of the uniaxial 10 anisotropy field , 𝐻∥, it is import ant to note that the shape anisotropy from magnetostatic effects in \na long thin rod would create an easy axis parallel to the nanowire axis, rather than the anisotropy \nperpendicular to the nanowire axis observed in Fig. 2. As a result, 𝐻∥ must arise from a crystal \nfield anisotropy wherein the local exchange vector acquires some anisotropy due to either lattice \nsymmetry or strain. \nThe difference in the coefficients of thermal expansion for organic and inorganic materials often \nvary by an order of magnitude or more , and have been reported to affect electronic properties of \nthe organic materials .37,38 As a result, it is likely that an anisotropic strain field is created in the \nnanowire structures due to the continuous contact with the SiO 2 substrate along the nanowire axis \nand the ability for the nanowires to relax along the radial direction due to the presence of the \ngrooves. This p henomenology , along with successful fitting using Eq. 3, suggests that the highe r \nintensity set of peaks in Fig. 2 do result from the quasi -uniform FMR mode of the nanowire and \nthat the easy axis is indeed, surprisingly, perpendicular to the patterning axis. We also note that \nthe magnetic material within the trenches is not subjected to the anisotropic strain field induced by \nthe ridges in the nanowire s, and therefore become s a leading candidate for the second set of \nresonance peaks present in Fig. 2. \nFurther insight can be gained by comparing these results to thin films grown on unpatterned \nsubstrates, wherein no in -plane anisotropy is observed (see supplementary material). Previously, \nthis lack of in-plane anisotropy has been interpreted to mean no anisotropy fields exist . However , \nthis data suggests an alternative explanation. In films deposited as a uniform thin film, strain due \nto differential thermal expansion would be a pplied uniformly in the plane of the film , yielding \nsignificantly simpler , modified forms of Eq. 3: \n𝜔\n𝛾=√𝐻 (𝐻+(4𝜋𝑀𝑆−𝐻A) , (𝜃 = 90°) (4a) 11 in which the 𝜙 dependence has dropped out . And f or the full in -plane to out of plane rotation , \n𝜔\n𝛾=√(𝐻−(4𝜋𝑀𝑆−𝐻A)cos2𝜃) (𝐻−(4𝜋𝑀𝑆−𝐻A) cos2𝜃) . (4b) \nHere , we have included the contribution from the demagnetization fields of a thin film as 4𝜋𝑀𝑆 \nand have conformed to the sign convention typical of 𝐻A, which arises from uniformly applied \nstrain in the plane of the film . Note that while this reproduces the lack of in -plane anisotropy \nobserved for thin films, it implies an additional anisotropy field in the out of plane direction. \nCoincidentally, this anisotropy field has the same symmetry, but not necessarily the same sign, as \nthe shape anisotropy for a thin film. This in turn implies that prior measurements of the anisotropy \nof thin films are in fact measuring 4𝜋𝑀𝑆−𝐻A=4𝜋𝑀eff rather than the bare 4𝜋𝑀𝑆 as previously \nassumed .27,28 In the literatur e, 𝐻A often presents itself as 𝐻⊥ because it is responsible for inducing \nperpendicular anisotropy . In the present study, the width of the rid ges on which the nanowires are \ntemplated is not zero, and therefore there may be some residual in -plane strain perpendicular to \nthe nanowire axis generating a residual anisotropy field, 𝐻A. However, as with previous studies of \nuniform thin films , there is no straightforward way to disentangle this residual anisotropy from \n4𝜋𝑀𝑆, leading us to use the more general 𝐻eff≡4𝜋𝑀eff in defining Eq. 3. \nWhile this analysis resolves several long-standing mysteries in the nature of magnetic ordering \nand anisotropy in V[TCNE] x, it only describes the primary peak visible in Fig. 2 and does not \ndescribe either the additional resonances or the anti -crossing behavior noted above. In order to \nanswer these questions, the effective field analysis de scribed in Eqs. 1–3 is used to inform \nquantitative time -domain micromagnetic simulations using the open -source GPU -accelerated \nsimulation software MuMax3 .39 The results of the simulations are sho wn in Fig 3, with the \ngeometry determined by the real structure of the nano wires as extracted from the corresponding \nSEM images (Figs. 3e–j). Figure 3a shows the in-plane experimental FMR data previously shown 12 in Fig. 2b, while Fig. 3b shows a plot of the simulated FMR data over the same field range as the \nintegrated spectra . Figures 3c–d show experimental spectra compared directly to the \ncorresponding simulated spectra for geometries with the field applied perpendicular ( 𝜙 = 90°) and \nparallel ( 𝜙 = 0°) to the nanowire array. The fit values of 𝐻eff and 𝐻∥ extracted from the dataset \n(Fig. 2) and literature values for the Gilbert damping constant and exchange constant28,29 are used \nas the materials parameters in put into the simulation . The simulation s are run using a n out of plane \ncontinuous -wave microwave excitation, as was the case in the real experiment .40,41 Consistent with \nthe pheno meno logy proposed above, the magnetic material in the nanowires themselves are \nsimulated with a uniaxial magnetic anisotropy. However, the material in the trenches does not \ninclude this anisotropy and instead, only includes contributions from the demagnetizing fields. \nAdditionally, t he simulations were found to most faithfully reproduce the experimental data when \nthe top and bottom surfaces are ( perfectly ) pinned . \nFigures 3e–j show resonant microwave excitation mode ma ps created by overla ying the change \nin the z -component of the reduced magnetization (Δm z) onto the SEM micrograph used in the \nsimulation. The colored bezel around each mode map corresponds to the color of the dashed line \nin Fig 3c or d to which the mode map corresponds. Each panel shows the structure of the resonant \nexcitations supported by the V[TCNE ]x array at the indicated position on the corresponding \nspectra. \nThe simulation results reveal that the two sets of peaks in the data are the result of a quasi -\nuniform mode supported in the nanowire and a second resonantly excited mode supported by the \nmagnetic material within the trench es of the SiO 2 template , consistent with the initial assumptions \nabove . Furthermore, a s the angle of the field is rotated from perpendicular to the wires (e.g. \n𝜙 = -90°) to parallel to the wires ( e.g. 𝜙 = 0°), the excitation mode structure hybridize s as the two 13 dominant peaks come together . At 𝜙 = 50° in the simulated data (see Fig. S1 in the supporting \nmaterial) , a node forms between what was formerly a pure trough mode excitation and the quasi -\nuniform FMR mode of the nanowire. These two intimately linked but spatially distinct regions \nexist in substantially different magn etic environments ; they represent two high-Q magnon cavities , \neach with its own magnetic anisotropy, connected by a continuous low -damping magnetic \nmaterial. As a result, t he two cavities can be tuned into and out of resonance with each other using \nan applied magnetic field , and w hen their individual resonant conditions approach the point where \nthey would coincide, the modes hybridize, and the result is an avoid ed crossi ng in the FMR data . \nThe gap between the mode branches in this regime is 14 Oe, corresponding to an energy (μB𝐵) \nof 0.081 µeV, while the half -width of the gap in terms of a frequency (γ) corresponds to a spacing \nof 20 MHz . This gap is approximately seven times the peak -to-peak linewidth and 10% of the full \nfield variation of the quasi -uniform mod e, indicating that these two excitations are in the strong \ncoupling regime.42–45 While there is also a significant change in the intensity of these lines as the y \nproceed through the crossing, it is difficult to disentangle effects due to the intrinsic strength of \nthese resonances from the efficiency of their detection due to complicating factors such as the fact \nthat this data is acquired by the physical rotation o f the sample within the microwave cavity (which \ncan perturb the cavity mode) and the varying spatial symmetries of the modes (which can affect \ntheir coupling efficiency to the microwave cavity and therefore their detection ). \nIn conclusion, this work demons trates the ability to engineer the magnetic anisotropy in thin \nfilms deposited on patterned substrates and to engineer both the dispersion and anisotropy of \nconfined spin wave modes in templated V[TCNE] x nanowires. Nanowires with a diameter of \napproximatel y 300 nm are grown on the plateaus between grooves, exhibit the high -Q quasi -\nuniform FMR mode, and display anisotropy with a shift in resonant field of 23.527 Oe ± 0.083 Oe. 14 Finally, when the trough spin-wave mode and quasi -uniform mode are brought into resonance by \nvarying the orientation of an in -plane magnetic field , we observe anti -crossing behavior and the \nopening of a gap of 14 Oe, indicating strong coherent coupling between these two excitations at \nroom temperature . These results position V[TC NE] x as a leading candidate for the development of \ncoherent magnonics, with potential applications ranging from microwave electronics28 to quantum \ninformation .46–49 \n \nEXPERIMENTAL SECTION \nThe samples in this study consist of organic -based magnetic nanostructures of vanadium \ntetracyanoethylene (V[TCNE] x) that assemble along the ridges of a grooved substrate. To fabricate \nthe grooved SiO 2 substrates , 35 nm of Cr was sputtered onto SiO 2(1 µm)/Si(100) wafers and \nsubsequently coated with photoresist in preparation for laser interference lithography . The samples \nwere exposed using a custom, home -built laser interf erence lithography setup equipped with a \n266 nm laser and then etched using an Oxford Plasmalab 100 system ICP 180 to produce a regular \nalternating pattern of trenches and ridges with pitch ranging across different samples from 200 nm \nto 350 nm and ridge w idth varying from 63 nm to 180 nm. The resist was then stripped using a \nstandard O 2 plasma clean and the Cr mask was removed using the reactive ion etching system \npreviously noted. \nThe V[TCNE] x layer was grown in a custom chemical vapor deposition (CVD) setup via the \nreaction of vanadium hexacarbonyl (V(CO) 6) with tetracyanoethylene (TCNE) in argon carrier \ngas. The precursors were prepared according to standard techniques in the literature and t he \nnanowire samples were synthesized using the same CVD growth process that has previously been \noptimized for thin films. 15 All samples were mounted so as to prevent unwanted rotation and sealed in evacuated electron \nspin resonance ( ESR) grade quartz tubes immediately after growth and without exposure to air. \nWhen not being measured, the sealed samples were stored in a -35 °C freezer within an argon \nglovebox . When necessary, samples were manipulated and remounted within an argon glovebox \nbefore being resealed in quartz tubes for additional measurements. \nSuperconducting quantum interference device (SQUID) m agnetometry measurements were \ncollected using a Quantum Design Magnetic Property Measurement System using the \nReciprocating Sample Option (RSO) and with an applied field of 100 Oe . Cavity ferromagnetic \nresonance (FMR) measurements were made using a n X-band Bruker ESR spectrometer at room \ntemperature with an applied microwave power of 200 μW . The microwav e frequency was tuned \nbetween 9 GHz and 10 GHz for optimal microwave cavity performance before starting the \nmeasurement. \nAll of the FMR data was collected from a single high-quality 3 mm × 3 mm sample (patterned \nwith a nominal pitch of 300 nm and trench width of 1 70 nm), from which the four edges were \ncleaved to avoid spurious effects introduced by V[TCNE] x growth on the substrate edges ; the \ncleaved edges were used for the cross -sectional scanning electron microscopy seen in Figs. 3 and \nS1. The SQUID data s hown in Fig. 1c was collected on a separate sample grown on a substrate \ncleaved from the same template as th at used for Figs. 2 and 3 (again, with a nominal pitch of 300 \nnm and trench width of 1 70 nm) . The micrograph in Fig. 1b shows a nanostructured V[TCNE] x \ngrowth on a template with a nominal pitch of 350 nm and a nominal trench width of 220 nm. \nMicromagnetic simulations were performed using the open -source GPU -accelerated \nmicromagnetic simulation so ftware MuMax3. The simulations were performed using the real 16 geometry of the nanostructures as extracted from SEM micrographs of the sample, with periodic \nboundary conditions applied in order to simulate an array. The simulations were run using a \ncontinuou s wave approach, with an out of plane continuous -wave microwave excitation, as was \nthe case in the real experiment. Except for the Gilbert damping constant (for which an artificially \nhigh value was used to reduce simulation time; α = 8.0 × 10-4) and the exchange constant (for \nwhich l iterature values were used), all materials parameters used in the simulation were obtained \nfrom fitting the experimental FMR data. The magnetic material on top of each SiO 2 ridge was \nmodeled to include a contribution from the un iaxial anisotropy obtained by fitting the FMR data. \nThe magnetic material within the trenches was modeled without the inclusion of any additional \nanisotropy beyond standard contributions from the demagnetizing fields. The top and bottom \nsurfaces were perfe ctly pinned during all simulations. The mode maps in Fig. 3 show the change \nin the z -component of the reduced magnetization (Δm z) overla id on SEM micrograph s. \nSUPPLEMENTARY MATERIAL \nSupplementary material , available from AIP Publishing or the corresponding author , contains \nadditional m icromagnetic simulation mode maps of the quasi -uniform mode , the full equation for \nthe angular dependence obtained from the alternate form of the Smit -Beljers -Suhl equation, thin -\nfilm control data, additio nal data on a sample with a thicker magnetic layer, and data on the \ndependence of the magnetic anisotropy as a function of both the thickness of the deposited \nV[TCNE] x layer and the pitch of the underlying SiO 2 templated substrate. \nACKNOWLEDGMENTS \nThis wor k was supported by NSF Grant No. DMR -1507775 and DFG Grant No. SFB762. 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Nakamura, \nScience 349, 405 (2015). \n49 P. Andrich, C.F. de las Casas, X. Liu, H.L. Bretscher, J.R. Berman, F.J. Heremans, P.F. Nealey, \nand D.D. Awschalom, Npj Quantum Inf. 3, 28 (2017). \n 21 Figure 1 . (a) Plan view of the chemical vapor deposition schematic. (b) Cross -sectional s canning \nelectron micrograph of V[TCNE] x nanowires grown on a patterned SiO 2 substrate. The inset shows \nan alternate view of the same sample demonstrating that the nanowires extend the length of the \nsample (scale bar: 1 m). (c) Magnetization as a function of applied field at room temperature for \nV[TCNE] x grown on a patterned substrate. The inset shows the coordinate system used across all \nexperiments, with the nanowires aligned parallel to the x -axis. The magnetization is normalized \nby the saturati on magnetization and plotted for Happ perpendicular ( 𝐻app⟂; = 90° , = 90°) to the \nsubstrate pattern (∆) and parallel ( 𝐻app∥; = 90° , = 0°) to the substrate pattern (■). \nFigure 2 . (a) Shows FMR spectra for the magnetic field applied in -plane as the sample is rotated \nfor values of ranging from -90° to 270° (shown from bottom to top and as labeled on the right \nside of the data ) for a V[TCNE] x nanowire array. (b) Shows the integrated microwave absorption \n(numerical integration of the data as shown in (a)), as opposed to the synchronously detected \nderivative spectra. (c) Shows the extracted center fields of the in -plane angular series shown in (a) \nand (b). The solid line is a fit to the data (see text). Panels (d) and (e) show the FMR spectra for \nrotati ons from in -plane to out of plane both for orientations parallel ( = 0°) and perpendicular \n( = 90°) to the nanowire axis, respectively. (f) Shows the extracted center fields from the angular \nseries shown in (d) and (e) with fits shown as solid and dashed lines. The inset shows the coordinate \nsystem with respect to the sample geometry. \nFigure 3 . (a) Shows the integrated in -plane experimental FMR data previously shown in Fig 2b. \n(b) Shows a plot of the simulated FMR data over the same field range for values of ranging from \n-90° to 270° (shown from bottom to top and as labeled on the right side of the data ) for a V[TCNE] x \nnanowire array. The spectrum simulated for 0° is shown in red, and the spectrum simulated for 90 ° 22 is shown in blue. (c) Shows the e xperimental spectrum collected at 90 ° in the top panel and the \nsimulated spectrum for 90 ° in the bottom panel in red. The four dashed lines in the bottom panel \ncorrespond to the mode maps shown in (e), (f), (g), and (h). (d) Shows the experimental spectrum \ncollected at 0 ° in the top panel and the simulated spectrum for 0 ° in the bottom panel in blue. The \ntwo dashed lines in the bottom panel correspond to the mode maps shown in (i) and (j). (e) --(j) \nShow mode maps overlaid on the SEM micrograph used in the s imulation (see text ; the change in \nthe z -component of the reduced magnetization , Δm z, is shown ). The colored bezel around each \nmode map corresponds to the color of the dashed line in (c) or (d) to which the mode map \ncorresponds. Scale bar: 500 nm. 23 \nFigure 1 \n \n \n \n \n 24 Figure 2 \n \n 25 Figure 3 \n \n \n" }, { "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. (c) \nVariation of the re-switched fraction with t12 for different pump fluences.Figure 4: Magnetization of the 4 a and 4c sublattices during SP-AOS trajectory (red dashed \nline). The first stage of exchange driven demagnetization switches the 4 a sublattice while \nkeeping the net magnetization constant; a transient ferromagnetic like state is reached \nbetween 0.1 ps and 0.3 ps. In the second stage, between 0.3 ps and 3.0 ps, the 4 c sublattice \nreverses and the system relaxes toward equilibrium on the black dotted line. Between 3 ps \nand 1 ns, the system cools down. At 10 ps, the net magnetization changes sign and the system\ncan be than be re-switched. The inset in the third quadrant illustrates the proposed spin \nconfigurations.[1]I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. Ostler, J. Barker, R. F. L. \nEvans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Rasing, and A. V. Kimel, Nature \n472, 205 (2011).\n[2]T. A. Ostler, J. Barker, R. 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Stamenov, and K. Rode, Appl. \nPhys. Lett. 106, 122402 (2015).\n[14]J. M. Liu, Opt. Lett. 7, 196 (1982).\n[15]V. N. Gridnev, Phys. Rev. B 98, 014427 (2018).\n[16]R. D. Cowan, The Theory of Atomic Structures and Spectra (University of California Press, \n1981).\n[17] We consider on-atom direct Coulomb interaction and spin-orbit coupling only. We speculate \nthat new materials exhibiting SP-AOS are likely to contain half-filled d or f shells.\n[18]B. Koopmans, M. van Kampen, J. T. Kohlhepp, and W. J. M. de Jonge, Phys. Rev. Lett. 85, 844 \n(2000).\n[19]J. Barker, U. Atxitia, T. A. Ostler, O. Hovorka, O. Chubykalo-Fesenko, and R. W. Chantrell, Sci. \nRep. 3, 3262 (2013).\n[20]K. E. Siewierska, G. Atcheson, A. Jha, K. Esien, R. Smith, S. Lenne, N. Teichert, J. O’Brien, J. M. D.\nCoey, P. Stamenov, and K. Rode, ArXiv:2012.05736 (2020).\n[21]S. Lenne, Y.-C. Lau, A. Jha, G. Y. P. Atcheson, R. E. Troncoso, A. Brataas, J. M. D. Coey, P. \nStamenov, and K. Rode, ArXiv 1903.04432 (2019).\n[22]A. 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Johnston-Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210 \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853 \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242 \n \n* These authors contributed equally to this work. \n \nAbstract: Quantum information science and engineering requires novel low-loss magnetic \nmaterials for magnon-based quantum-coherent operations. The search for low-loss \nmagnetic materials, traditionally driven by applications in microwave electronics near \nroom-temperature, has gained additional constraints from the need to operate at cryogenic \ntemperatures for many applications in quantum information science and technology. \nWhereas yttrium iron garnet (YIG) has been the material of choice for decades, the \nemergence of molecule-based materials with robust magnetism and ultra-low damping has \nopened new avenues for exploration. Specifically, thin-films of vanadium \ntetracyanoethylene (V[TCNE] x) can be patterned into the multiple, connected structures \nneeded for hybrid quantum elements and have shown room-temperature Gilbert damping \n(α = 4 × 10-5) that rivals the intrinsic (bulk) damping otherwise seen only in highly-polished \nYIG spheres (far more challenging to integrate into arrays). Here, we present a \ncomprehensive and systematic study of the low-temperature magnetization dynamics for \nV[TCNE] x thin films, with implications for their application in quantum systems. These \nstudies reveal a temperature-driven, strain-dependent magnetic anisotropy that \ncompensates the thin-film shape anisotropy, and the recovery of a magnetic resonance \nlinewidth at 5 K that is comparable to room-temperature values (roughly 2 G at 9.4 GHz). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 2 We can account for the se variations of the V[TCNE] x linewidth within the context of \nscattering from very dilute paramagnetic impurities, and anticipate additional linewidth \nnarrowing as the temperature is further reduced. \n \nThe search for low-loss magnetic materials dates to the early days of radio and \nmicrowave electronics [1–3], and the study of elementary excitations, or magnons, in these \nmagnetically-ordered materials has proven to be a rich area of research for both \nfundamental physics and their potential technological applications. More recently, interest \nin these low-loss systems has expanded to include applications in the field of quantum \ninformation technology such as quantum sensing and quantum transduction [4 –7], wherein \nlow-temperature operation allows for the freeze-out of thermal excitations and access to \nthe single-quantum regime . In this regime the field of quantum magnonics utilizes hybrid \narchitectures for coupling magnons to other quantum degrees of freedom, such as \nmicrowave photons, with the aim of extending their functionality in the quantum limit \n[8,9]. It has been demonstrated that magnons can be resonantly excited over a wide range \nof microwave frequencies, allowing for precise control of qubit states mediated by coherent \nexchange via cavity-mode photon excitations [4 ,7]. Magnons also exhibit the potential to \ncoherently couple localized spin-qubits with high cooperativity [10] . However, while \nmagnons exist in a wide range of materials, the same delocalized electrons that are most \noften responsible for stabilizing ferromagnetic order also contribute to electron-magnon \nscattering [11], leading to substantial losses in most metallic ferromagnets. As a result, the \nstudy of low-dissipati on magnon dynamics for quantum applications has focused on \ninsulating ferromagnets and ferrimagnets, with yttrium iron garnet (YIG) and its close \nrelatives holding pride of place as the benchmark low-loss materials for more than 50 \nyears [ 4,12–14]. As a result, despite these longstanding and emerging needs, applications \nare still constrained by the materials limitations of YIG; namely the need for growth or \nannealing at high temperatures (typically 800° C) [15–17] and the resulting difficulty in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 3 integrating and patterning YIG thin-films with other microwave electronic structures and \ndevices. \nIn this context, the emergence of the molecule-based ferrimagnet vanadium \ntetracyanoethylene (V[TCNE] x) has dramatically expanded the playing field for low-loss \nmagnets. Despite what one might expect from its molecular building blocks, V[TCNE] x \nhas a magnetic ordering temperature of over 600 K and shows sharp hysteresis at room-\ntemperature [ 18–20]. Moreover, its dynamic properties are exceptional, showing ultra-\nnarrow ferromagnetic resonance (FMR) linewidth (typically ~ 1 – 1.5 G at 9 .4 GHz) with \na Gilbert damping parameter, of 4 × 10-5 for thin-films [ 18,21]. As a comparison, the \nbest YIG thin-films typically show = 6.5 × 10-5 [22] and a value of 4 × 10-5 is competitive \nwith the intrinsic damping of bulk YIG = 3 × 10-5 [15,23]. From an applications \nperspective, V[TCNE] x has been shown to deposit on a wide variety of substrates without \ncompromising material quality [24–26], facile encapsulation allows for direct integration \nwith pre-patterned microwave structures for operation under ambient conditions [ 27], and \nrecent work has demonstrated patterning at length scales down to 10 m without increased \ndamping [21]. However, while these properties clearly establish the potential of \nV[TCNE] x for new applications in traditional microwave electronics, very little is known \nabout its low-temperature magnetization dynamics and therefore its potential for \napplications in quantum information science and engineering (QISE ). \nHere we present a detailed study of the low-temperature magnetic resonance of \nV[TCNE] x films. We identify two regimes. In the high-temperature regime, extending from \n300 K down to 9 K, we observe a monotonic shift in the resonance frequency consistent \nwith a temperature-dependent strain. This strain results in a crystal-field anisotropy that \nincreases with decreasing temperature with a magnitude of at least 140 Oe and the same \nsymmetry, but opposite sign, to the shape anisotropy of the thin-film. In addition, we \nobserve an increase in linewidth consistent with magnon scattering from paramagnetic \nimpurities similar to what has been observed in YIG [23,28,29], but with an amplitude 3 \ntimes smaller ( i.e. an increase in linewidth by 9 times in V[TCNE] x as compared to 28 \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 4 times in YIG [23, 30]). In the low-temperature regime, starting at 9 K and extending to 5 \nK, we observe a discontinuous change in both anisotropy and linewidth: the anisotropy \nabruptly reverts to the room-temperature symmetry (in-plane easy-axis) and the linewidth \napproaches room-temperature values (2.58 G) at 5 K. This linewidth variation can be \nexplained using a model for scattering between magnons and paramagnetic impurities that \ntakes into account the finite spin-lifetime of the impurity spins [23,31]. At high \ntemperatures (above 100 K) the spin-lifetime is sufficiently short that changes in \ntemperature do not lead to significant changes in scattering rate, and at low-temperatures \n(below 9 K) the spin-lifetime becomes long with respect to the spin-magnon scattering \ntime, resulting in a saturation of the excited state. At intermediate temperatures (from 9 K \nto 100 K) this spin-magnon scattering dominates relaxation due to the increase of the \nground state impurity population, which results in a local maximum in the linewidth that \nis 9 times larger than the room-temperature value. These results are extremely promising \nfor low-temperature applications of V[TCNE] x magnonics, promising low-temperature \nmagnon resonators with unprecedented low-loss that can be integrated on-chip into \nmicrowave electronic circuits and devices [20,21]. \nFor this study, thin-films of V[TCNE] x are deposited on sapphire (Al 2O3 (0001)) \nsubstrates using chemical vapor deposition (CVD) growth process consistent with prior \nreports [18,19]. Briefly, argon gas transfers the two precursors tetracyanoethylene (TCNE) \nand vanadium hexacarbonyl (V(CO) 6) into the reaction zone of a custom-built CVD reactor \n(Fig. 1(a)) where V[TCNE] x is deposited onto polished sapphire substrates. The system is \ntemperature controlled to maintain the TCNE, V(CO) 6 and the reaction zones at 65° C, 10° \nC and 50° C respectively . After growth the sample is mounted on a custom, microwave-\ncompatible sample holder and sealed using a septa cap in an electron paramagnetic \nresonance (EPR) grade quartz tube in an argon environment. When the sample is not being \nmeasured, it is stored in a - 35° C freezer housed in an argon glovebox and is stable for over \none month [ 27]. \nFerromagnetic resonance (FMR) measurements are performed using a Bruker EMX \nPlus X-band EPR spectrometer at temperatures ranging from 300 K down to 5 K. The \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 5 microwave frequency of the spectrometer is tuned between 9 and 10 GHz for optimal \nmicrowave cavity performance before the measurement, and then the frequency is fixed \nwhile the DC field is swept during data collection. Figure 1(b) shows a representative \nroom-temperature FMR measurement of a typical V[TCNE] x thin-film with the external \nmagnetic field applied in the plane of the sample. The resonance feature is consistent with \npreviously reported high-quality V[TCNE] x thin-film growth, showing a peak- to-peak \nlinewidth of 1.5 G at 9.4 GHz [18,19]. \nComparing this data to FMR measurements at temperatures of 80 K and 40 K \n(Figure 1(c)) shows an increase in the resonance field of over 40 G (roughly half of the \nsaturation magnetization, 4𝜋𝑀 𝑠) as the temperature decreases. Since the applied \nmicrowave frequency is held constant at 9.4 GHz , this shift must arise from fields internal \nto the V[TCNE] x film, i.e. magnetic anisotropy fields. Note that since the value of the DC \napplied field varies between 3350 G and 3450 G, well above 4𝜋𝑀 𝑠, changes in the \nmagnetization of the film are not expected to contribute to this field shift. In a similar \nfashion, changes in the shape-dependent anisotropy fields can be ruled out, leaving only \nchanges to the crystal-field anisotropy as a potential source of this phenomenon. Crystal-\nfield anisotropy originates from the interaction of a material’s mean exchange field and the \nangular momenta of neighboring atoms (ions) in the material , indicating that there is a \ntemperature dependence to the local atomic environment within the V[TCNE] x films , e.g. \ndue to a temperature-dependent strain within the film. \nIn order to more comprehensively map out this phenomenon angle dependent FMR \nmeasurements are performed to quantitatively track changes in the magnetic anisotropy at \ntemperatures of 300 K, 80 K, and 40 K (Fig. 2). Variation of the magnetic resonance field \nas a function of the angle between the applied field and the princip al axes of the film can \nbe modeled by considering the free energy of the magnetic system with anisotropic \ncontributions. If we consider the case of a uniaxial anisotropy with the hard-axis \nperpendicular to the easy-axis, and where the magnetization is parallel to the external field \n(i.e. external field is much larger than the saturation magnetization, as is the case here) the \ntotal magnetostatic energy is as follows [ 32]: \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 6 \n 𝐸 = −𝑴 · 𝑯 + 2𝜋 (𝑴 · 𝒏 )𝟐− 𝐾 (𝑴 · 𝒖 𝑀 ⁄)2 (1) \n \nwhere M is the magnetization, H is the applied magnetic field, n is the unit vector parallel \nto the normal of the magnetic sample , u is the unit vector parallel to the easy-axis and K is \nan anisotropy constant. For the case of in-plane uniaxial anisotropy, this simplifies to \n \n 𝐸 = − 𝑀𝐻 (sin𝜙sin2𝜃 + cos2𝜃)+ 2𝜋𝑀2cos2𝜃 − 𝐾 sin2𝜃sin𝜙2 (2) \n \nwhere 𝜃 is the angle between M and the sample normal and 𝜙 \nis the azimuthal angle. Minimizing the magnetostatic energy with respect to 𝜃 , one will \nfind that the easy-axis orientation occurs when 𝜃 = 2 𝑛𝜋± 𝜋\n2, where n is an integer . Using \nthis simple symmetry analysis, we can see that the data in Fig. 2 indicates that the easy-\naxis lies in-plane at a temperature of 300 K (i.e. the resonance field is smallest when the \napplied magnetic field lies in-plane) and out- of-plane at a temperature of 40 K (i.e. the \nresonance field is smallest when the applied magnetic field is out- of-plane). In this context, \nthe lack of variation in resonance field at 80 K indicates a nearly isotropic magnetic \nresponse. This switch in magnetic easy-axis from in-plane to out- of-plane further supports \nthe proposition that there is an additional temperature-dependent crystal-field contribution \nto the magnetic anisotropy. \nIn previous studies, templated growth of V[TCNE] x resulting in nanowire \nmorphologies induced an additional in-plane magnetic anisotropy with easy-axis \nperpendicular to the long-axis of the nanowires, strongly suggesting the presence of a \nstrain -dependent contribution to the crystal-field anisotropy [ 33]. In the thin-films studied \nhere, such a strain-dependent crystal-field effect would be expected to generate anisotropy \nparallel to the surface normal, i.e. in the out- of-plane direction. The anisotropy field would \nthen be parallel to the expected shape anisotropy from a thin-film, though not necessarily \nwith the same sign. As a result, if there is a difference in the coefficient of thermal \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 7 expansion between the V[TCNE] x film and the sapphire substrate then the temperature \ndependence of magnetic anisotropy can potentially be understood as a proxy for a \ntemperature dependence of strain in the thin- film; such variations in strain leads to changes \nin the local atomic structure, leading to the observed changes in magnetic anisotropy. We \nnote that while the coefficient of thermal expansion for V[TCNE] x has not yet been \nmeasured, the value for sapphire is 5.4 ppm/K and typical values for molecular-based solids \ncan range somewhere between 28 –500 ppm/K [ 34]. Assuming no strain at room-\ntemperature, this would then imply a compressive strain between 0.67% to 15% at the \nsapphire –V[TCNE] x interface at 5 K , leading to an out- of-plane distortion whose symmetry \nis consistent with the observed anisotropy. \nA schematic describing how these two anisotropy fields would be expected to \ninteract as a function of temperature can be found in Fig. 3(a). At a temperature of 300 K \n(Fig. 3(a), upper panels), the orientation of the easy-axis is determined by the shape \nanisotropy, resulting in an in-plane easy-axis for thin-films. But at a temperature of 40 K \n(Fig. 3(a) lower panels), there is an additional crystal-field anisotropy, 𝐻⊥, proposed that \ndominates the shape anisotropy, reorienting the easy-axis to be out-of-plane. This \nsymmetry analysis also explains the lack of orientation dependence at a temperature of 80 \nK, which is apparently the temperature at which the strain-driven crystal-field anisotropy \nperfectly cancels out the shape anisotropy. We note that similar phenomenology is also \nobserved in vanadium methyl tricyanoethylenecarboxylate (V[MeTCEC] x) thin-films (see \nsupplementary materials), indicating that this temperature- and strain-dependent \nanisotropy is a general property of this class of metal-ligand ferrimagnets. \nThe fact that the shape and proposed crystal-field anisotropies have the same \nsymmetry make it challenging to distinguish between the two; therefore, an effective field \nis defined as 𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓= 4𝜋𝑀 𝑠− 𝐻 ⊥, where 𝑀𝑠 is the saturation magnetization and \n𝐻⊥is the crystal-field anisotropy. Figure 2 shows the effects of this net anisotropy field in \nthe form of resonance field shifts and a change in the easy-axis orientation . Quantitatively \nextracting the magnitude and direction of this anisotropy field provides detailed insight \ninto the role of crystal-field anisotropy in tuning the magnetic response of V[TCNE] x thin-\nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 8 films. To this end, each scan is fit to the sum of the derivatives of absorption and dispersion \nfrom a Lorentzian function to extract the resonance frequency and linewidth (experimental \ndata are obtained using a modulated-field technique that yields the derivative of the \nexpected Lorentzian resonance lineshape). For scans showing an out- of-plane easy-axis a \nsingle derivative sum provides good agreement with the data, while for scans showing in-\nplane easy-axis more complex structure is observed requiring the addition of up to three \nderivative sums. In the results discussed below we focus on the behavior of the primary , \ni.e. largest amplitude, peak (a full description of the fitting and resulting phenomenology \ncan be found in the supplemental material). \nFigure 3(b) shows the extracted resonance field plotted against sample rotation \nangle for the high-and low-temperature data shown in Fig. 2, 300 K and 40 K, respectively . \nTaking into account a uniaxial out- of-plane anisotropy defined by 𝐻𝑒𝑓𝑓, as described \nabove, the angular dependence for in-plane to out- of-plane rotation of a thin-film sample \nis given by [19,35,36]: \n \n𝜔\n𝛾 = √(𝐻 − 𝐻 𝑒𝑓𝑓cos2𝜃)(𝐻 − 𝐻 𝑒𝑓𝑓cos 2𝜃)\n= √(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥) cos2𝜃)(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥)cos 2𝜃) \n \nwhere is the resonance frequency and is the gyromagnetic ratio. As a result, the \nphenomenology of the data presented in Fig. 2 can be understood as an 𝐻𝑒𝑓𝑓 that is positive \nat 300 K and negative at 40 K, as 𝐻⊥ increases with decreasing temperature, consistent \nwith the mechanism for anisotropy switching described in Fig. 3(a). This qualitative \nunderstanding can be made quantitative by fitting the data in Fig. 2 using Eq. (3) to extract \n𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓 of 91.2 G 1.6 G and -22.8 G 0.4 G, respectively. \nFigure 3 (c) shows this 𝐻𝑒𝑓𝑓 plotted against temperature over the temperature range \nfrom 300 K to 5 K, extracted from angular dependencies such as the measurements \npresented in Fig. 2 . It should be noted that each anisotropy point in Figure 3(c) represents (3) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 9 a fit to a complete angular dependence such as the data shown in Figure 3(b).The effective \nfield mak es a smooth transition through zero from positive (in-plane) to negative (out- of-\nplane) at a temperature of roughly 80 K . This behavior is qualitatively consistent with the \nphenomenological model presented above and reveals a magnitude of the variation in 𝐻𝑒𝑓𝑓, \nfrom +91 .2 G ± 1.6 G at 300 K to - 45.2 G ± 1.1 G at 10 K, that is roughly 150% of the \nroom-temperature value. \nNotably, this more comprehensive study also reveals new phenomenology at the \nlowest temperature of 5 K, where the anisotropy abruptly shifts back to in-plane with a \nvalue of +26.2 G ± 0.6 G (roughly 25% of the room-temperature value). This behavior \nreproduces across all samples measured and is quantitatively reproduced upon temperature \ncycling of individual films. The abruptness of this change is distinct from the broad and \nmonotonic behavior observed for temperatures greater than 9 K . The origin of this abrupt \nchange is unclear, but there are two potential explanations consistent with this \nphenomenology. First, it is possible that the increase in strain results in an abrupt relaxation \nthrough the creation of structural defects. This explanation would require some level of \nself-healing upon warming in order to explain the reproducibility of the transition. Given \nthe lack of long-range structural order in V[TCNE] x films as-grown [ 37] it is possible that \nany residual structural defects do not contribute to additional magnetic loss (damping). \nSecond, it is possible that there exist paramagnetic spins in the system that magnetically \norder at temperatures below 9 K. If such spins were preferentially located in an interface \nlayer, for example, their ordering could create an exchange bias that would then pull the \neasy-axis back to an in-plane orientation. \nThe temperature dependence of the linewidth of the magnetic resonance provides \nan additional avenue for evaluating these potential explanations. Figure 4 shows the \nlinewidth for the in-plane magnetic resonance from 300 K to 5 K, with additional data to \nmore clearly resolve the sharp change between 5 K and 9 K. The linewidth data presented \nin Figure 4 is extracted from a single (in-plane applied magnetic field orientation) scan. As \na result, the initial dataset underlying Figure 3 was supplemented by a second temperature \ndependent scan at fixed angle in Figure 4. This data reveals a monotonic increase in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 10 linewidth with decreasing temperature from 300 K down to 9 K followed by a dramatic \ndecrease in linewidth between 9 K and 5 K, coincident with the abrupt change in magnetic \nanisotropy. We note that in studies of YIG thin-films broadly similar phenomenology is \nobserved, though with a maximum in linewidth that is both higher amplitude (roughly 28 \ntimes the room-temperature value) and at higher temperature (typically 25 K) than is \nobserved here [23,30]. Prior work [23,28 ] has explained this behavior using a model of \nmagnon scattering from paramagnetic defect spins (also referred to as two-level \nfluctuators, TLF) wherein the scattering cross-section at high temperature increases with \ndecreasing temperature as the thermal polarization of the spins increases. This \nphenomenology competes with magnon-pumping of the paramagnetic spins into their \nexcited state, a process that saturates as the spin-lifetime of the defects becomes long \nrelative to the spin-magnon scattering time. The competition between these two processes \nyields a local maximum in the damping (linewidth) that depends on the temperature \ndependent spin lifetime, ts, the energy separation between majority and minority spin states, \nℏωeg, and the difference between that energy splitting and the uniform magnon energy, \n(ℏω - ℏωeg ). \nIn this model, the linewidth expression is proportional to the square of the exchange \ninteraction energy between V[TCNE] x atoms and the impurity level ( ℏωint)2 ~ (ℏωeg)2, a \nline-shape factor accounting for the finite spin lifetime, 1/𝑡 𝑠/(ℏ2/𝑡𝑠2 + (ℏω - ℏωeg )2ts2), and \nthe ratio between the ground and excited impurity states for fast impurity relaxation , \ntanh(ℏω/2k BT) [23, 28], \n \nΔ𝐻 = 𝑆\n𝛾 𝑁𝑖𝑚𝑝\n𝑁 (ℏ𝜔 𝑖𝑛𝑡)2 1/𝑡 𝑠 \nℏ2/𝑡𝑠2+ (ℏ𝜔 − ℏ𝜔 𝑒𝑔)2 𝑡𝑠2tanh (1\n2 ℏ𝜔\n𝑘𝐵𝑇) + 𝐻 𝑂 \n \nwhere Nimp/N is the ratio between number of impurit ies and number of V[TCNE] x atoms, \nand S is the averaged V[TCNE] x spin per site , 𝛾 is the gyromagnetic ratio and 𝐻𝑂 is a \nconstant offset due to other relaxation mechanisms. In addition, we assume spin lifetime ts (4) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 11 = t∞ 𝑒𝐸𝑏𝑘𝐵𝑇⁄ [31, 38, 39] where t∞ is the spin lifetime limit at very high temperatures, and \nEb is a phenomenological activation energy. Figure 4 includes a fit of Eq. (4) to the \nexperimental linewidth (orange line) that yields for S ~ 1 and ωint ~ ωeg the parameters: \nωegt∞ = 0.98, Eb = 1meV, ω egNimp/N = 36.5GHz and 𝐻𝑜= 1 G . Interestingly, if we assume \na reasonable value for ℏωeg of 1.3 meV, a value of Nimp/N = 0.1 follows, thus indicating \nthat V[TCNE] x is an exceptional low-loss magnetic material even if we assume an impurity \nconcentration as high as 10%. This observation is consistent with the hypothesis of \ninsensitivity to structural defects discussed above. \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out- of-plane easy-axis to an in-plane easy-axis. This \nchange in magnetic anisotropy has the potential to have a substantial impact on spin-\nmagnon scattering efficiency. For example, this change will result in a shift of the energy \nof the magnon bands (see Eq. 1), and if this change involves a commensurate change in the \nstrain there will also be a modification to the spin-orbit coupling and exchange parameters \nat the paramagnetic defects. It should be noted that although this reentrant anisotropy is an \nintriguing feature, the fits to our model for TLFs in Figure 4 are able to reproduce our \nlinewidth data without reference to this effect. As a result, we interpret this fit as an upper \nbound on Eb. This is represented by the additional fits shown in Figure S7 within the \nSupplemental Material wherein we assume a lower temperature for the nominal peak in \nlinewidth occurring due to spin-magnon scattering that is experimentally preempted by the \nchange in magnetic anisotropy. These alternate fits agree with experimental observations \nat temperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>> ℏω) to populate their excited states, which is \nunlikely to happen. Hence, magnetic ordering of the paramagnetic spins would also \nenhance the suppression of spin-magnon scattering, resulting in the sharp linewidth \nsuppression for T < 9 K. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 12 When considering the expected behavior as the temperature is further reduced below \n5 K, as would be the case for many applications in QISE, it is useful to consider recent \nmilliKelvin-range measurements of YIG films [40]. That work confirms the expected \ncontinued narrowing down to 500 mK followed by a modest increase from 500 mK down \nto 20 mK, for an overall line narrowing of roughly a factor of 2. The model of scattering \nfrom TLFs described above is consistent with this result in YIG if one supposes a second \npopulation of TLFs that are dipole coupled to the magnons rather than exchange coupled , \nfor example dilute magnetic impurities in the substrate or environment. We note that \nextending this model into V[TCNE] x requires taking into account: i) the substantial \ndifference in structure and chemistry between V[TCNE] x and YIG, and ii) the fact that Ms. \nin V[TCNE] x is roughly 20 times smaller than in YIG. The former consideration indicates \nthat the presence of these dipole coupled TLFs need not correlate between the two systems, \nwhile the latter predicts that any relaxation associated with their presence should be \nreduced by a factor of 20 from Ref. [40]. As a result, the overall factor of 2 decrease in \nlinewidth observed in YIG between temperatures of 5 K and 20 mK should be taken as an \nextremely conservative lower bound on the performance of V[TCNE] x. Given that the \nlinewidth in V[TCNE] x at 5 K is already on par with its room temperature value, these \nresults firmly establish the suitability for this material for applications in quantum \nmagnonics and related aspects of QISE. \nIn conclusion, this work presents the first systematic study of the magnetization \ndynamics of V[TCNE] x at low temperatures. A strong variation in resonance frequency and \nanisotropy with temperature is observed , and attributed to a temperature-dependent strain \narising from the mismatch in thermal expansion coefficients between V[TCNE] x films and \ntheir sapphire substrates. The resonance linewidth of these films is found to increase with \ndecreasing temperature up to a maximum value of 15 G (roughly 9 times the room-\ntemperature value) and is well fit by a model based on magnon scattering from \nparamagnetic defect spins. At 5 K the magnetic anisotropy reverts to in-plane, coinciding \nwith a nearly complete recovery of the resonance linewidth to room-temperature values; \nquantitative modeling suggests the linewidth behavior arises from scattering from \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 13 paramagnetic defect spins that is suppressed at very low-temperature. This suppression of \nspin-magnon scattering is expected to strengthen as temperature is further decreased into \nthe milli-Kelvin range due to freeze-out of thermal magnons and phonons, providing a \ncompelling case for the utility of V[TCNE] x for low-temperature microwave applications, \nsuch as those emerging in the field of quantum information science and technology. \n \nAcknowledgements: The authors would like to thank A. Franson for providing a \nsoftware suite for fitting FMR spectra as well as general fitting assistance, and G. Fuchs \nfor fruitful discussions . The work presented in the main text, both experiment and theory, \nwas primarily supported by the U.S. Department of Energy, Office of Basic Energy \nSciences, under Award Number DE-SC0019250. S. Kurfman was supported by NSF \nEFMA-1741666 and grew V[TCNE] x calibration samples used for preliminary \nmeasurements not explicitly included in this paper . Work on V[MeTCEC] x presented in \nthe supplementary material was performed by M. Chilcote and Y. Lu with the support of \nNSF Grant No. DMR- 1741666. \n \nData availability statement: See supplementary material at URL will be inserted by AIP \nPublishing for datasets pertaining to temperature-dependent anisotropy of V[MeTCEC] x, \nmethod for extracting linewidth of V[TCNE] x from FMR scans and additional fits to \nexperimental data highlighting temperature dependence of V[TCNE] x linewidth. \n \nReferences: \n \n[1] A. Raveendran, M. T. Sebastian, and S. Raman, “Applications of Microwave \nMaterials: A Review” J. Electron. Mater. 48, 2601 (2019). \n[2] Ü. Özgür, Y. Alivov, and H. Morkoç, “Microwave ferrites, part 1: Fundamental \nproperties” J. Mater. Sci. Mater. Electron. 20, 789 (2009). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 14 [3] J. M. Silveyra, E. Ferrara, D. L. Huber, and T. C. 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However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 18 Risko, J. E. Anthony, A. Amassian and O. D. Jurchescu, “Crossover from band -\nlike to thermally activated charge transport in organic transistors due to strain-\ninduced traps” P NAS 114, 33 (2017). \n[35] H. Suhl, “Ferromagnetic Resonance in Nickel Ferrite Between One and Two \nKilomegacycles” Phys. Rev. 97, 555 (1955). \n[36] J. Smit and H. G. Beljers., “Ferromagnetic resonance absorption in BaFe 12O19” \nPhilips Res. Rep. 10, 113 (1955). \n[37] M. Chilcote, Y. Lu, and E. Johnston-Halperin, Organic-Based Magnetically \nOrdered Films (World Scientific, 2018). \n[38] J. K. Galt and E. G. Spencer, “Loss Mechanism in Spinel Ferrites ” Phys. Rev. 127, \n1572, 1962. \n[39] H. Maier-Flaig , S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. \n Huebl, and S. T. B. Goennenwein , “Temperature dependent damping of yttrium \n iron garnet spheres ” Phys. Rev. B 95, 214423 (2017). \n[40] S. Kosen, A. F. van Loo, D. A Bozhko, L. Mihalceanu, R. Gross, and A. D. \n Karenowska , “Microwave magnon damping in YIG films at millikelvin \n temperatures ” APL Mater. 7, 101120 (2019 ). \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 19 Figure Legends: \n \nFigure 1 \n \n(a) Schematic (planar view) of the CVD growth system; (b) FMR scan of V[TCNE] x \nthin film at 300 K with the applied magnetic field applied in the plane (IP) of the \nsample with 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. ΔH pp denotes the peak-\nto-peak linewidth measured as the difference between the positive and negative peak \npositions; (c) FMR line scans for in-plane field orientation at 300 K, 80 K and 40 K \nwith 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. \n \nFigure 2 \n \nAngle-dependent FMR spectra at temperatures of 300 K, 80 K and 40 K at different \nfield orientations with respect to the sample normal. Nominally the sample is rotated \nfrom 𝜃 = − 10𝑜 to 𝜃 = 100𝑜 in increments of 10𝑜, where 𝜃 = 90𝑜 and 𝜃 = 0𝑜 \nare in-plane and out- of-plane field orientations respectively. Angle corrections have \nbeen taken into account (through fitting with Eq. ( 3)) to reflect the actual rotation \nangles, denoted by the black arrows to the right of each of the temperature-labeled \npanels. \n \nFigure 3 \n \n(a) Schematic of the changes in anisotropy at 300 K and 40 K. 𝑯𝒂𝒑𝒑 denotes the \nexternal magnetic field, 𝑯𝒅𝒆𝒎𝒂𝒈 represents the demagnetizing field of the \nV[TCNE] x film and 𝑯𝒄𝒓𝒚𝒔𝒕𝒂𝒍 is the crystal-field anisotropy. It should be noted that \na finite thin-film has a (negligibly) small demagnetization field when the external \nfiled is applied in the plane since this is not a truly infinite film; (b) Resonance field \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 20 at different field orientations plotted against sample rotation angles for 300 K (open \ncircles) and 40 K (filled circles) and fits to Eq. (3) (dashed and solid line, \nrespectively) to extract the effective field 𝑯𝒆𝒇𝒇; (c) 𝑯𝒆𝒇𝒇 plotted against temperature \nranging from 300K – 5K. The inset shows the FMR lineshapes at 300 K and 5 K; \nfitting the data to extract the linewidth at FWHM gives 1.63 G and 2. 58 G \nrespectively , this shows that the two linewidths are indeed comparable with the \nlinewidth at 5 K only about 1.66 times larger than the room-temperature value . For \nboth (b) and (c), experimental errors are smaller than the point size. \n \nFigure 4 \n \nV[TCNE] x linewidth as a function of temperature (black points) and \ncorresponding curve fit (orange line) using Eq. (4). \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 21 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 1 H. Yusuf et al. \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 22 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2 H. Yusuf et al. \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 23 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3 H. Yusuf et al. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 24 \n \n \n \n \n \n \n \nFigure 4 H. Yusuf et al. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 25 \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 1 Supplementary Materials for “ Exploring a quantum -information -relevant \nmagnonic material: ultralow damping at low temperature in the organic \nferrimagnet V[TCNE] x” \nH. Yusuf *1, M. Chilcote *1,2, D. R. Candido3, S. Kurfman1, D. S. Cormode1, Y. Lu1, M. E. \nFlatté3, E. Johnston -Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210 \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853 \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242 \n \n* These authors contributed equally to this work. \n \n1. Temperature -dependent anisotropy of V [MeTCEC ]x \nHere, we investigate the magnetic properties of vanadium methyl tricyanoethylene \ncarboxylate V[MeTCEC ]x thin-films using temperature -dependent cavity ferromagnetic \nresonance (FMR). The MeTCEC ligand is similar to the TCNE described in the main text, \nand these results demonstrate that strain -dependent anisotropy is a general feature of this \nclass of metal -ligand materials. Figure S1a shows the molecular structures of both the \nTCNE molecule and the MeTCEC molecule discussed below. Figure S1b shows \ntemperature -dependent magnetization data for zero field -cooled (ZFC; open black squares) \nand field -cooled (FC; open red circles) measurements , and electron transport data (filled \nblack squares) collected for V[MeTCEC] x thin-films on the same temperature axis. Notice \nthat the maximum in the ZFC magnetization curve – sometimes referred to as the blocking \ntemperature[13,14] – corresponds to the rapid rise observed in the resistance data. This \nchange in electronic and magnetization properties has been associated with carrier freeze \nout and a magnetic phase transition in related materials such as magnatites, but in light of 2 the results presented in the main text we note \nthat a structural transition associated with \nincreased strain in the films may also play a role \nin these measurements. \nV[MeTCEC] x samples are deposited on \nAl2O3(0001) substrates using a previously \nreported synthesis and chemical vapor \ndeposition (CVD) growth process.[4,16] During \nthe deposition, argon gas carries the two \nprecursors, MeTCEC and V(CO) 6, into the \nreaction zone where V [MeTCEC ]x is deposited \nonto one or more substrates. The system \nemploys three independently temperature -\ncontrolled regions for the MeTCEC , V(CO) 6, \nand reaction zone with typical setpoints of \n55 °C, 10 °C, and 50 °C, respectively and with \ntypical flow rates for each precursor of 50 \nsccm. Sample growth, manipulation, and \nhandling is p erformed in an argon glovebox \n(O2 < 1.0 ppm; H 2O < 1.0 ppm). \nAfter growth, samples are mounted onto custom microwave compatible sample \nholders in the appropriate orientation, protected from undesired rotation, and flame -sealed \nin evacuated electron paramagnetic resonance (EPR) grade quartz tubes without exposure \nto air. When not being measured, the sealed samples are stored in a -55 °C freezer and are \nfound to be stable for weeks. \nFigure S2 shows four ferromagnetic resonance (FMR) spectra of V[MeTCEC] x \noriented both in plane (90 °; see inset to Fig. 3c) and out of plane (0 °) at 140 K and 80 K. \nThe FMR response of magnetic materials is sensitive to the local field environment of the \nFigure S1 (a) The molecular structures of \ntetracyanoethylene (TCNE) and \ntricyanoethylenecarboxylate (MeTCEC). (b) \nMagnetization vs. temperature curves for zero field -\ncooled (ZFC; open black squares) and zero field -\ncooled (FC; open red circles) measurements. On the \nsame temperature axis, resistance vs. temperature \ndata is shown for a V[MeTCEC]x thin -film (filled \nblack squares). The corresponding dependent -axis is \nshown on the right axis. Note the maximum in the \nmagnetization data corresponds to the rapid rise \nobserved in the resistance data. 3 sample and therefore allows for sensitive characterization of the anisotropy fields in \nV[MeTCEC] x. FMR measurements are performed using a Bruker electron paramagnetic \nresonance spectrometer setup for X -band measurements with 200 µW of applied \nmicrowave power and fitted with an Oxford Instruments ESR900 cryostat insert. The \ncryostat is cooled by flowing liquid nitrogen and operates at tempe ratures ranging from \n80 K to 300 K with better than 50 mK stability during FMR measurements. In standard \noperation, the microwave frequency of the spectrometer is tuned between 9 and 10 GHz \nfor optimal microwave cavity performance before the measurement, a nd then the frequency \nis fixed while the DC field is swe pt during the measurement. \nFigure S2a shows FMR spectra collected at 140 K for the magnetic field applied in \nplane (𝜃 = 90°) and out of plane ( 𝜃 = 0°). Consistent with prior FMR measurements of \norganic -based magnetic materials,[6,16,17] the center field associated with th e resonant \nfeature in the in -plane spectrum is at a lower field than that of the out -of-plane spectrum, \nand therefore the easy magnetization axis is oriented in the plane of the film. This easy -\naxis orientation is the expected outcome resulting from the sh ape anisotropy present in \nthin-film samples. Figure S2b also shows FMR spectra collected with the magnetic field \napplied in plane ( 𝜃 = 90° ) and out of plane (𝜃 = 0°). However, this data is collected at 80 K, \nfurther below the maximum in the V [MeTCEC ]x ZFC magnetization curve than the data \nshown in Fig . S2a. Surprisingly, the center field of the dominant resonance feature in the \nin-plane spectrum is at a higher field than that of the out -of-plane spectrum. This behavior \nseems to indicate that th e sample has an easy axis oriented out of the plane of the sample; \nthe spectra show signs of a switch in the magnetic easy axis from in plane to out of plane \nas it is cooled from 140 K to 80 K. 4 To investigate this behavior in greater \ndetail, angular -dependent data is collected in 10 ° \nincrements as the ap plied field is rotated from in \nplane ( 𝜃 = 90° ) to out of plane (𝜃 = 0°) of the \nsample. The data is shown in Figs . 3a and b for \n140 K and 80 K respectively. A gray dashed line \nis overlaid on the data to serve as a guide to the \neye. The field shifts shown in Figs . S3a and S3b \nare consistent with those shown in Fig . 2 above. \nFigure S3c shows the center fields extracted from \nthe two -angle series, emphasizing the magnitude \nof the change in the anisotropy. \nThis switch in the magnetic easy axis from \nin plane to out of plane present in the data \nsuggests the presence of an additional \ncontribution to the anisotropy beyond simply \nshape anisotropy. Previously, given the isotropic \nin-plane response of thin films at room \ntemperature, additional contributions to the anisotropy had been excluded. However, the \nresults here warrant the inclusion of an additional term 𝐻#, which is responsible for \ninducing perpendicular anisotropy in thin films. This phenomenology is consistent with the \nmeasurements of V[TCNE] x thin films presented in the main text. Following that \ndevelopment, the angular dependence of the FMR response for in plane to out of plane \nrotation of a thin -film sample can therefore be described by,[17–19] \n𝜔\n𝛾='(𝐻−4𝜋𝑀-..cos2𝜃)\t(𝐻−4𝜋𝑀-..\tcos2𝜃)\t\n='(𝐻−(4𝜋𝑀6−𝐻7)cos2𝜃)\t(𝐻−(4𝜋𝑀6−𝐻7)\tcos2𝜃), (1) \nFigure S2 (a) Single FMR line scans at 140 K for \na sample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield. (b) Single FMR line scans at 80 K for a \nsample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield. 5 where ω is the resonance fequency, γ is the gyromagnetic ratio, 𝐻 is the applied field, and \n𝜃 is the polar angle of the magnetization. The FMR resonance fields are more than an order \nof magnitude larger than the typical saturation field for V[TCNE] x, and therefore we have \nassumed that the magnetization is effectively p arallel to the applied magnetic field (i.e. \n𝜙≈𝜙> and 𝜃≈𝜃? where 𝜃, 𝜙 and 𝜃?, 𝜙? are the polar and azimuthal angles of the \nmagnetization 𝑀 and the applied bias field 𝐻, respectively). Also, note that the in -plane \nFMR response remains isotropi c, with the 𝜙 dependence dropping out: \n𝜔\n𝛾='𝐻\t(𝐻+4𝜋𝑀-..)\t\n='𝐻\t(𝐻+(4𝜋𝑀6−𝐻7)) \nThe data in Fig. S3c are then fit according to the dispersion relation in Eq. 1. The fits \nare shown as solid and dashed lines in Fig. 3c. The 140 K data yields an 𝐻eff=4π𝑀eff= \n15.4 Oe ± 0.1 Oe while fi tting to the 80 K data result in an 𝐻eff value of -28.3 Oe ± 1.0 Oe. \nThe negative value of 𝐻eff for the 80 K data means that 𝐻#>4π𝑀S and that the film has \nperpendicular magnetic anisotropy. Note that the magnetic energy landscape, and therefore \nthe angular dependence contained in Eq. 1, does not allow for the easy magnetization axis \nto take on an intermediate vector between in plane or out of plane for this set of anisotropy \nfields. This result also implies that prior measurements of the anisotropy of thin films are \nin fact measuring 4π𝑀eff rather than the bare 4π𝑀S as previously assumed.[17,20] However, (2) \n (𝜃=90°). 6 as with previous studies of uniform thin films, \nit is challenging to disentangle this form of \nanisotropy from 4π𝑀S, leading us to use the \nmore general 𝐻eff=4π𝑀eff. Temperature -\ndependent FMR studies combined with careful \nDC magnetization measurements provide a \npromising avenue to decoupling the two \nanisotropy fields. \nIn comparing the data shown in Fig. S3a \nand b, also note that at lower temperatures, the \nresonance response becomes markedly multi -\nmodal and appears to broaden. To investigate \nthis behavior in greater detail, FMR data is \ncollected over a range of temperatures with the \napplied field oriented in the plane of the \nsample. The dat a is shown in Fig. S4a. Note \nthe clear shift of the resonant features towards \nhigher field at lower temperatures as the in -\nplane orientation, which is the geometry being \nmeasured in this data set, changes from the \neasy magnetization axis to the hard \nmagnet ization axis. \nThe effective magnetization, 4π𝑀eff, \nextracted from the data shown in Fig. S3 \ncontains contributions from a perpendicular \nmagnetic anisotropy energy. This 𝐻# does not \narise from shape anisotropy in thin films and \nFigure S3 (a) Shows FMR spectra as the sample is \nrotated from in -plane ( 𝜃 = 90°) to out of plane \n(𝜃 = 0°) with respect to the externally applied \nmagnetic field at 140 K. (b) Shows the FMR \nspectra as the sample is rotated from in -plane \n(𝜃 = 90°) to out of plane ( 𝜃 = 0°) with respect to \nthe externally applied magnetic field at 80 K. (c) \nShows the extracted center fields from the angular \nseries shown in (a) and (b) with fits shown as solid \nand dashed lines. The inset shows the coordinate \nsystem with respect to the sam ple geometry. 0.1 Oe \n1.1 Oe 7 \n \n \n \nmust instead come from a crystal -field anisotropy wherein the local exchange vector \nacquires some anisotropy due to some combination of lattice symmetry and strain. Given \nthe large differences in the coefficients of thermal expansion for organic and in organic \nmaterials (often varying by an order of magnitude or more) , stain due to differential thermal \nexpansion at the interface between the substrate and organic -based materials is likely \ncreating an anisotropic strain field in the magnetic material. As the sample temperature is \nlowered, this strain field increases until 𝐻# becomes larger in magnitude than 4π𝑀S and \n4π𝑀eff takes on a negative value. The result is a magnet with an easy -axis out of plane as \nshown in Fig. S3b. \nWe note that qualitatively similar results were obtained for vanadium ethyl \ntricyanoethylene carb oxylate ( V[ETCEC ]x). V[ETCEC ]x is a third member of this class of \nmetal ligand ferrimagnets[23,24], supporting the thesis that strain -dependent anisotropy is a \ncommon feature of this class of materials. \n2. Method for extracting linewidth from FMR scans \nThe FMR scans are obtained through phase -sensitive detection, where in addition to the \nstatic DC magnet ic field the sample sees a sinusoidally modulated field component that is Figure S4 (a) Shows FMR spectra of V[MeTCEC ]x sample mounted in-plane \n(𝜃 = 90°) with respect to the externally applied magnetic field as a function of \ntemperature . (b) Shows the extracted peak -to-peak linewidths from the \ntemperature -dependent spectra shown in (a) \n 8 varied at the same frequency as the amplitude modulation of the microwaves reflected from \nthe cavity. If there is an EPR signal , that signal is converted into a sine wave whose \namplitude is proportional to the derivative of the signal (change in microwave power \nrelative to field modulation) and appears as the first derivative of a Lorentzian function . In \naddition, it should be noted that some FMR scans show multi peaks (for examp le, the 300 \nK scans shown in Figure 2 of the main text) and a possible reason for that could be \ninhomogeneous strain. As discussed in our main text, strain in our films is induced by \ndifference in thermal expansion coefficients between V[TCNE] x and the substrate. Given \nthat we have taken no special precautions to prevent it, we believe it is likely that this strain \nwill be inhomogeneous, resulting in regions of our sample with differing magnetic \nanisotropy, and therefore the potential for additi onal peaks in FMR spectra. It has been \nreported that strain -induced distortions can alter the local electronic and crystal -field \nenvironment by changing the orbital occupancy, tilt angle between neighboring spins[25] or \nmagnetocrystalline anisotropy[26,27], for instance, leading to local changes in magnetic \nanisotropy which result in the appearance of additional resonance peaks . \n Since the asymmetry of the FMR lineshape and the multi -peaks need to be \naccounted for , scans are not simply fit by the derivative of a symmetric Lorentzian . In \nphase -sensitive measurements the microwave electric field generates oscillati ng electric \ncurrents in the sample ; the oscillating magnetization due to the microwave magnetic field \nresults in oscilla ting angles between the current flow and magnetization, leading to local \nlattice distortions which may c ause the observed asymmetry in signal lineshape due to \ninhomogeneous broadening[28,29]. Another possible source of this asymmetry could be the \nresult of high cavity loading[30] and the resulting phase error introduced by the automatic \nfrequency controller of the EPR spectrometer when the sample is resonantly excited. This \nwarrants the inclusion of a dispersion or antisymmetric term that takes int o account this \nasymmetry, therefore the FMR scans are fitted to the sum of the derivative of an absorption \n(symmetric term) and dispersion (antisymmetric term ) from a Lorentzian . The derivative s \nhave the following form : (3) (4) 9 absorption derivative =\t−32\t√3\t𝐴\t𝐹𝑊𝐻 𝑀K(𝐵−𝐵M)\n9\t[FWHM \t2+4(𝐵−𝐵M)\t2]\t2 \n \ndispersion derivative =\t−4\t𝐷\t𝐹𝑊𝐻𝑀 \t(𝐵−𝐵M)\nFWHM \t2+4(𝐵−𝐵M)\t2 \nwhere FWHM is the full -width at half -max, A is the height of the absorption derivative, D \nis the height of the dispersion derivative, 𝐵M is the location of the resonance (center) field \nand B is the amplitude of the magnetic field that is being swept at each data point. \nTherefore, the resulting li ne shape depends on the relative contributions of these two terms . \nFor scans with an out -of-plane easy \naxis fitting with a single derivative sum \nprovides good agreement with the data \n(Figure S5). But for scans with in -plane \neasy axis, due to the appearance of a \nmodest satellite peak , obtaining a good fit \nto the data requires addition of up to three \nderivative sum s. For FMR scans in the \nrange 9 K – 80 K (out -of-plane easy axis \nbetween 9 K – 100 K and negligible \nanisotropy at 80 K) the date is well fit with a single derivate sum . On the other hand, fits \nfor scans in the high temperatures between 120 K – 300 K (in -plane easy axis) give good \nagreement with data when two derivative sums are used , a few requiring up to three \nderivative sums (Figure S6b). However, FMR scans at 5 K (in-plane easy axis) and 6 K \n(out-of-plane easy axis) mimic the high temperature fits by requiring two derivative sums. \nFor the purposes of this study, which explores the fundamental FMR mode, in scans \nshowing multiple peaks we focus on the contribution fr om the peak that persists to low Figure S5 Shows a single derivative fit to the \nFMR data collected at 2 2 K \n 10 temperatures. If we plot the individual Lorentzian \ncomponents of the FMR fit, we find that the first \ncomponent YL1 (component with the highest \noverall peak -to-peak magnitude) is present in all \nthe temperatures being consid ered in the range 300 \n– 5 K. Therefore, the linewidth date plotted against \ntemperatures in Figure 4 of the main text is the \nlinewidth at f ull width half max (FWHM) of YL1. \nIn Figure S6a it can be seen that fitting the \nFMR scan at 300 K with a single derivative sum \ndoes not provide a great fit to the data. However , \nfrom Figure S6b it becomes clear that fitting the \nsame data with the superposition of three \nderivative sums or components (each with their \ndistinct A, D and FWHM ) gives a decent fit. In \nFigure S6c the amplitude of each individual \ncomponent is plotted against magnetic field sweep \nrange to provide a visual understanding of how \neach component contributes to the overall FMR \nline shape. \n3. Temperature dependent linewidth \nThe V [TCNE ]x linewidth dependence on \ntemperature can be well explained from the \ninteraction between magnons and defects or Figure S6 (a) Shows a single derivative fit to the \nFMR data collected at 300 K. (b) Shows FMR \nscan at 300 K fitted to superposition of three \nLorentzian derivative sums. (c) Amplitude of \neach sum or component plotted against magnetic \nfield sweep range. YL1, YL2 and Y L3 are the \nfirst, second and third components respectively. \n 11 impurities in V [TCNE ]x. The \ndefects or impurities are \nconsidered to be a two -level spin \nsystem s. These experience spin -\nflip transitions excited by the \nannihilation of a uniform -magnon \nmode [31,32]. This process \nintroduces a finite magnon \nlifetime, which in turn leads to the \nlinewidth expression Eq. (4) in the \nmain text. In Fig. S7, we use four \ndiffere nt parameter sets to fit the \nhigh temperature experimental \ndata using Eq. (4). All the different \nsets yield a good fitting for T> 9 K, \nalthough the smaller the E b, the \nsmaller the nominal peak in \nlinewidth. As discussed in the main text, this imposes an upp er bound on E b ~ 1meV. \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out -of-plane easy axis to an in -plane easy axis. This change \nin magnetic anisotropy has the potential to have a substantial impact on spin -magnon \nscattering efficiency. For example, this change will result in a shift of the energy of the \nmagnon bands (see Eq. 1 in main text ), and if this change involves a commensurate change \nin the strain there will also be a modifi cation to the spin -orbit coupling and exchange \nparameters at the paramagnetic defects. This is represented by the fits shown in Fig. S 7 \nwherein we assume a lower temperature for the nominal peak in linewidth occurring due \nto spin -magnon scattering that is experimentally preempted by the change in magnetic \nanisotropy. As a result, we interpret th e fit in Fig. 4 of the main text as an upper bound on \nFigure S7 (a) (b), (c) and (d) show V [TCNE ]x linewidth as a \nfunction of temperature and the corresponding fit curves using \nfitting parameters of Eq. (4) \n 12 Eb. The alternate fits presented in Fig. S7 agree with experimental observations at \ntemperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>> ℏω) to populate their excited states, which is \nunlikely to happen. Hence, magnetic ordering of the paramagnetic spins would also \nenhance the suppression of spin -magnon scattering, resulting in the sharp linewidth \nsuppression for T < 9 K. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 13 References \n[1] C. J. Brabec, Sol. Energy Mater. Sol. Cells 2004 , 83, 273. \n[2] H. Shirakawa, E. J. Louis, A. G. MacDiarmid, C. K. Chiang, A. J. Heeger, J. Chem. \nSoc. Chem. Commun. 1977 , 578. \n[3] C. W. Tang, S. A. Vanslyke, Appl. Phys. Lett. 1987 , 51, 913. \n[4] Y. Lu, M. Harberts, C. -Y. Y. Kao, H. Yu, E. Johnston -Halperin, A. J. Epstein, Adv. \nMater. 2014 , 26, 7632. \n[5] Y. Lu, H. Yu, M. Harberts, A. J. Epstein, E. Johnston -Halperin, J. Mater. Chem. C \n2015 , 3, 7363. \n[6] Y. Lu, H. Yu, M. Harberts, A. J. Epstein, E. Johnston -Halperin, RSC Adv. 2015 , 5, \n82271. \n[7] J. L. Arthur, S. H. Lapidus, C. E. Moore, A. L. Rheingold, P. W. Stephens, J. S. \nMiller, Adv. Funct. Mater. 2012 , 22, 1802. \n[8] J. P. Fitzgeral d, B. B. Kaul, G. T. 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Re v. 1955 , 97, 555. \n[20] N. Zhu, X. Zhang, I. H. Froning, M. E. Flatté, E. Johnston -Halperin, H. X. Tang, \nAppl. Phys. Lett. 2016 , 109, 082402. \n[21] Y. Li, V. Coropceanu, J. -L. Brédas, J. Phys. Chem. Lett. 2012 , 3, 3325. \n[22] W.-C. Wang, C. -H. Wang, J. -Y. Lin, J. Hwang, IEEE Trans. Electron Devices 2012 , \n59, 225. \n[23] Y. Lu, H. Yu, M. Harberts, A.J. Epstein and E. Johnston -Halperin, RSC Adv. 2016, \n 5, 82271 . 15 [24] Y. Lu, H. Yu, M. Harberts, A.J. Epstein and E. Johnston -Halperin, J Mater. Chem. \n C 2015 , 3, 7363 . \n[25] Y. Tokura, Colossal magneto -resistive oxides, Advances in condensed matter \n sciences, v. 2 (Amsterdam, The Netherlands : Gordon and Beach Science \n Publishers, 2000 ). \n[26] K. Steenbeck and R. Hiergeist, Appl. Phys. Lett. 75, 1778 (1999) . \n[27] F. Tsui and M. C. Smoak, Appl. Phys. Lett. 76, 2421 (2000). \n[28] Z. Celinski and B. Heinrich , Journal of Applied Physics 1991 , 70, 5935 . \n[29] Y. Li, F. Zeng, S. -L. Zhang, H. Shin, H. Saglam, V. Karakas, O. Ozatay, J. E. \n Pearson, O. G. Heinonen, Y. Wu, A. Hoffman and W. Zhang , Phys. Rev. Lett. \n 2019 , 122, 117203 . \n [30] I. B. Goldberg and H. R. Crowe, Anal. Chem. 1977 , 49, 9, 1353 -1357. \n [31] M. Sparks, Ferromagnetic -Relaxation Theory (McGraw Hill, New York, 1964 ). \n [32] P. E. Seiden, Phys. Rev. 1964 , 133, A728. \n \n \n " }, { "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": "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": "1506.06585v3.Exchange_scattering_as_the_driving_force_for_ultrafast_all_optical_and_bias_controlled_reversal_in_ferrimagnetic_metallic_structures.pdf", "content": "arXiv:1506.06585v3 [cond-mat.str-el] 28 Feb 2016Exchange scattering as the driving force for ultrafast all- optical\nand bias-controlled reversal in ferrimagnetic metallic st ructures\nA. M. Kalashnikova and V. I. Kozub∗\nIoffe Physical-Technical Institute of the Russian Academy of Sciences, 194021 St. Petersburg, Russia\n(Dated: today)\nExperimentally observed ultrafast all-optical magnetiza tion reversal in ferrimagnetic metals and\nheterostructures based on antiferromagnetically coupled ferromagnetic d−andf−metallic layers\nreliesonintricateenergyandangularmomentumflowbetween electrons, phononsandspins. Herewe\ntreat the problem of angular momentum transfer in the course of ultrafast laser-induced dynamics in\na ferrimagnetic metallic system using microscopical appro ach based on the system of rate equations.\nWe show that the magnetization reversal is supported by a cou pling of d−andf−subsystems to\ndelocalized s−orp−electrons. The latter can transfer spin between the two subs ystems in an\nincoherent way owing to the ( s;p)−(d;f) exchange scattering. Since the effect of the external\nexcitation in this process is reduced to the transient heati ng of the mobile electron subsystem, we\nalso discuss possibility to trigger the magnetization reve rsal by applying a voltage bias pulse to\nantiferromagnetically coupled metallic ferromagnetic la yers embedded in point contact or tunneling\nstructures. Weargue thatsuchdevices allow controlling re versal withhigh accuracy. Wealsosuggest\nto use the anomalous Hall effect to register the reversal, thu s playing a role of reading probes.\nPACS numbers: 75.78.Jp, 75.40.Gb, 75.50.Gg\nI. INTRODUCTION\nThe possibility of ultrafast control of the magnetic\nstate of nanostructures is an important constituent of\nferromagnet-based spintronics. Due to the problem of\nnon-localityand difficulty to createstrongyet shortmag-\nnetic field pulse,1–3the natural idea to use the latter is\nbecoming incompatible with the request for further in-\ncrease of storage densities and operation speed of novel\nspintronic devices.4Thus a great enthusiasm arose fol-\nlowingthe proposal5,6to usespininjection forcontrolling\nthe magnetization state of ferromagnetic specimen with\nthe help of an applied voltage. Later such a possibil-\nity was profoundly studied both theoretically and exper-\nimentally (for the critical review see e.g. [7]). However,\ntwo important obstacles were found. First, the switch-\ning time of the magnetization reversal by spin injection\nis defined by magnetization precession damping and is\nrather long (around ∼10−9s). Second, relatively large\ncurrents required for effective switching inevitably lead\nto unreasonable heat losses.\nThus a great attention8was attracted by recent exper-\niments demonstrating extremely fast ( ∼10−12s) magne-\ntization reversal triggered by a single femtosecond laser\npulse in ferrimagnetic metallic rare-earth (RE) - tran-\nsition metal (TM) alloy GdFeCo.9,10Very recently, ex-\nperimental observation of ultrafast laser-induced switch-\ning was reported in a variety of the engineered ferrimag-\nnetic structures, showing that this process is not specific\nfor the RE-TM single phase alloys, but can be realized\nin exchange coupled RE-TM multilayers, as well as het-\nerostructures comprised by two TM layers antiferromag-\nnetically coupled through 0.4nm nonmagnetic metallic\ninterlayers.11\nMost importantly, experimental studies have demon-\nstrated that the all-optical reversal of magnetizationis not precessional and relies solely on subpicosec-\nond quenching of the magnetizations of RE and TM\nsublattices.10Furthermore, as it was revealed by the\ntime-resolved X-ray experiments and supported by the\natomistic simulations,12the laser-induced quenching of\nthe TM and RE sublattice magnetizations occurs on dis-\ntinct time-scales. As a result, the magnetization rever-\nsal proceeds via non-equilibrium transient ferromagnetic\nstate.13Suchnon-equilibriumdynamicsallowsforthede-\nterministic magnetization reversal, without any need for\nother stimuli defining the magnetization direction. We\nnote, that circularly polarized laser pulse polarization\nwas mostly used for triggering the all-optical magnetiza-\ntion reversal.9–11,14–18However, it has recently been pro-\nposed that the difference in the magnetization reversal\nprocesses triggered by left- and right-handed laser pulses\ncan be explained to a large extent by a magnetic circular\ndichroism possessed by the studied samples.19\nNaturally, microscopical mechanism underlying such\nunconventional response of magnetization of a ferrimag-\nnetic metallic system to a femtosecond laser pulse is\nthe subject of intense discussion nowadays. In Refs.\n13,14,20–22 atomistic and multiscale calculations based\non the Landau-Lifshitz-Bloch equation23for the ensem-\nble of the exchange-coupled spins have been successfully\nemployed to account for the main features of the all-\noptical magnetization reversal. This approach allowed\ndescribing the all-optical reversal in both single phase al-\nloys and exchange-coupled multilayers.24In Ref.25 com-\nprehensive phenomenological model based on the On-\nsager’s relations suggested by Baryakhtar26was devel-\noped in order to account for the reversal via transient\nferromagnetic-like state. This theoretical study intro-\nduced the exchange-dominated regime of laser-induced\ndynamics in a ferrimagnet, which allows the reversal of\nmagnetization solely due to the ultrafast heating. This2\nwork highlighted the importance of the angular momen-\ntum exchange between the sublattices. Understanding\nmicroscopical processes responsible for this angular mo-\nmentumexchangebecame, therefore,thekeyissueinthe-\noretical studies of the laser-induced magnetization rever-\nsal. In Ref.22the two-magnonbound statewasproposed\nto mediate the angular momentum transfer. In Ref. 27\ndissipationless energy and angular momentum exchange\nbetween TM and RE sublattices mediated by 5 d-4fex-\nchange coupling in RE ions was analyzed as the driv-\ning mechanism for the all-optical magnetization rever-\nsal. The role of the exchange electron-electron scattering\nin the magnetization reversal was recently discussed in\nRef.28.\nHere we consider the problem of the angular momen-\ntum exchange between two nonequivalent magnetic sub-\nlattices in a metal in the frameworks of a general micro-\nscopic model based on the rate equations. This model\ndescribes evolution of the occupation numbers of two\ndifferent ferromagneticsublattices coupled antiferromag-\nnetically. They are formed either by nearly localized d-\nelectrons in a case of a transition metal sublattice or lo-\ncalizedf-electrons in a case of a rare-earth metal sublat-\ntice. The coupling between the sublattices is mediated\nby delocalized s- orp-electrons. In the frameworks of\nthis model we demonstrate that the spin exchange be-\ntween the localized ferromagnetic subsystems is medi-\nated by delocalized electrons and is triggered by ultra-\nfast increase of the temperature of the latter. This leads\nto the switching of the net magnetization without any\nadditional stimuli, such a external magnetic field or cir-\ncular polarization of light. Importantly, the model we\npropose is not restricted to the case of RE-TM alloys or\nheterostructures, but is also applicable for the case of the\nstructures composed by two different transition metals.\nFurthermore, since the laser pulse only plays a role\nof the stimulus supplying energy to the delocalized elec-\ntrons, we consider feasibility of the magnetization re-\nversal triggered by a short pulse of external electric\nbias in the ferrimagnetic system either imbedded into\nmetallic point contact or sandwiched between two tun-\nnel junctions. We show that, first, this alternative ap-\nproachfor driving the magnetic system into the strongly-\nnonequilibrium state enables one to tune the demagne-\ntization times by variation of the bias. This is impor-\ntant since the reversal depends on a delicate interplay\nbetween demagnetization time and cooling time of the\nmobile electrons. Second, in this case one deals with a\ncompact nanoscale device compatible with existing spin-\ntronics applications.\nThe paper is organized as follows. In Section II we\nintroduce the microscopical model describing the evolu-\ntion of the ferrimagnetic metallic system in response to\nthe rapid increase of the delocalized electrons tempera-\nture. In Section III we discuss the applicability of the\nproposed model to the process of the all-optical reversal\ndemonstrated experimentally. In Section IV we consider\nthe electric bias induced reversal either in point contactsor in tunnelling structures.\nII. THEORETICAL MODEL OF\nMAGNETIZATION REVERSAL IN A METALLIC\nFERRIMAGNET\nA. Model of a metallic ferrimagnet\nWe start our consideration by introducing three inter-\nacting electronic subsystems (Fig.1). We denote two fer-\nromagnetic sublattices as A and B. For a sake of clarity\nA is the transition metal d-electrons subsystem, while B\nis either d-electrons or the rare-earth metal f-electrons\nsubsystem. The A and B subsystems could comprise ei-\nther single phase alloy or exchange-coupled layers. The\nthird subsystem is the mobile s- orp-electrons(e). In the\nstructures where both A and B sublattices are based on\nthe TM elements these mobile electrons do not give an\nimportantcontributiontotheferromagnetismofeitherof\nA and B sublattices. By contrast, mobile electrons sup-\nport ferromagnetism of the rare-earth sublattice B via\nthe indirect exchange with the ferromagnetic TM sublat-\ntice A. In our model these are the mobile electrons that\nplay a decisive role in energy and angular momentum ex-\nchange within the sample. In particular, we assume that\ntheir coupling to d- andf- electrons controls the energy\ndistribution in the corresponding subsystems. For a sake\nof convenience in the following discussion we consider s-\nelectronsasthemobileelectrons,whilealltheconclusions\nare valid for the case of mobile p-electrons as well.\nFIG. 1: (Color online) The A, B and s-subsystems comprising\nferrimagnetic metal. ∆0\nA,Bare the exchange splittings in the\nequilibrium. NA,B\n↑↓,n↑↓are the occupation numbers of the\nsubsystems A, B and e. Subscripts ↑,↓denote up and down\nspin states with respect to the initial net magnetization di rec-\ntion.γexare the exchange constants between corresponding\nsubsystems\nForasakeofsimplicityweconsiderthespinsubsystems\nAandBcharacterized by pronounced peaks in energy\ndistribution. Both subsystem are assumed to be strong\nferromagnets and, thus, the exchange splittings ∆0\nAand\n∆0\nBforthesesubsystemsarelargerthanthe widths ofthe3\ncorresponding energy peaks, as shown in Fig.1. ∆0\nA,B\ndescribe exchange between neighboring ions comprising\nthe subsystems A and B, and are equal to the averageex-\nchangecouplingscorrespondingtothe Weissfield. Under\nassumptionofAandBbeingstrongferromagnets,theoc-\ncupation numbers NA\n↓,NB\n↑= 0, as illustrated in Fig.1.\nHere the subscripts ↑,↓correspond to the up and down\nspin directions with respect to the net magnetization di-\nrection. Here we consider the case, when the magnetiza-\ntion of the A subsystem is larger than the magnetization\nof the B subsystem.\nB. Rate equations for the ferrimagnetic metallic\nsystem\nThe excitation of the described above system is intro-\nducedin ourmodelasarapidincreaseofthe temperature\nTeof the mobile electrons. We consider instantaneous\nincrease of Teat the time t= 0 followed by the slow de-\ncrease, governed by the processes specific for the system\nin consideration.\nIn order to simulate response of the A and B subsys-tems to the rapid increase of the temperature of the mo-\nbile electrons we exploit the fact that at temperatures\nclose to the critical ones the suppression of ferromag-\nnetism of the TM sublattice A occurs mainly via the\nStoner excitations which are created by a transfer of\nthed-electron from a majority band to a minority band\n(Fig.2(b)). Such a transfer leads to a decrease of the\nsubsystem magnetization and is naturally related to an\nenergy and angular momentum cost which is supplied by\nthe mobile s-electrons. The decrease of magnetization\nof the A subsystem is compensated by the spin reversal\n↓→↑of thes-electron mediating the excitation. If the\ns-electrons are simultaneously coupled to both A and B\nsubsystems, they can effectively lead to spin exchange\nbetween A and B subsystems. Thus there is the indirect\ninteraction between total spins of A and B subsystems,\nwhich is spin conserving in a natural way.\nTo describe this interaction we write down the rate\nequationsfortheoccupationnumbersofsitescorrespond-\ning to subsystems A ( NA\n↓↑), B (NB\n↓↑) and occupation\nnumbers of s-electrons states ( n↑↓) participating in the\nexchange scattering:\ndNA\n↓↑\ndt=−/integraldisplaydε\nTe/bracketleftbigg1\nτAe/parenleftbig\nn↑↓(1−n′\n↓↑)NA\n↓↑(1−NA\n↑↓)−n↓↑(1−n′\n↑↓)NA\n↑↓(1−NA\n↓↑)/parenrightbig/bracketrightbigg\n; (1)\ndNB\n↓↑\ndt=−/integraldisplaydε\nTe/bracketleftbigg1\nτBe/parenleftbig\nn↑↓(1−n′\n↓↑)NB\n↓↑(1−NB\n↑↓)−n↓↑(1−n′\n↑↓)NB\n↑↓(1−NB\n↓↑)/parenrightbig/bracketrightbigg\n; (2)\ndn↑↓\ndt=−[n↑↓(1−n′\n↓↑)/parenleftbigg1\nτeANA\n↓↑(1−NA\n↑↓)+1\nτeBNB\n↓↑(1−NB\n↑↓)/parenrightbigg\n+n↓↑(1−n′\n↑↓)/parenleftbigg1\nτeANA\n↑↓(1−NA\n↓↑)+1\nτeBNB\n↑↓(1−NB\n↓↑)/parenrightbigg\n]+1\nτs(n↓↑−n↑↓). (3)\nThe r.h.s of the Eqs.1-3 are the standard collision inte-\ngralsoftheBoltzmannequationsdescribingthe exchange\nscattering between three components of the electronic\nsystem. Each equation corresponds to the pair of the in-\nteracting subsystems. nandn′are the functions of ener-\ngiesεandε′, respectively. The values of εandε′are con-\nnected by the energy conservation relations and include,\nin particular, the exchange splittings ∆ A,Bwithin the\nsubsystems A and B. Here we take into account that, in\ncourse of demagnetization the exchange splittings ∆ A,B\ndiffer from the equilibrium values ∆0\nA,B.\nIn the Eqs.1,2 integration is performed only over the\nenergy of the delocalized electrons energies. The inte-\ngration over the energies of the states within the A- and\nB-subsystems distributions is omitted under the assump-\ntion that A and B subsystems are the strong ferromag-\nnets. By contrast, the subsystem of mobile s-electrons\nhas a broad energy distribution with Fermi energy much\nlarger than Te. Nevertheless, Eq.3 is written only forthoses-electrons which are effectively coupled to A and\nB subsystems and their energy is within the band of a\nwidth∼TearoundtheFermilevel. The latterfact means\nthat the phase volume of the s-electrons involved in the\nexchange scattering is smaller than the phase volume of\neither of the ferromagnetic subsystems A and B. Fur-\nthermore, it allows to assume the s-electrons densities of\nstates within corresponding energy interval to be nearly\nconstant. In addition, in what follows we do not not\ntake into account the energy dependencies of nas well as\nenergy dependencies of relaxation times τ.\nτA,B;eare the effective electron-electron relaxation\ntimes, characterizing A−sandB−sexchange scat-\ntering processes. The factors 1 /Teτ(A,B)ein Eqs.1,2 are\nthe probabilities of the exchange scattering involving s-\nelectronsnormalizedwithrespecttoenergy ε. Thevalues\n1/τ(A,B)eareofthe orderoftotalscatteringprobabilities,\nsince the integration over εis only within the energy in-\nterval∼Te.4\nThe values 1 /τe(A,B)describe the probabilities of ex-\nchange scattering of s−electrons by AandBsubsys-\ntems. The effective exchange scattering probability of\ns-electrons including both relaxation channels is given\nby 1/τee= 1/τeA+1/τeB.\nCharacteristic time τsdescribes the angular momen-\ntum exchange between the s-electrons and the external\nbath.\nEqs.1-3 take into account spin balance within the sys-\ntem only and thus do not include processesleading to the\nthermal equilibrium. We assume that the characteristic\ntimes for electron-electron processes, responsible for the\nthermalization within considered subsystem are smaller\nthan spin relaxation times. The evolution of the temper-\natureTefollowingthe instantaneousincrease, is governed\nbyelectron-phononprocessesandheatwithdrawal,which\nis specific for different systems. These processes are con-\nsidered to be slower than the introduced above charac-\nteristic times τresponsible for the angular momentum\ntransfer.\nFinally, we stress that in this model the effect of direct\nA-A and A-B or indirect B-B exchange couplings is not\nincluded. The processes involving these interactions are\nexpected to be related to spin reversals leading to ferro-\nmagnetic or antiferromagnetic ordering in corresponding\nsubsystems. We believe that at highly-nonequilibrium\nstate the spin-conserving processes considered above are\nmore efficient and fast than the ones including spin dis-\nsipation and, thus, are the dominant mechanism of the\nspin-redistribution. In order to set the criterion for a\nrange of the mobile electrons temperatures Tewhere the\nexchange scattering governs the evolution of a particular\nferromagnetic system subsystem, we take into account\nthat this process is effective only for Te>∆0\nA,B. Con-\nsequently, we introduce effective partial critical tempera-\ntures ofthe A and B subsystems, which arerelated to the\ncorrespondingexchangesplittings TA;B\nC∼∆0\nA,B. Strictly\nspeaking the concept of critical temperature holds only\nforthermodynamicallimit. Intheequilibriumthecritical\ntemperatures of A and B sublattices coupled via mobile\nelectrons should be considered equal, in agreement with\nthe experimental data.29In the strongly non-equilibrium\nstate of the medium, if the rate of electron-electron in-\nelastic scattering within given subsystem is higher than\nthe rate of the corresponding sublattice magnetization\nevolution, one can introduce the partial electron temper-\nature. Electron-electron inelastic scattering, responsible\nfor the electron thermalization is, typically, in the range\nof 50-300fs.30–33In the non-equilibrium state the A and\nB subsystems can be also considered as partly decoupled\nfrom each other and, therefore, we can discriminate be-\ntween partial values of the critical temperatures TA;B\nCof\nthese subsystems. We will consider the Curie tempera-\ntures for uncoupled A (pure TM metal) and B (pure TM\nor RE metals) systems as the partial critical tempera-\nturesTA\nCandTB\nC, respectively.C. Exchange scattering probabilities and\nrelaxation times\nFrom the Eqs.1-3 it follows that the efficiency of spin\ndecay within a given subsystem is related to the purely\nspin-conserving s−dors−fexchange scattering and\nis controlled by relaxation rates 1 /τ(A,B)e. Correspond-\ningly, the decay rate is higher for a subsystem where\nthis parameter is larger, i.e. the exchange coupling with\nmobiles-electrons is stronger. One expects that the\nexchange scattering between s-electrons and the corre-\nsponding ferromagnetic subsystem is more pronounced if\nthe latter possesses strong exchange interaction within\nitself. Although the evolution of magnetization in any of\nthe subsystems includes not only spin transferbetween A\nor B subsystem and s-subsystem, controlled by 1 /τ(A,B)e\nbutalsothespindecaywithin s-subsystem(1 /τs), weex-\npect that it is the difference between the values of τ(A,B)e\nthat leads to the distinct times of spin decay within A\nand B subsystems.\nNumerical estimation of the relaxation rates 1 /τ(A,B)e\nand 1/τe(A,B)requires knowledge of the electron spec-\ntrum of all involved systems. Here we use simplified es-\ntimates. As it is known, for electron-electron scattering\nin standard metals relation 1 /τ∼T2\ne/εF¯hholds, where\nεFis the Fermi energy (see, e.g., Ref.34). This expres-\nsionmakesuseofthe momentum conservationlawforthe\nelectron system where Te<< εF. In the case considered\nherethe situation is differentsince the electronscattering\ntakes place between two different electronic subsystems,\nwith one of them (A or B) characterized by very narrow\nenergy band, and, thus, the effective mass of electrons in\nwhich is much larger than in the subsystem of the mo-\nbile electrons. In this case the momentum conservation\nlaw can be disregarded and thus the electron-electron\nscattering time has a form close to the one for electron\nscattering by impurities. Therefore, one can estimate the\nrelaxationtime as1 /τ∼σneffvrwhereσisthescattering\ncross-section, neffistheeffective concentrationofscatter-\ners, and vris the relative velocity of scattered electron\nwith respect to the scatterer (see, e.g., Ref.34). Tak-\ning these considerations into account, one obtains for the\ncharacteristic time of the exchange scattering of the A\nor B subsystem electrons by the s−electrons with the\nspherical Fermi surface:\n1\nτ(A,B)s∼γ(A,B)e\nex¯h2nTe\nε3/2\nFm3/2, (4)\nwherenis the concentration of the s-electrons, εFis the\nFermi energy of the s-electrons, mis their mass. γ(A,B)e\nex\nis the dimensionless A−sandB−sexchange constant,\nwhich absorbs the dependence of the exchange scattering\nprobability on the exchange splittings ∆0\nA,B. Here the\nestimates σ(A,B)e∼¯h2/mεF,vr∼ε1/2\nF/m1/2,neff∼\nnTe/εFare used.\nInits turn, forthe probabilityofexchangescatteringof\ns−electrons by the ones of A or B subsystems we obtain5\nfor the case Te>∆A,B:\n1\nτs(A,B)∼γ(A,B)e\nex¯h2NA,B\nε1/2\nFm3/2, (5)\nwhereNA,Bis the concentration of the subsystem A or\nB. Thus, according to these expressions, the values of\nτ(A,B)sandτs(A,B)are different for the same value of the\nexchange scattering crossections and exchange constants\nbetween the A(B) subsystem and the mobile electrons.\nAs Eqs.4,5 demonstrate, the relaxation probabilities\n1/τ(A,B)eand 1/τee= 1/τeA+ 1/τeBpossess different\ntemperature dependencies. For the exchange scattering\ntimeτ(A,B)ewe have1 /τ(A,B),e∝Te. The exchangescat-\ntering time τeeis, in turn, temperature -independent.\nThis is in contrast to standard electron-electron scatter-\ning where 1 /τ∼¯hT2\ne/εF. The later relation holds for\nelectron-electron scattering between s-electrons respon-\nsible for thermalization at the initial stage.\nD. Evolution of the ferrimagnetic system with\nnearly quenched sublattice magnetizations\nUsing Eqs.1-3 we consider the evolution of the mag-\nnetizations of the A and B sublattices triggered by an\ninstantaneous increase of the temperature of the mobile\nelectrons in the metallic ferrimagnet. If one would deal\nwith asingle ferromagneticsubsystem, e.g. A, coupled to\nthe mobile electrons, the rapid increase Te> TC\nAwould\ntrigger the decrease of the sublattice magnetization, i.e.\ndecrease of NA\n↑and increase NA\n↓due to spin transfer\nto the mobile electrons via exchange scattering and the\nfollowing spin decay within s-subsystems. This would\nfinally lead to total suppression of magnetization.\nIt is important to stress that the values of ∆ A,Bde-\ncrease in the course of the demagnetization process. In\nparticular, when the averagemagnetization of the A sub-\nlattice tends to zero, the same holds to the average ex-\nchange fields. As a result, in the mean field approxima-\ntion ∆ Atends to zero as well. However, locally, given\nmagnetic ions from the A sublattice is exchange coupledto the nearest neighboring ions. The distribution of local\nmagnetic moments does not possess any long range or-\nder, and the sum magnetization of the neighboring ions\nfluctuates depending on the spatial position. Since in the\nequilibrium ∆0\nA∝ NA, whereNAis the number of neigh-\nbours, the mean exchange splitting ∆ A∼∆0\nA/(N)1/2\nA\nwhen the average magnetization of the A sublattice is\nzero.\nFor the antiferromagnetically coupled A and B subsys-\ntems the evolution of their magnetizations is somewhat\nmoredelicate than in the caseofthe single sublattice sys-\ntem. The exchange scattering with the mobile electrons\nleads to the redistribution of the total spin between the\nsubsystems according to the factor τA,e/τB,e. The sup-\npression of total magnetization would occur only via the\nspin non-conserving process, which is described by the\nterm (n↓↑−n↑↓)/τsin Eq.3. Without this term the total\nsuppression of magnetization in the system is possible\nonly if the magnetizations of A and B subsystems are\nequal initially.\nIn order to illustrate the evolution of the magneti-\nzations of the A and B subsystems which follows the\ninstantaneous increase of the s−electrons temperature\nTe> TA,B\nCwe consider the situation when the creation\nof Stoner excitations completely suppresses magnetiza-\ntion of the one of the subsystems. We consider the case,\nwhenτA,e< τB,e, which is consistent with the nota-\ntions we accommodated, i.e. A and B subsystems are\nformed by d−andf−electrons, respectively. Then mag-\nnetization of the A sublattice is quenxhed, NA\n↑=NA\n↓,\nwhile the magnetization of the B subsystem remains fi-\nnite,NB\n↓> NB\n↑(Fig.2(c)). In this case the rate equation\nfor the A subsystem takes a form:\ndNA\n↓↑\ndt|0=1\n4τAe(n↓↑−n↑↓). (6)\nHere the notation |0means the configuration where\nNA\n↓=NA\n↑. The further calculations give that at NA\n↑≃\nNA\n↓≃1/2,|n↑−n↓|<< n ↑the rate equation of the\ns-electrons takes a form:\ndn↑↓\ndt=1\nτs(n↓↑−n↑↓)+1\n4τee(n↓↑−n↑↓)+1\n4τBe(NB\n↑↓−NB\n↓↑)+1\n4τAe(NA\n↑↓−NA\n↓↑), (7)\nHere we assumed n↑∼n↓∼1/2.\nThen we recall that the phase volume of the s-\nsubsystem is smaller than the phase volumes related to\nthe subsystems A and B. Therefore, the characteristic\ntime of evolution of the magnetic system as a whole is\nmuch larger than the characteristic relaxation times τeA,\nτeB,τs, relevant for the mobile electrons. Consequently,we neglect time derivative in the l.h.s. of Eq.7. This\nleads to the expression\n(n↓↑−n↑↓)≃ −(τBe)−1\n(4/τs)+(1/τee)(NB\n↑↓−NB\n↓↑).(8)\nSince at the time moment when NA\n↓=NA\n↑the magne-\ntization of the subsystem B is non-zero and the r.h.s. of6\nEq.8 is positive, Eqs.8,6 show that the magnetization of\nthe subsystem A changes its sign. Therefore, after this\nmoment the total configuration of the A-B system starts\nto beferromagnetic ((Fig.2(d))). Note that this happens\nin course of decay of total magnetization of the system,\nprovidedthat spin non-conservingprocessesdescribed by\nτsaretakenintoaccount. Wedenotethemomentoftime,\ncorresponding to reversal of the magnetization of the A\nsubsystem, as the ”reversal point” tr. It is important to\nnote that at the times t > trthe former minority elec-\ntrons of the A subsystem become to be majority and vice\nversa. Correspondingly, the reference for the Stoner ex-\ncitations is changed - now they are referred with respect\nto the ”new” orientation of magnetization. Therefore,\nthe exchange with B subsystem via s-electrons leads to a\ndecrease of the excitations number in the A subsystem.\nIf the electron temperature Tewould be kept constant\naftertr, the magnetizations in both subsystems would\nvanish, provided finite τs. SinceTegradually decreases\naftert=trat some moment it reaches the critical tem-\nperature TA\nC, while is still above TB\nC(Fig.2(e)). At this\nmoment, asshownabove,themagnetizationoftheAsub-\nsystem is non-zero and is aligned along the initial mag-\nnetization direction of the subsystem B. Then, the pres-\nence of a gap between new majority and minority bands\nin the A subsystem is restored, and the electron-electron\ns−dexchange scattering can not support anymore some\npairs of the Stoner excitations with the ”new” reference.\nThis leads to the increase of the magnetization of the A\nsubsystem aligned to the direction of initial magnetiza-\ntion of B subsystem. We note that, simultaneously, the\ninter-A exchange interactions are also restored. To the\ncontrary, magnetization of the B subsystem, character-\nized by the smaller critical temperature TB\nCcontinues to\ndecrease due to the Stoner excitations supported by the\ns-electrons subsystem, according to Eqs.1-3.\nWhen the electron temperature Tedecreases down to\nthe value TB\nCthe subsystem A already acquired the mag-\nnetization large enough to force the subsystem B to re-\nconstruct its magnetization state according to the new\nmagnetization state of the subsystem A via indirect an-\ntiferromagnetic exchange (Fig.2(f)).\nThe critical value of Tecorresponding to irreversible\nswitching can be estimated from ∆0\nAorTC\nA. At this crit-\nical value of Te=TC\nAthe self-consistent character of ex-\nchangeis restoredand the standardWeiss field is formed.\nSuch an estimate is mostly a semi-qualitative one since\nthe process of transition from strongly non-equilibrium\nregimetoanequilibriumhasacomplexcharacter.Itcould\nbe calculated with a help of numerical methods provided\none has a detailed information concerning temperature\nbehavior of τs, the heat withdrawal processes etc.\nWe would like to emphasize that at the strongly non-\nequilibrium state considered above the main processes\ndefining the spin kinetics within the system are related\ntospin-conserving exchange scattering. With lowering\nthe temperature below critical temperatures TA,B\nCof the\nsubsystems A and B this scattering becomes suppressed,\ns\nA B\ns\nA BSE SE\ns\nA Bs\nA B\nsinitial state\nfinal states\nA Bferromagnetic\nstateT=T0\nT>TCA;B\nT>TCA;BT>TCA;B\nTC1/τBe, which are correlated to the exchange\nsplittings ∆0\nA,Bpossessed by the A and B subsystems.\nIn the presented model the magnetization reversal is\ndriven by the exchange of angular momentum between\nthe A and A sublattices mediated by the mobile elec-\ntrons, while the transfer of the angular momentum to\nother reservoirs (lattice) is only responsible for overall\ndecay of magnetization of the whole system. Earlier, it\nhas been suggested, based on the studies of the ultrafast\nlaser-induced demagnetization in GdFeCo alloys, that\nthe angular momentum transfer from TM to RE sublat-\ntice plays an important role in the process.40Spatially-\nand element-resolved studies of the reversal dynamics in\nGeFeCo have shown that there is the angularmomentum\ntransfer between Gd-rich and Fe-rich nanoscale areas in\nthe GdFeCo sample which accompanies the reversal.41\nRecently, has been reported that there is a transfer of\nthe angular momentum between RE and TM sublattices\nof the metallic ferrimagnetic alloys CoGd and CoTb in-\nduced by the action of the laser pulse and monitored by\nthe spin- and orbital-resolved X-ray technique.42The re-\nsults of the element-specific studies of the laser-induced\ndemagnetizationand reversalinTbCoalloys43supported\nfurther the involvement of the exchange between the RE\nand TM sublattices in these processes.\nWhen introducing our model we did not specify\nwhether the A-e-B ferrimagnetic system should be sin-\ngle phase one or comprised by coupled A and B layers.\nThus, we argue that this model accounts well for the re-\nsults reported in Ref.11, where the all-optical reversal\nwas observed in four distinct types of single-phase and\nmultilayered synthetic metallic ferrimagnets.\nTherefore, the model consideredherecaptures the gen-\neralpictureofthelaser-inducedmagnetizationreversalin\na metallic ferrimagnet. However, due to a number ofsim-\nplifications applied and since our model does not include\nthe realistic band structure of a ferrimagnetic metal, it\ncannot account for a number of experimental evidences,\nwhich we discuss below.\nOngoing studies of the magnetization reversal reveal\nvery diverse and even contradictory features of the pro-\ncess in the RE-TM metallic alloys of various composi-\ntions. The important issue of the role of the magne-\ntization compensation point possessed by ferrimagnets\nhas been studied experimentally in both alloys9,13,14,17,44\nandengineeredmultilayerstructures.11Numberofexper-\niments have demonstrated,13,14,44that the reversal canbe realized for ferrimagnets which equilibrium tempera-\nture either below compensation point or above it, which\nagrees well with the proposed model. On the other hand,\nexperiments reported in Ref.17 suggest that for the re-\nversal it is essential that the compensation point is above\nthe equilibrium sample temperature. The recent study\nof the reversal in the series of specially engineered fer-\nrimagnetic structures showed that this condition holds\nfor the majority but not for all structures.11Despite of\nthese controversies, all the studies confirm that the re-\nversal does not occur in TM-RE alloys, which are either\nTM-richorRE-rich. Castingthelightonthis problem, in\nRef.45 the importance of the low remanence, possessed\nby ferrimagnets in a vicinity of the compensation point,\nhas been revealed. This is in agreement with the earlier\nreporteddata,14showingthat the closerthe sample to its\ncompensation point, the less laser fluence is required for\nthe reversal. Our model does not treat such details of the\nferrimagnetic metal as the equilibrium ratio between the\nsublatticemagnetizationsandthereforeitcannotaccount\nfor the role of the magnetization and angular momentum\ncompensation points.\nAnother issue which is still to be comprehended is the\nlaser pulse duration required for the reversal. In Ref.14\nthe reversal in GdFeCo alloys of certain compositions\ncould not be realized by the pulses longer than 1.7ps,\nwhile in Ref.17 the reversal by the laser pulses as long as\n10ps was reported. Based on the present knowledge, the\nreversal scenario treated in the frameworks of our model\nrequires femtosecond laser pulses which could bring the\nRE-TM alloys, studied in the reported experiments,14,17\ntothehighlynon-equilibriumstateonthetimescalecom-\nparable to the relaxation times τAE< τBE<1ps. We\nnote, however,thatthemaximalpulselengthrequiredfor\nthe reversalis dependent on the balance between the rate\nandthedegreeoftheelectronictemperatureincrease, the\nexchangerelaxationtimes τAe,τBeandtherateoftheen-\nergy and angular momentum withdrawal τs. Therefore,\nthe knowledge of the details of the spin-conserving and\nspin-nonconserving relaxation processes in a particular\nferrimagnetic samples for the particular pulse durations\nisessentialforunderstandingthe restrictionsonthe max-\nimal pulse duration. We are not aware about the time-\nresolved studies of the laser-induced reversal by pulses\nlonger than 100fs.\nFinally, we note that recently the switching effects for\npurelyferromagnetic structures were reported.46In this\ncaseonly the laser-pulsehelicity dependent switching has\nbeen reported, reopening the discussion about the role\nof the light polarization open. We believe that further\nexperimental studies, clarifying this issues are required\nbefore any conclusions regarding the mechanism of the\nreversal in the ferromagnets can be evaluated.9\nIV. MAGNETIZATION REVERSAL INDUCED\nBY AN ELECTRIC BIAS PULSE\nAccording to the model described in Sec.II the rapid\nheating of the mobile electrons is sufficient for triggering\nthe magnetization reversal. Therefore, we suggest that\nan electric bias pulse used as an alternative to femtosec-\nond optical pulses and can drive the ferrimagnetic metal-\nlic system into strongly-nonequilibrium state, where the\nmagnetization reversal can be realized. We consider a\npossibility of switching within the A-e-B system formed\neither the two ferromagnetic layers AandBor by the\nA-B metallic alloy imbedded within the metallic point\ncontact (see Fig.3(a)).\nNM\nNMIV\nI(a)\n(b)M\nFIG. 3: (Color online) (a) Structure formed by the two ferro-\nmagnetic layers, A and B, separated by normal metal inter-\nlayer NM, which is imbedded into point contact on the base of\nnormal metal NM. The interlayer NM is chosen in a way that\nit supports antiferromagnetic coupling between the layers A\nand B. (b) Structure formed by two ferromagnetic islands A\nand B, separated by normal metal interlayer NM, imbedded\nbetween two normal metal electrodes NM and separated from\nthem by two tunneling interlayers I. The normal metal inter-\nlayer NM supports antiferromagnetic coupling between A and\nB.\nAgain, for a simplicity we consider the model of strong\nferromagnetswhere the minority spins do not exist in the\nequilibrium (Fig.1). Furthermore, we neglect the energy\ndistribution ofboth spin subsystems thus reducingA and\nB subsystems to the two spin sublevels separated by the\nexchange energy ∆0\nAand ∆0\nB, respectively. The energy\ndistribution of the s-electrons is controlled by a voltage\napplied to the point contact. Namely, if the width of\nthe contact Lis smaller than the diffusive length with\nrespect to energy relaxation, the distribution of the s-\nelectrons is formed as a mixture of electrons coming from\nthe left bank of the contact and those coming from the\nright bank, and is controlled by corresponding chemical\npotentials. In the center of the contact the distributionFhas a double-step form:47\nF=1\n2(F0(εF−eV/2)+F0(εF+eV/2)),(9)\nwhereVis the applied voltage, εFif the Fermi energy,\nandF0(ε) = [1+exp( ε−εF)]−1. At some distance from\nthe center to the left or to the right the weight of the cor-\nresponding”left” or”right”contributionincreasesandat\nlargedistances the equilibrium distributions of”left” and\n”right” types are restored. Note that the distribution\n(9) holds near the center of the contact even for diffu-\nsive transport provided that inelastic mean free path is\nsmaller than the size of the contact L.\nAs it is seen, the energy eV, defined by the applied\nvoltageV, can play a role of effective temperature of the\ns-electron system, and, therefore, trigger the magnetiza-\ntion dynamics described in Sec.II. In contrast to the case\nofall-opticalreversal,in this casefollowingthe excitation\nthenon-equilibriumspin occupationsof s-electronsdecay\ndue to ballistic or diffusive transport from the contact to\nthe banks. Thus in this case the spin relaxation time τs\nin Eq.8 is the escape time, which is defined as\nτs=τb\nesc∼L\nv;τs=τd\nesc∼L2\nD, (10)\nfor the case of ballistic and diffusive transport, respec-\ntively. Here Lis the characteristic size of the contact,\nv∼108cm/s is the electron velocity, and Dis the diffu-\nsion constant.\nIn this point-contact device one can control both the\nexcitation intensity (by the value of the bias V) and\nthe parameters of the excitation pulse (including the\nswitching-on/off times). The switching-on time - if small\nenough - is of no great importance. In contrast, the\nswitching-off time should be comparable with the time\nscale of the magnetization reversal. The latter, as we\ndiscussed in Sec.II is controlled by electron-electron ex-\nchangerelaxationtimes τ(A,B)e, which areexpected to be\nof the order of 10−12s. Importantly, as we discuss below,\nthe time requiredforthe magnetizationreversal trcanbe\nincreased both by the choice of the bias and by a proper\nposition of the layers with respect to the contact center,\nsince an increase of the corresponding distance decreases\nthe phase volume of the electron-electron scattering and,\nthus suppressing the magnetization reversal.\nLet us consider an effect of the s-electron distribution\ngiven by Eq.9 on the ferromagnetic layer A imbedded\ninto the contact near its center. We consider the case of\nzero equilibrium temperature, T0= 0. IfeV >∆0\nA, than\nthe occupation NA\n↓of the minority level ofthe subsystem\nA is described by an equation48\ndNA\n↓\ndt+τ−1\nAe\n2∆A(∆A+eV)NA\n↓=τ−1\nAe\n4∆A(eV−∆A).(11)\nNow we take into account that the occupation of the\nminority level leads to a decrease of the exchange field,10\ni.e. ∆ A= (α/2)(NA\n↑−NA\n↓), where αis the propor-\ntionality factor. As we discussed above, although the\naverage exchange splitting vanished at NA\n↑=NA\n↓, the\nexchange splitting of the given ion is ∆ A∝ NAdue to\nlocal fluctuations. Then NA\n↑−NA\n↓= 1−2NA\n↓and thus\n∆A= ∆0\nA−αNA\n↓, whereα= 2∆0\nA.\nIfwedenote eV= 2∆0\nA+δ,thenEq.11canberewritten\nin a form:\ndNA\n↓\ndt=−τ−1\nAe\n4∆0\nA(1−2NA\n↓)\n×/parenleftbig\n4∆0\nA(NA\n↓−1/2)2+2δ(1/2−NA\n↓)/parenrightbig\n.(12)\nForδ <−∆0\nA, i.eeV <∆0\nA, there is no non-zero so-\nlution of the Eq.12. The occupation of the minority\nlevel starts naturally at the threshold value eV= ∆0\nA, or\nδ=−∆0\nA. The gradual increase of NA\n↓with an increase\nofeVterminates at the value NA\n↓= 1/2 which is reached\natδ= 0. Thus one concludes that the transition to the\nferromagnetic-like state with parallel A and B magneti-\nzations takes place at eV= 2∆0\nA. ForeV >2∆0\nAthe\nsteady state of the layer A corresponds to normal metal.\nClose to the critical value of the bias Vc= 2∆0\nA/e,\nthe evolution of NA\n↑↓(NB\n↑↓) and of the exchange field ∆ A\n(∆B) with time is controlled by the difference V−Vc,\nsince the non-vanishing part of r.h.s. of Eq.12 scales with\nthis difference. In the vicinity of NA\n↓= 1/2the difference\n1/2−NA\n↓can be considered as a variable. In the r.h.s. of\nEq.12theleadingtermislinearin1 /2−NA\n↓, andthecoef-\nficient at this term gives the rate of the evolution. Such a\nslow evolution is expected, roughly speaking, in the case\nwhen (1/2−NA\n↓)<|eV−eVc|/∆0\nA. Correspondingly, the\nevolution of ferromagnetic order parameter near critical\npoint is defined by a characteristic time\ntr∼τAe∆0\nA\n|eV−eVc|. (13)\nThus the time required for suppression the ferromagnetic\nstate of the sublattice A can be tuned by a proper choice\nof the bias.\nNow we would like to note that a specific feature of the\npoint contact is a possibility to apply voltage pulses with\na sharp form. As a result, one can operate the device in\na threshold way which was demonstrated above. Thus\nwe believe that our predictions including Eq.13 hold also\nfor more realistic models including finite energy width of\nthe ferromagnetic subsystem.\nAnother design of the bias-controlled switching device\ncanbe basedonthe tunnel junctions(Fig.3(b)). Namely,\nwe assume that the ferrimagnetic structure A−s−Bdis-\ncussed above is fabricated on the base of thin film tech-\nnology, and the corresponding thin film device is sand-\nwiched between two tunnel junctions. Note that for the\ncase of the device formed by two ferromagnetic region\nseparated by normal metal all three components are con-\nsidered to be fabricated within the same plane, as shown\nin Fig.3(b).In this case the external bias is applied to external\nmetallic electrodes of the tunnel junctions. If the trans-\nparencycoefficient, assumed to be the same for both tun-\nneling barriers, is k, than the effective time spent by a\nnon-equilibrium electron within the device, which is the\nmeasure of τs, is\nτs=τd∼d\nvFk−1, (14)\nwheredis athickness ofthe film forming the device while\nvFis the Fermi velocity within the metal structure. Thus\nwe conclude that if τdis comparable to the characteristic\nelectron-electron relaxation time and is larger than the\ncharacteristic electron-phonon relaxation time, then the\nelectron distribution function within the device is given\nin a same way as in Eq.9.\nWe note that for such a design the picture is to some\nextent similar to the one corresponding to optical exci-\ntation In particular, here we also deal with vertical ge-\nometry of excitation. Then, in contrast to the point-\ncontact design (Fig.3(a)), here we have no limitation\nin horizontal size of the device. The important differ-\nence is that in the tunnel junction-based device the sharp\nform of the electron distribution allows effective control\nof the switching process by controlling the form of the\nbias pulses.\nThe point-contact or tunnel junctions scheme where\nthe reversal is driven by the electric bias allows to avoid\na number of drawbacks which are often considered as\nthe limitations for the all-optical reversal. As it fol-\nlows from the considerations in the Sec.II, the magne-\ntization reversal depends on a balance between magne-\ntization decay time τA,B;einTe→ ∞limit and the de-\ncay time of the electron temperature Te. In a case of\noptical excitation achieving such a balance may require\na delicate choice of the pulse duration and pulse inten-\nsity and sample characteristics.14Furthermore, bringing\nthe optical excitation to the nanoscale is a challenging\ntask.49,50An additional factor which requires accurate\ncontrol for application of the all-optical magnetization\nreversalis related to a rate of the cooling time, controlled\nby a heat withdrawal from the laser-excited spot. Thus,\nin Ref.10, where the record-short all-optical write-read\ntime ofτw−r=30ps has been reported, in experiments\nthe residual heating resulted in only 83% magnetic con-\ntrast restored at τr−w. In the case of the bias driven\nmagnetization reversal, better control of the decay of the\nelectron temperature and the cooling can be achieved by\ntuning the duration and the recovery time of the voltage\npulse.\nNaturally, the question arises about an approach al-\nlowing generating bias pulses of required strength and\nduration. The most plausible solution for this problem\nis the photoconducting switch,51employing the illumina-\ntion of the semiconductor by a femtosecond laser pulse.\nThe rise time of such switches is mostly defined by the\nrise time of the laser pulse, and the RCcharacteristic of\nthe circuit. The controllable and short decay time of the11\nvoltage pulse is, however, the challenging issue. We are\naware of the reports where the voltage pulse durations\ndown to several hundreds femtoseconds were achieved by\ndesigningspecialphotoswitchcircuitsandusingthesemi-\nconducting media characterized by short electron decay\ntimes.52–54\nAs for the decay time, they can be very small in\nmetal-based structures. In the limit of ballistic trans-\nport estimates based on Eq.10 give the escape time as\nsmall as τs∼10−14s for s-electrons in the point contact\n(Fig.3(a)) of a size of L∼10nm. This value is the esti-\nmateoftheupperlimitofasharpnessofanyequilibration\nprocesswithintheballisticpointcontact. Inparticular,it\ndescribes the cooling rateafter the externalbias has been\nswitched off. In the diffusive point contact (Fig.3(a)) of\na sizeL∼30nm the escape time will be τs∼10−12s for\ntheelectronmeanfreepathof3nm. Forthe tunnelstruc-\nture presented on Fig.3(b) one can control τs(Eq.14) by\na proper choice of the tunneling transparency coefficient\nk. Certainly, to ensure effective control of the magne-\ntization the time τsshould not be smaller than the ex-\nchangescatteringtimes τe(A,B). Thus, weemphasizethat\nthe advantage of the devices suggested is a possibility to\ncontrol the process of switching in rather broad region\nby a proper choice of the device parameters including its\ngeometry and the bias applied.\nTherefore, we can conclude that the mechanism of for-\nmation of very short electric pulses seems to be the only\nrestriction of operation times for the metal-based devices\nin question. However the same restriction concerns any\nelectronicdevice while many otherrestrictionstypical for\nsemiconductor-based devices seem to be lifted.\nAn important ingredient of any spintronic device is a\npossibility of read-out of the magnetization controlled by\nsome external stimulus. We believe that such a read-out\nin the case of the electric bias driven reversal can be pro-\nvided by the well-known anomalous Hall-effect55(AHE).\nIndeed, the Hall voltage VHis related to the current I\nthrough the structure as VH/I=RH+R1M, whereR\nandR1are some material-dependent parameters, His\nan external magnetic field , and Mis the sample magne-\ntization. In the absence of external field the magnitude\nand the sign of VHis controlled by the sample magneti-\nzation. Thus a presence of Hall probes attached to the\ncorresponding ferromagnetic layer allows to detect the\nstate of the layer magnetization. Note that, although the\nAHE is often used to detect a presence of ferromagnetic\nordering, when other technique possessed poor sensitiv-\nity for the decisive conclusion. However, to the best of\nour knowledge it was not used to detect the sign of mag-netization since typically the samples had multidomain\nstructure.\nV. CONCLUSIONS\nTo conclude, we proposed the general microscopical\nmodeloftheultrafastmagnetizationreversalinantiferro-\nmagnetically coupled ferromagnetic metallic subsystems.\nIn the proposed model the rapid increase of the temper-\nature of mobile s−orp−electrons triggers effective ex-\nchange scattering between these electrons and the ferro-\nmagneticsubsystems oflocalized f−and nearlylocalized\nd−electrons. Then incoherent spin exchangebetween the\ntwo (nearly-)localized ferromagnetic subsystems is medi-\nated by the mobile electrons. Owing to the different ex-\nchange relaxation times for two involved ferromagnetic\nsubsystems, there is a moment after the excitation, when\none of the subsystems is completely demagnetized, while\nanother one still possess finite magnetization. It is this\nmomentwhen the reversalofthe ”faster”sublatticetakes\nplace. The model succeeds to explain most of the main\nfeatures of the all-optical magnetization reversal in ferri-\nmagneticmetallicsinglephaseormultilayeredstructures,\nreported recently by several groups. An important argu-\nment in favor of the model is the fact that the switching\nwas observed only for conducting structures inevitably\ncontaining mobile carriers. Since the effect of the exter-\nnal excitation in the considered here process is limited\nto a transient heating of the mobile electron system, we\nalso analyze a possibility to trigger the magnetization re-\nversal by application of the voltage bias. The relevant\nstructures are metallic point contacts or tunneling struc-\ntures with embedded ferrimagnetic metallic systems. It\nisshownthatsuchdevicesallowtocontrolswitchingwith\na great accuracy. We also suggest to use the anomalous\nHall effect to register the switching thus playing a role of\nreading probes.\nVI. ACKNOWLEDGEMENTS\nWe thank Dr. A. V. 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Phys. 82, 1539 (2010)." }, { "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\nL=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": "1504.01951v1.Magnetic_and_nonmagnetic_phases_in_doped_AB2_t_J_Hubbard_chains.pdf", "content": "arXiv:1504.01951v1 [cond-mat.str-el] 8 Apr 2015Magnetic and Nonmagnetic Phases in Doped AB2t-JChains\nR. R. Montenegro-Filho and M. D. Coutinho-Filho\nLaborat´ orio de F´ ısica Te´ orica e Computacional, Departa mento de F´ ısica,\nUniversidade Federal de Pernambuco, 50670-901, Recife-PE , Brazil\nWe discuss the rich phase diagram of doped AB2t-Jchains using data from DMRG and exact\ndiagonalization techniques. The Jvsδ(hole doping) phase diagram exhibits regions of itinerant\nferrimagnetism, Incommensurate, RVB, and Nagaoka States, Phase Separation, and Luttinger Liq-\nuid (LL) Physics. Several features are highlighted, such as the modulated ferrimagnetic structure,\nthe occurrence of Nagaoka spin polarons in the underdoped re gime and small values of J= 4t2/U,\nwheretis the first-neighbor hopping amplitude and Uis the on-site repulsive Coulomb interaction,\nincommensurate structures with nonzero magnetization, an d the strong-coupling LL physics in the\nhigh-doped regime. We also verify that relevant findings are in agreement with the corresponding\nones in the square and n-leg ladder lattices. In particular, we mention the instabi lity of Nagaoka\nferromagnetism against Jandδ.\nI. INTRODUCTION\nThet-Jversion of the Hubbard Hamiltonian [1] is a\nkey model for the understanding of strongly correlated\nelectron systems. The model is defined through only\ntwocompeting parameters: the hopping integral t, which\nmeasures the electron delocalization through the lattice,\nand the exchange coupling J= 4t2/U, whereU >> tis\ntheon-siteCoulombrepulsion. In fact, severalversionsof\nthe simplest Hubbard Hamiltonian, with a single orbital\nat each lattice and the on-site Coulomb repulsion, have\nbeen extensively used to model a variety of phenomena,\nsuch as: metal-insulator transition [2], quantum mag-\nnetism [3] and High- Tcsuperconductivity [4]. Moreover,\nexact solutions [1] and rigorous results [5, 6] have played\na central role in this endeavor.\nWe emphasize Lieb’s theorem [7], a generalization of\nthe one by Lieb and Mattis [8] for Heisenberg systems,\nwhich asserts that the ground state (GS) total spin of\na bipartite lattice at half filling and U >0 is given\nbySGS=|NA−NB|/2, where NA(NB) is the num-\nber of sites on sublattice A(B); indeed, Lieb’s theorem\nhas greatly enhanced the investigation of new aspects of\nquantum magnetism [6]. In particular, we mention the\noccurrence of ferrimagnetic GS, in which case we select\nstudiesusingHubbardor t-Jmodels[9–15], includingthe\nHeisenberg strong-coupling limit [16–18], on chains with\nAB2orABCtopological structures with SGS= 1/2 per\nunit cell [9–13, 16, 17], which implies ferromagnetic and\nantiferromagnetic long-range orders [10]. Further, the\ninclusion of competing interactions or geometrical and\nkinetic frustration [19–21], enlarge the classes of models,\nthereby allowing ground-states not obeying Lieb or Lieb\nandMattis theorems. Thesestudieshaveprovedeffective\nin describing magnetic and other physical properties of a\nvariety of organic, organometallic, and inorganic quasi-\none-dimensional compounds [19, 22].\nOf particular physical interest are doped systems, al-\nthough in this case rigorous results are much rare [6].\nOne exception is Nagaoka’s theorem [23], which asserts\nthat for J= 0 (U→ ∞) thet-Jmodel with one holeadded to the undoped system (half-filled band) is a fully\npolarized ferromagnet , favored by the hole kinematics, if\nthe lattice satisfies the so-called connectivity condition\n[24]. A long-standing problem about this issue is the sta-\nbility of the ferromagnetic state for finite hole densities\nand finite values of J. Numerical results have indicated\n[25, 26] that two-dimensional lattices display a fully po-\nlarized GS for J= 0 and δ/lessorsimilar0.2, where δ=Nh/N, with\nNh(N) the total number of holes (sites); while, analyti-\ncal studies [27, 28] havesuggested that this state is stable\nup toJt∼δ2.\nFurther, an ubiquitous phenomenon in doped strongly\ncorrelated materials is the occurrence of inhomogeneous\nstates, particularly spatial phase separation in nano- and\nmesoscopic scales [29] and incommensurate states [29,\n30]. In underdoped High- Tcmaterials, dynamical and\nstatical stripes in copper oxide planes has been the focus\nof intensive research [31]. Concerning two-dimensional t-\nJorHubbardmodels, phaseseparationintohole-richand\nno-hole regions was discussed in the large −and small −J\nlimits [32]. Howeverthe precisechargedistributionin the\nground state remains controversial. The use of distinct\nand refined numerical methods have pointed to striped\n[33] or uniform phases [34]; recently, it was claimed that\nthe origin of this issue relies on the strong competition\nbetween these phases [35]. For the linear t-JHubbard\nchain the physics is more clear [36], and phase separation\ntakes place for J= 2.5−3.0, depending on the doping\nvalue, but it is absent in the small −Jregime.\nIn this work, we use Density Matrix Renormalization\nGroup (DMRG) [37] technique and Lanczos exact diago-\nnalization(ED)toobtainthegroundstatephasediagram\nand the low-energy excitations properties of the doped t-\nJmodel on AB2chains [9] for J= 0.0−0.4. We verify\nthe occurrence of an itinerant modulated ferrimagnetic\n(FERRI) phase in the underdoped regime, regions of in-\ncommensurate (IC) states and Nagaoka ferromagnetism\n(F), and two regions of phase separation (PS), in which\nIC and F states coexist with the resonating valence bond\nstate (RVB), respectively. In addition, we find that the\nRVB state is the stable phase at δ= 1/3, and identify\na crossover region that ends at the onset of a Luttinger2\n0 0.23 1/30.1δ00.10.20.30.4 J\nFERRI PS\nFIC\nPS(IC - RVB)\n(F - RVB)δPS, J\nδFERRI, J\nJF, δ\n2/31CrossoverLL\n(RVB)~~(a)\n(b)\nB\nB\nδ = 1/3RVB at F\nδ = 2/3A\n21FERRI\nOnset of LL at IC\n00.050.10.15\nδ00.51\nSGS / SLJ = 0.1\nJ = 0.3(c)\nFIG. 1. (Color online). (a) GS phase diagram for the AB2\nt-Jmodel (error bars account for the discrete values assumed\nbyδin a finite-size system). The phases are illustrated in\n(b): modulated ferrimagnetism (FERRI), incommensurate\n(IC), Nagaoka ferromagnetism (F), short-range resonating va-\nlence bond (RVB) states, phase separation (PS), and Lut-\ntinger liquid (LL). The estimated transition lines δFERRI,J,\nδPS,J, andJF,δare also pointed out. (c) Ground state total\nspin,SGS, normalized by its value in the undoped regime:\nSL≡(Nc/2)−0.5, as function of δfor the indicated values\nofJandN= 3Nc+1 = 100.\nliquid (LL) phase at δ= 2/3, above which the LL physics\n[38] sets in.\nII. PHASE DIAGRAM\nThet-Jmodel reads:\nHt−J=−t/summationdisplay\n,σPG(c†\niσcjσ+H.c.)PG(1)\n+J/summationdisplay\n(Si·Sj−1\n4ninj),\nwhereciσannihilateselectronsofspin σat sitei,niisthe\nnumber operator at site iandPG=/producttext\ni(1−ni↑ni↓) is the\nGutzwiller projector operator that excludes states with\ndoubly occupied sites. In our simulations, we set t= 1\nand have consideredchains with Nc(N) unit cells (sites).\nIn ED calculations closed boundary conditions are used\nwithNc= 8 (N= 3Nc), while in the DMRG simulations\nopen boundary conditions are used and the system sizesB+B2 1B+B2 1B+B2 1B+B2 1A(a)\nA A A A4321l = a\n00.5 (b)δ = 0.04 J = 0.1 \n-0.500.5(c)δ = 0.18 J = 0.1\nFIG. 2. (Color online). (a) Effective linear chain (spacing\na≡1) associated with N= 3Nc+ 1 = 100 sites for J= 0.1\nused to illustrate the hole, /angbracketleftnh,l/angbracketright, and spin, /angbracketleftSz\nl/angbracketright, profiles: (b)\nδ= 4/100 (FERRI phase) and (c) δ= 18/100 (IC phase).\nranged from Nc= 33 (N= 3Nc+1 = 100) to Nc= 121\n(N= 364). We retain from 243 to 364 states in the\nDMRG calculations, and the typical discarded weight is\n1×10−7.\nThe ground state (GS) phase diagram, shown in Fig. 1\n(a), displays the regions of the above-mentioned phases,\nillustrated in Fig. 1(b), including the estimated transi-\ntion lines and the crossover region. A special feature of\ntheAB2chain is its symmetry [12, 13, 17] under the ex-\nchange of the labels of the Bsites in a given unit cell l\n[identified in the FERRI state, Fig. 1(b)]. This symme-\ntry implies in a conserved parity pl=±1 in each cell of\nthe lattice. The phase diagram of a chain with Ncunit\ncells is calculated by obtaining the lowest energy for all\nsubspaces with xcontiguous cells of parity −1 and the\nothersNc−xcells with parity +1, with x= 0...Nc,\nfor fixed δandJ. In the phase diagram shown in Fig.\n1(a),p≡/summationtextNc\ni=1pl= +1 for δ≥1/3,p/negationslash=±1 in the PS\nregion, and p=−1 forδ < δPS,J. The magnetic configu-\nration of a phase is identified by the total spin SGS, local\nmagnetization, magnetic structure factor, and spin cor-\nrelation functions. In what follows, we shall characterize\nthe phases shown in Fig. 1(a).\nIII. FERRIMAGNETISM AND TRANSITION\nTO IC STATES\nAtδ= 0 and J/negationslash= 0, the insulating Lieb ferrimag-\nnetic state with total spin quantum number SGS=SL≡\nNc/2−0.5≡SLis found for a chain with open bound-3\n0 0.5q/π01S(q)0\n0.02\n0.04\n0.06\n0.08\n0.10\n0.12(a)δJ = 0.3\n0.5 1q/π0123S(q) 00.05 0.1δ00.10.20.30.4\n∆q/π(b)\nFIG. 3. (Color online). Chain with N= 3Nc+1 = 100 and\nJ= 0.3. (a) and (b): Magnetic structure factor S(q) for the\nindicated values of δ. Inset of (b): ∆ q≡qmax−π, whereqmax\nis the value of qat which the local maximum of S(q), near\nq=π, is observed.\nary conditions, N= 3Nc+ 1 = 100, with an Asite on\neach side. In order to evaluate the stability of this state\nagainst doping, we calculate SGSas a function of δfrom\nthe energy degeneracy in Sz. As shown in Fig. 1(c),\nas hole doping increases from δ= 0 to a critical value\nδ=δFERRI,J, the value of SGSdecreases linearly from\nSLto 0 or a residual value, signaling a smooth transi-\ntion to the IC phase. However, for low enough J,SGSof\nthe IC phase increases linearly with δup toδ=δPS,J,\nthe line at which PS occurs [see Fig. 1(a)], or up to\nthe boundary, JF,δ, of the Nagaoka F phase. This unex-\npected behavior claims for an explanation.\nIn order to understand the behavior of SGSfor low\nJwe have calculated the profiles of the magnetization,\n/angbracketleftSz\nl/angbracketright, in the spin sector Sz=SGS, and of the hole den-\nsity,/angbracketleftnh,l/angbracketright, forJ= 0.1 (see Fig. 2). To help in the data\nvisualization, we use a linearized version of the lattice,\nas illustrated in Fig. 2(a). As shown in Fig. 2(b), for\nδ= 0.04 the holes distort the ferrimagnetic structure,\nwhich display a modulation with wavelength λ≈17, in15 30 45 60l\n-0.4-0.200.20.40.6 0.53IC(p = −1)\nRVB\n(p = +1)\n J = 0.30.280.44\n0.20\n\n\nFIG. 4. (Color online). Phase separation (IC-RVB) for a\nchain with N= 3Nc+ 1 = 100 sites, J= 0.3, andNh= 18\nholes: spin correlation function between Bspins at the same\ncell,/angbracketleftSB1,l·SB2,l/angbracketright, and hole density profile, /angbracketleftnh,l/angbracketright.\nanti-phase with that exhibited by the hole (charge) den-\nsity wave. We have thus identified a modulated itinerant\nferrimagnetic phase in this underdoped regime. On the\nother hand, as shown in Fig. 2(c), for δ= 0.18 the mag-\nnetization has local maxima in coincidence with those of\nthe holedensity profile. In this case, the IC phaseis char-\nacterized by the presence of ferromagnetic Nagaoka spin\npolarons [28, 39] due to hole density wave with λ≈4.\nOur results point to a value of J(∼0.2) below which\nferromagnetic “bubbles” appear as precursors of the F\nphase found for J < JF,δ[see Fig. 1(a)].\nForJ= 0.3,SGS= 0 in the IC phase, as shown in\nFig. 1(c). In Figs. 4 (a) and (b) we present the magnetic\nstructure factor\nS(q) =1\nSL(SL+1)2Nc+1/summationdisplay\nl,meiq(l−m)/angbracketleftSl·Sm/angbracketright,(2)\nwherel,mandSrefertothe latticerepresentationshown\nin Fig. 2(a), for this value of Jand doping ranging from\nδ= 0 up to δ= 0.12. In a long-range ordered ferrimag-\nnetic state, sharp maxima at q= 0 (ferromagnetism) and\nq=π(antiferromagnetism) are observed in the curve\nS(q) forδ= 0. Adding two holes to the undoped state,\nsharp maxima at q= 0 and πare also observed, while\nbroad maxima occur for δ= 0.04, indicating short-range\nferrimagnetic order which evolves to the IC phase by in-\ncreasingdoping, beforephaseseparation(IC-RVB)atthe\nlineδ=δPS,J[see Fig. 1(a)]. In the inset of Fig. 4(b) we\nshow the departure of the maximum of S(q) fromq=π.\nIV. PHASE SEPARATION, RVB STATES AND\nLUTTINGER LIQUID\nIn Fig. 1(a) the dashed line inside the PS regionfix the\nboundary between two types of phase separation: in one\ncase, the separation occurs between Nagaoka ferromag-\nnetismandshort-rangeRVBstates(F-RVB); whileinthe\nother, it occurs between IC and short-range RVB states4\n2/3 1\nδ0.81R0.00\n0.05\n0.10\n0.15\n0.20\n0.25\n0.30\n0.35\n0.40\n2/3 1\nδ0.51\nKρ(a) (b)J\nFIG. 5. (Color online). Luttinger liquid behavior for a chai n\nwithN= 3Nc= 24 (ED results). (a) Ratio R=uρ//radicalbig\nDχ/π\nas a function of δfor the indicated values of J. (b) Exponent\nKρas a function of δ.\n(IC-RVB). Indeed, for 0 ≤J/lessorsimilar0.063 and δF−RVB≤\nδ <1/3, the GS phase separates with F and short range\nRVB states under coexistence, where δF−RVBdenotes\nhole density values along the phase separation line F-\nRVB, thereby extending our previous result [13] valid\nonly for J= 0. However, for 0 .063/lessorsimilarJ≤0.4 the sys-\ntem behaves differently. The new PS (IC-RVB) region is\nhere illustrated for J= 0.3,N= 3Nc+ 1 = 100 sites,\nandNh= 18 holes: we thus find that there are 26 cells\nwith odd parity ( pl=−1), associated with the IC phase,\nand the remaining 7 cells with even parity ( pl= +1),\nassociated with the RVB phase. In this case, as shown\nin Fig. 4(c), the hole-poor IC phase presents a local\nspin correlation function /angbracketleftSB1,l·SB2,l/angbracketright ≈0.2, average\nhole density per site ≈0.16, estimated from the sites\nindicated by arrows [one Asite and two Bsites in the\ncontext of the effective linear chain shown in Fig. 2(a)],\nand hole-density wave with λ≈4; while the hole-rich\nRVB phase presents /angbracketleftSB1,l·SB2,l/angbracketright ≈ −0.4 and average\nhole density per site ≈1/3, estimated from a cell with A\nandBsites indicated by arrows. Therefore, apart from\nboundary effects, the above results thus indicate that the\nphase separation for a given Jvalue is defined by the co-\nexistence ofthe two phaseswith the hole densities δIC-PS\n(≈0.16 forJ= 0.3) andδPS-RVB(≈1/3 forJ= 0.3)\nfixed at the IC-PS and PS-RVB boundaries, respectively,\nwhile the size of the phases are fixed by the chemical\ndopingδ=Nh/N(= 0.18 forN= 100 and Nh= 18).\nWe also remark that the stable RVB phase observed at\nδ= 1/3 and 0 ≤J≤0.4, which has finite charge and\nspin gaps, is in agreement with predictions for J= 0.35\n[12] and J= 0 [13].\nFor 0≤J≤0.4 and 1/3< δ <2/3, a crossover\nregion with the presence of long-range RVB states after\nhole addition away from δ= 1/3 is observed [see Fig.\n1(a)]. At the commensurate filling δ= 2/3, the system\npresentsachargegap,whilethespinexcitationisgapless,\nalso extending our previous result for J= 0 [13].\nWith the aim of investigating the LL behavior as a\nfunction of Jandδ≥2/3, we have calculated, through0 0.2 0.4 0.6 0.8 11.2\nJ / JF,δ(Nc)-0.002-0.0010(EGS− EF) / Nc\nNc = 33, δ = 0.100\nNc = 67, δ = 0.109\nNc = 121, δ = 0.104(a)\n00.05 0.1\nδ00.010.020.03\nJF,δNc=33\nNc=67\nNc=121\n0.6 δ2 + 3.3 δ3(b)\n0 10 20 30 40 5060l\n-0.500.51\nSz = 45, J = 0.0000\nSz= 39, J = 0.0025\nSz = 39, J = 0.0050\nSz = 39, J = 0.0075(c)\nδ = 0.1\n0 10 20 30 40 5060\nl00.050.10.150.2\n\nSz= 39, J = 0.0025\nSz = 39, J = 0.0050\nSz = 39, J = 0.0075\nNon-interacting \nspinless fermions(d)\nδ = 0.1\nFIG.6. (Color online). (a)Shift EGS−EFperunitcell, where\nEFis the energy of the fully polarized ferromagnetic state, as\na function of J/JF,δfor the indicated values of Ncandδ. (b)\nInstability line of the Nagaoka ferromagnetic phase. (c) Sp in\nand (d) hole profiles, /angbracketleftnh,l/angbracketrightand/angbracketleftSz\nl/angbracketright, respectively, for a chain\nwithN= 3Nc+1 = 100, δ= 0.1, and the indicated values of\nSzandJ.\nED techniques, the ratio R=uρ//radicalbig\nDχ/π,where\nχ=Nc\n4[E(Nh+2)+E(Nh−2)−2EGS(Nh)] (3)\nis the charge susceptibility, and E(Nh±2) is the total\nenergy for Nh±2 holes;\nD=Nc\n4π/bracketleftbigg∂2E(Φ)\n∂Φ2/bracketrightbigg\nΦmin(4)\nis the Drude weight, where E(Φ) is the lowest energy for\na magnetic flux Φ through a closed chain, and Φ minits\nvalue at EGS;\nuρ=E(kGS+∆k,S= 0)−EGS(kGS,SGS= 0)\n∆k(5)\nis the charge excitation velocity, where ∆ k= 2π/Nc, and\nE(kGS+∆k,S= 0) is the lowest energy with wavenum-\nberk=kGS+ ∆kand total spin S=SGS= 0. If\nthe low-energy physics of the system is that of a LL, we\nshould find R= 1 [40]; moreover, the exponent govern-\ning the asymptotic behavior of the correlation functions,\nKρ, satisfies the relation Kρ=πuρ/2χ. As shown in\nFig. 5(a), Ris indeed very close to 1 for δ >2/3; in\naddition, as shown in Fig. 5(b), we find 0 .7/greaterorsimilarKρ/greaterorsimilar0.5\nforδ >2/3. Remarkably, as shown in Figs. 5(a) and (b),\nthe data for RandKρexhibit data collapse as a function\nofδfor 0≤J≤0.4. In short, the results above clearly\nindicate that for δ >2/3 and 0 ≤J≤0.4 the system\nbehaves as a LL in the strong coupling regime.5\nV. STABILITY OF NAGAOKA\nFERROMAGNETISM\nIn this Section, we shall provide strong evidence that\nfor0≤J≤JF,δand 0< δ≤δF−RV B, the kinetic energy\nofholesisloweredinafullypolarizedferromagneticstate,\nan extension of Nagaoka ferromagnetism [23, 24], with\nthe GS energy equal to that of non-interacting spinless\nfermions: EGS=EF.\nThe estimate of JF,δis based on the data for the shift\n(EGS−EF)/Ncas a function of J, as illustrated in Fig.\n6(a) forδclose or equal to 0.1. We stress that the shift\ndecreases as Ncincreases for 0 <(J/JF,δ)<1, and goes\nto zero in the thermodynamic limit. In addition, one\nshould notice that, by examining the data above and be-\nlowJ=JF,δ, particularly for N= 3Nc+1 = 364 sites,\n∂EGS/∂Jappears to be discontinuous at J=JF,δin the\nthermodynamic limit, thus suggesting a first-order tran-\nsition to the IC phase at ( J/JF,δ) = 1. In Fig. 6(b) we\nshow that our estimated transition line, JF,δ, [see also\nFig. 1(a)] is almost not affected by finite size effects and\nimpliesδF,J∼√\nJasδ→0, as found from analytical\nresults [27, 28] for the t-Jmodel in a square lattice. In\nparticular, for J= 0 the instability of the Nagaoka state\noccurs at δ≈0.23, which is very close to the values of\nhole doping estimated for n-leg ladder systems [25] and\nthe square lattice [25, 26].\nThe spin profile for a chain with N= 3Nc+1 = 100,\nJ/greaterorsimilar0andδ= 0.1isalsoin verygoodagreementwith the\nNagaoka state, as shown in Fig. 6(c), although boundaryeffects are visible for J/greaterorsimilar0; in fact, Szchanges from 45\nto 39 (on average, three spins at each boundary are not\nfully polarized), but one should notice that the change\nsaturates as Jslightly increases above zero. This fact\nis corroborated by the hole density shown in Fig. 6(d),\nwhosedataforthe referredstateswith Sz= 39atδ= 0.1\nare very well described by the Nagaoka profile.\nVI. DISCUSSION AND CONCLUDING\nREMARKS\nThe presented phase diagram of doped AB2t-Jchains\nis remarkably rich. Indeed, several magnetic and non-\nmagnetic phases manifest themselves in a succession of\nsurprising relevant features, some of which are similar to\nthose observed in the square and n-leg ladder lattices:\nall in a simple doped chain. In particular, we empha-\nsize the modulated ferrimagnetic structure, the occur-\nrenceof Nagaokaspin polaronsin the underdoped regime\nand small values of J, incommensurate structures with\nnonzero magnetization, the strong-coupling LL physics\nin the high-doped regime, and the instability of Nagaoka\nferromagnetism against Jand doping. 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By artificially generating isolated vacancies on a graphite surface and \nmeasuring the ir local density of states on the atomic scale, we have show n how single vacancies modify \nthe electronic properties of this graphene -like system. Our scanning tunneling microscopy experiments, \ncomplemented by tight binding calculations, reveal the presence of a sharp electronic resonance at the \nFermi energy around each single graphite vacancy, which can be associated with the formation of local \nmagnetic moments and implies a dramatic reduction of the charge carriers' mobility. While vacancies \nin single laye r graphene naturally lead to magnetic couplings of arbitrary sign, our results show the \npossibility of inducing a macroscopic ferrimagnetic state in multilayered graphene samples just by \nrandomly removing single C atoms. \n \nPACS numbers: 73.22.Pr, 75.70.Rf, 73.20.Hb, 68.37.Ef \nGraphite is a semimetal, where the low density of states \nat the Fermi level, and its high anisotropy induces \nsignificant diffe rences from conventional metals (1). It \nconsists of weakly van der Waals coupled graphene layers \nand thus it shows a strong 2D character sharing many \nproperties with graphene. Graphite´s unusual features are \nenhanced in single layer graphene (2,3) , where the density \nof states vanishes at t he neutrality point, and carriers show \na linear, massless, dispersion in its vicinity. In graphene, \nthe existence of localized el ectronic states at zigzag edges \n(4,5) and vacancies (6) leads to an extreme enhancement of \nthe spin polarizability, and model calculations suggest that \nmagnetic moments will form in the vicinity of these \ndefects (7-9). It seems likely that similar phenomena also \ntake place in other sp2 bonded carbon materials, such as \ngraphite. An enhanced density of states has been observed \nnear zigzag steps in graphite surfaces (10). The existence \nof these localized states suggests that magnetic moments \n(11-13), and possibly magnetic ordering may exist in \nsingle layer graphene and graphite. In the case of irradiated \ngraphite, where lattice defec ts are expected to exist, \nmagnetic order has been reported even at room \ntemperature (14,15 ). \nIntroducing vacancies in graphene -like systems by \nirradiation has been shown to be an efficient method to \nartificially modify their properties (14-17). While the role \nplayed by these vacancies as single entities has been \nextensively addressed by theory (6-8,18), experimental \ndata available (14-17) refer to statistical properties of the \nwhole heterogeneous collection of vacancies generated in \nthe irradiati on process (19,20 ). The main goal of the \npresent work is to overcome this limitation ; thus, we first \ncreate perfectly characterized single vacancies on a \ngraphite surface by Ar+ ion irradiation and then, using low \ntemperature scanning tunneling microscop y (LT-STM) , we \nindividually investigate the impact of each of such \nvacancies in the electronic and magnetic properties of this graphene -like system . We identify well localized electronic \nresonances at the Fermi energy around graphite single \nlattice vacancies. The existence of these states is in good \nagreement with theoretical expectations, and it can be \nassociated with the formation of magnetic moments in this \nall-carbon material. Using simple extensions of these \nmodels, as well as the similarities between the properties \nof a clean graphite surface and single layer and \nmultilayered graphene, we can extrapolate our results to \nthose systems. In addition, w e also show that contrary to \nthe single layer graphene case, ferrimagnetism is favored \nin multilayered graphene samples. \nWe use highly ordered pyrolytic graphite (HOPG) \nsamples, which present the AB Bernal stacking. Thus, one \natom of the honeycomb unit ce ll (\n ) is located directly \nabove a C atom of the second layer and the other one (\n ) is \non top of a hollow site (see Fig. 1 B). A key point of the \npresent work is the atomistic control of the samples, which \nwas obtained by performing all the preparation proc edures \nand measurements under UHV conditions. We created \nsingle vacancies by irradiating with 140 eV Ar+ ions \npreviously in -situ exfoliated HOPG surfaces. At these low \nion energies, just above the threshold value for the \ndisplacement of surface atoms, the ion irradiation mainly \nproduces atomic point defects (19,20 ). After further sample \nannealing at 650ºC, the remaining defects were mostly \nsingle vacancies as revealed by STM images. Fig . 1A \nshows the general morphology of our samples after the \nirradiation and annealing procedure. The previous perfect \nand pristine graphite surface, now presents several point \ndefects surrounded by threefold (\n 3x\n3) patterns, R3 in the \nfollowing (see Fig.S1 in EPAPS (21) for a more general \noverview) . The comparison of our atomically resolved \nSTM images of these defects (Fig. 1 A, C) with calculations \n(22, 23 ) shows that these defects correspond to single \nvacancies on both \n and \n sites of the graphite honeycomb \nlattice. 2 \nFig. 1. . A) 17x17 nm2 STM topography, measured at 6 K, showing the graphite surface after the Ar+ ion irradiation (for a larger scale \noverview of the same region see ( 21)). Data analyzed using WSXM (37). Single vacancies occupy both a and b sites of the graphite \nhoneycomb lattice. Sample bias: + 270 mV, tunneling current: 1 nA. B) Schematic diagram of the graphite structure. C) 3D view of a \nsingle isolated vacancy. Sample bias: +150 mV, tunneling cur rent: 0.5 nA. D) STS measurements of the LDOS induced by the single \nvacancy and of graphite. Black circles correspond to dI/dV spectra measured on pristine graphite and red circles correspond to dI/dV \nspectra measured on top of the single vacancy, showing the appearance of a sharp resonance at E F. dI/dV measurements were done \nconsecutively at 6 K with the same microscopic tip . \nWe use a home -made LT -STM (24) to investigate the \nlocal electronic structure of the single atomic vacancies \ncreated in graphite. This is an unrivaled technique to \nprovide local information about the surface electronic \nproperties, achieving atomic precision and very high \nenergy resolution (\n 1 meV at 4.2 K). Th e use of these \nunique capabilities in graphene -like systems has already \nallowed, for example, to detect the coexistence of both \nmassless and massive Dirac Fermions in a graphite surface \n(25, 26 ), or to prove the Dirac nature of the quasiparticles \nin epitaxial graphene on SiC (27, 28 ). Differential \nconductance ( dI/dV ) spectra were measured in open \nfeedback loop mode using the lock -In technique with \nfrequency 2.3 kHz and a.c modulation of 1 mV. Various \ntungsten (W) tips were used for the measurements. I n order \nto avoid tip artifacts, tip status was always checked by \nmeasuring reference spectra on pristine graphite; only tips \nshowing the standard featureless V -shaped spectra and a \nwork function of 4 -5 eV were considered in this work. \nSpectra remained unch anged for moderate tip -sample \ndistance variations (stabilization current was routinely \nmodified from 10 pA to 10 nA). Fig. 1 D shows \nconsecutive dI/dV spectra, measured at 6 K, summarizing \nour results. Far enough from any defect, spectra showed a \nfeatureles s V-shaped form as expected for the LDOS of \npristine graphite (black circles). Spectra acquired on top of \na single vacancy (red circles), both in \n and \n \nreveal the existence of a sharp resonance peak around the \nFermi level (E F) with a FWHM of \n5 mV. \nThe presence of this resonance is a fundamental result \nthat, although anticipated in many theoretical works, had \nnever been experimentally observed before. The formation \nof a magnetic moment can be associated to the resonance, \nsince electron -electron inter actions, and the fact that the \nlocalized level is very close to the Fermi energy, favour the polarization of this state. In addition, the narrowness of the \nresonance and the low electronic density of graphite at the \nFermi level imply a very poor screening of the magnetic \nmoment, which anticipates a very high Curie temperature \nfor the vacancies (see below). Our results also demonstrate \nthat the presence of these single vacancies should have a \nstrong impact on the electronic transport, since the \nexistence of a resonance in the vicinity of the Fermi level \ngives rise to a strong reduction of the mobility with a mean \nfree path which tends to the distance between impurities at \nthe neutrality point (29). In this way, the artificial \nintroduction of a chosen density of vacancies can be used \nas an effective method to tune the mobility of graphite and \ngraphene -like samples. \nThe existence of this sharp resonance at the neutrality \npoint in single layer graphene can be derived from the \nnearest neighbor tight -binding Hamil tonian which \ndescribes the \n bands (6), neglecting deformations near the \nvacancy. We have checked that the resonance is also \npresent in a semiinfinite graphite layer by extending the \ncalculation and using the Slonczewski -Weiss -McClure \nparametrization of th e \n bands, which is expected to \ndescribe well the electronic structure near the Fermi level \n(30-32). Results are shown in Fig. 3 B. Details of the \ncalculation are given in the supporting material ( 21). The \nresults are consistent with those in ref. 6), and w ith \nextensions of that model to multilayered graphene (33). A \nsharp resonance exists when the vacancy is at \n sites, \nwhile a lower peak is found near vacancies at \n sites. The \nwavefunction \n associated to this resonance is very \nextended, decaying as functi on of the distance to the \nvacancy as r-1 (6, 33 ). It shows a R3 modulation, \nassociated to the wavevector which spans the two valleys \nin the Brillouin Zone ( 21). \n\nA) B)\nC)D)\ns\ns\nCl\n3 \n s\n\nsA)\nC) B)\nH\nLDOS\nFig. 2. A) LDOS as a function of sample voltage V and position \nx along the blue line drawn in B). A green line has been drawn to \noutline the evolution of the resonance peak height, showing a \nclear R3 modulation. B) STM topographic image of a single \ngraphite vacancy. Image size 8 x 8 nm2; sample bias +200 mV \ntunneling current 0.6 nA. C) r-2 decay of the resonance intensity. \nBlack dots correspond to the maxima of the resonance peak \nheight and the red line is parabolic fit to the experimental data. \nWe have also analyzed experimentally the spatial \nextension of the states induced around single vacancie s by \nmapping the narrow resonance as a function of distance \nfrom the defect. Fig. 2 A shows a map of the local density \nof states (LDOS) vs energy, measured along the line across \nthe vacancy drawn in Fig. 2 B. The narrow resonance \nextends several nanometers away from the vacancy, \nindicating that it is indeed a quasilocalized state. The \nresonance shows an overall decreasing intensity with \nincreasing distance, consistent with the expected r-1 decay \n(STM probes \n2) as shown in Fig. 2 C, its height is \nmodulated with the R3 periodicity (Fig. 2 A) and its width \nremains approximately constant for all distances. \nThe agreement between the experimental results and \nthe theoretical model shows that the latter describes \ncorrectly the main electronic properties of the vacancy. \nElectron -electron interactions prevent double occupancy of \nthe resonance, and lead to the formation of a magnetic \nmoment near the vacancy. The resonance is built up from\n \norbitals, and it is orthogona l to the extended states. Hence, \nthe coupling between the magnetic moment and the \nconduction and valence bands is not due to virtual \ntransitions involving short lived zero or doubly occupied \nstates. These are the processes which describe the \nantiferromagne tic Kondo coupling induced by a magnetic \nimpurity hybridized with a metallic band. In our case, we \nexpect a ferromagnetic coupling mediated by the Coulomb \nrepulsion ( 21). Moreover, the magnetic moment is \nextended throughout many lattice cells around the va cancy, \nso that it interacts with many partial wave channels built \nup from the extended states, leading to a multicha nnel ferromagnetic Kondo system (34). The magnetic moment \nis not quenched at low temperatures. \nA)\nsite\nsiteB) Experiment Theory\n \n \n \nl\n\n\n\n\n \nl\n \n sss\n\n\n\nFig. 3. A) dI/dV spectra measured with the same tip on top of a \nsingle vacancy in an \n (red), \n (blue) site and on pristine graphite \n(black). The intensity of the resonance measured on top of the \n \nvacancy is much higher than the one of the \n one, indicating that \nremoving a C atom from an \n site generates a stronger magnetic \nmoment. Solid lines are fits to a Lorentizan function giving a \nFWHM of \n 5mV for both the resonance on the \n and \n site. \nSmall variations (of a couple of mV) in the position of the \nresonance peak maxim a were observed, which we attribute to the \nlocal environment of each specific vacancy. B) Calculated \ndensity of states in the atom nearest to vacancy. Red: vacancy in \n site Blue: Vacancy in \n site. \nThe interaction between magnetic moments induced by \nvacancies at different sites has been extensively stu died for \nsingle layer graphene (35). Its sign depends on the \nsublattice occupied by the vacancies. In graphite, the two \nsublattices are inequivalent. Our samples present vacancies \nin both \n and \n sites of th e honeycomb lattice, which gives \nrise to two R3 scattering patterns of different shape and \nextension ( 21). It is then natural to think that the \nquasilocalized resonance, and thus the magnetic moment, \ninduced by graphite single vacancies are also affect ed by \nthe underlying C layers. Our dI/dV spectra clearly \ndemonstrate that this is indeed the case. Fig. 3 A shows \nconsecutive spectra measured, with exactly the same \nmicroscopic STM tip, on both types of vacancies and on \nclean graphite. A sharp resonance of very similar width \n(FWHM of \n 5 mV) is induced by both types of vacancies; \nhowever, the intensity is much higher in the case of the \n \nvacancy, in agreement with our calculations (Fig. 3 B). This \ninequivalence in the magnetic moment induced by each \ntype of vacancy will reduce antiferromagnetic coupling, \ninhibiting complete frustration. Hence, we expect a \nferrimagnetic ground state at low temperatures. The fact \nthat the resonances form a narrow band of delocalized \nstates suppresses screening effects and fluctuations due to \nspin waves (36). A simple estimate based on the direct \nexchange coupling between moments localized around \nimpurities gives a Curie temperature Tc\ne2·\nnv, where e2 \nis the electric charge (\n 1-5 eV·Å) ), and nv is the vacancy \nconcentration. In the present experiment, nv \n 3·1011 cm-2 \nand this simple estimation suggest a Curie temperature Tc \n50-200 K (21). \n4 \nOur findings have strong implications both from an \napplied and a fundamental point of view. They provide a \nsignificant stimulus to the theoretical community \ndemonstrating that for atomistically controlled \nexperiments, tight -binding methods give an excellen t \ndescription of graphene -like systems physics. The \nobserved resonances indicate that vacancies should limit \nsignificantly the mobility of carriers in graphene, and \nenhance its chemical reactivity. The existence of sharp \nelectronic resonances at the Fermi energy, strongly \nsuggests the formation of magnetic moments around single \nvacancies in graphite surfaces, implying a magnetic phase \nfor this free of impurities carbon system with high Curie \ntemperatures and small magnetization moments, which \nindicates a su itable route to the creation of non -metallic, \ncheaper, lighter, and bio -compatible magnets \nWe are thankful to J.Y Veuillen and P. Mallet for \nproviding us with the sputtering parameters. I.B was \nsupported by a Ram ón y Cajal project of the Spanish MEC. \nM.M.U . acknowledges financial support from MEC under \nFPU Grant Nº. AP -2004 -1896. Financial support from \nSpain's MEC under grants No. MAT2007 -60686, FIS2008 -\n00124 and CONSOLIDER CSD2007 -00010, and by the \nComunidad de Madrid, through CITECNOMIK, is \ngratefully ack nowledged. \n \n1. Brandt, N. B., Chudinov, S. M., and Ponomarev, Y. G., in \nModern Problems in Condensed Matter Sciences , edited by \nAgranovich, V. M., and Maradudin A. A. (North Holland, \nAmsterdam, 1988), Vol. 20.1 \n2. Novoselov K. S., Geim, A. K., Morozov, S. V., Jiang D., \nZhang, Y., Dubonos, S. V., Grigorieva, I. V., and Firsov, A. \nA., Science 306, 666 (2004). \n3. Novoselov, K. S., Jiang, D., Schedin, F., Booth, T. J., \nKhotkevich, V. V., Morozov, S. V., and Geim, A. K, Proc. \nNatl. Acad. Sci. U.S.A. 102, 10451 (2005). \n4. M. Fujita M., Wakabayashi K., Nakada K., and Kusakabe, K., \nJ. Phys. Soc. Jap. 65. 1920 (1996). \n5. Enoki, T., Kobayashi, Y., Fukui, K.I., Int. Rev. Mod. Chem. , \n26, 609 (2007). \n6. Pereira, V. M., Guinea, F., Lopes dos Santos, J. M., Peres, N. \nM. R., and Castro Neto, A. H., Phys. Rev. Lett. 96, 036801 \n(2006) \n7. Lehtinen, P. O ., Foster, A. S ., Ma, Y. C ., Krasheninnikov, A. \nV., Nieminen R. M ., Phys. Rev. Lett. 93, 187202 (2004). \n8. Yazyev, O. V., Phys. Rev. Lett. 101, 037203 (1998). \n9. J. J. Palacios, J. Fernández -Rossier, L. Brey, Phys. Rev B 77, \n195428 (2008). \n10. Niimi, Y ., Matsui, T ., Kambara, H ., Tagami, K ., Tsukada, M ., \nand Fukuyama H ., Phys. Rev. B 73, 085421 (2007). 11. Shibayama, Y ., Sato, H ., Enoki, T ., Endo. M ., Phys. Rev. Lett. \n84, 1744 (2000). \n12. Harigaya, K., and Enoki, T., Mechanism of magnetism in \nstacked nanographite with open shell electrons, Chem. Phys. \nLett. 351, 128 (2002). \n13 J. Cervenka, M. I. Katsnelson and C. F. J. Flipse, Nature \nPhys ics, doi:10.1038/nphys1399 \n14. Esquinazi , P., Spemann , D., Höhne , R., Setzer , A., Han, K.-\nH., and Butz , T., Phys. Rev. Lett. 91, 227201 (2003). \n15. Ohldag, H., Tyliszczak, T., Höhne, R., Spemann, D., \nEsquinazi, P., Ungureanu, M., and Butz, T., Phys. Rev. Lett. \n98, 187204 (2007). \n16. Kra sheninnikov, AV and Banhart, F., Nature Materials 6, 723 \n(2007) \n17. Gomez -Navarro C., De Pablo PJ., Gomez -Herrero J., Biel B, \nGarcia -Vidal FJ., Rubio A., Flores F, Nature Materials 4, 534 \n(2005) \n18. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, \nK. S., Geim, A. K. , Rev. Mod. Phys. 81, 109 (2009). \n19. Hashimoto, A., Suenaga, K., Gloter, A., Urita, K., and Iijima, \nS., Nature 430, 870 (2004). \n20. Hahn, J. R., and Kang, H., Phys. Rev. B 60, 6007 (1999). \n21. See EPAPS Document No. for a large sample overview and \nfor details of the theoretical model. For more information on \nEPAPS, see http://www.aip.org/pubservs/epaps.html. \n22. Mizes H. A., Foster J. S., Science , 244, 559 (1989). \n23. Kelly K. F., Halas N. J., Surface Science , 416, L1085 (1998) \n24 M. M. Ugeda, Doctoral Thesis, in preparation. \n25. Matsui,T ., Kambara, H ., Niimi, Y ., Tagami, K ., Tsukada, M ., \nFukuyama. H., Phys. Rev. Lett. 94, 226403 (2004). \n26. Li, G., and Andrei E. Y., Nature Phys . 3, 623 (2007). \n27. Brihuega I, Mallet P, Bena C, Bose S. Michaelis C., Vitali L., \nVarchon F., Ma gaud L., Kern K., Veuillen J. Y , Phys. Rev. \nLett. 101, 206802 (2008). \n28. Miller, D. L., Kubista, K. D., Rutter, G. M., Ruan M., de \nHeer, W. A., Fir st, P. N., and Stroscio, J. A. , Science 324, 924 \n(2009). \n29 T. Stauber, N. M. R. Peres and F. Guinea, Phys. Rev. B 76, \n205423 (2007). \n30. Arovas, D. P., and Guinea F., Phys. Rev. B 78, 245416 \n(2008). \n31. Slonczewski, J. C., and Weiss, P. R., Phys. Rev. 109, 272 \n(1958). \n32. McClure, J. W., Phys. Rev. 108, 612 (1958). \n33. Castro, E. V., López -Sancho, M. P., and Vozmediano, M. A. \nH., arXiv :0906.4061. \n34. A. C. Hewson, The Kondo Problem to Heavy Fermions , \nCambridge University Press, Cambridge, 1993. \n35. Brey, L., Fertig, H. A. , and Das Sarma, S., Phys. Rev. Lett. \n99, 116802 (2007). \n36. Edwards, D. M., and Katsnelson M. I., Journ. Phys.: -\nCondens. Matter 4, 3289 (2006). \n37. Horcas I., Fernandez R., Gomez -Rodriguez J. M., Colchero \nJ., Gomez -Herrero J. and Baro A. M., Rev. Sci. Instrum. 78, \n013705 (2007) .\n 5 \nSUPPLEMENTARY INFORMATION \n \nSample oveview \n-600 -400 -200 0 200 400 6000.20.4dI/dV [a.u.]\nVoltage [mV] LDOS clean graphite\nE) A) B)\nD)\nC) F)\n0.0 0.5 1.0 1.5 2.00204060\n height [pm]\nDistance [nm]\n2ndC layer\nsite\nsite\n2ndC layer\n \nFig S1. Sample overview before and after the irradiation process. 6 \nFigure S1 shows an overview of the sample before (a) and after (d) the Ar+ irradiation process. In these 40x40nm2 images, it \ncan be seen that no defect is present in our surfaces previously to the irradiation procedure. Fig. S1 C shows a high \nresolution STM images of the HOPG surface where both atoms of the honeycomb lattice are resolved; the one on \n site \nshowing a higher intensity than the one on \n site (see profile in S1 B). \nThe exact atomic location of the vacancies in the graphite l attice can be inferred from STM measurements, as it is reflected \nin the complex R3 scattering patterns originated from them [ 22, 23]. Both \n and \n vacancies exhibit a 3 -fold scattering with \nthe presence of three “arms” at 120º each. In the case of Fig. S1 D, vacancies on a \n site have one of their arms pointing \ntowards left, while in the case of \n vacancies one of their arms points towards right. \nLDOS of pristine graphite surfaces showed a featureless V -shaped form as inferred from our dI/dV spectra (fig S1 F). \n \nTHEORETICAL MODEL. \n \n1. Electronic structure . \nWe describe the \n bands of graphite using the parametrization s uggested in refs. ( 30, 32), as modified because of recent \nresults for bilayer graphene in ref. (28). The tight binding parameters include hoppings between next and next nearest \nneighbor layers. The values used (in meV) are shown in Table I. \nThe calculation of the local density of states is done used the iterative procedure in (S1). The \nimplementation of the method requires a finite imaginary part of the energy. The vacan cy is modeled \nwith a large on site energy, \n2000\nvac\n eV. The large enhancement of the density of states near the \nimpurity implies that even the smallest broadenings used, \n1\n meV induce the peak shown in Fig. \n3B. A simpler mod el, which can Table I be solved analytically, is obtained by keeping only the \nin-plane hopping \n0\n , and nearest neighbor interlayer hopping, \n1\n . The analytical solution of this \nmodel gives, for vacancies in an \n site the same solution as that found for single layer graphene (6, \n32), localized in the layer where the vacancy sits. The density of states at bulk and surface states are \nshown in Fig. S2. \nThe resonance in single layer graphene (6) is very delocalized, \n))/ log((1),(2 2aD r r\n where a is the lattice \nconstant, and D is the distance between vacancies. It can be smoothly matched to a superposition of continuum solutions \nobtained from the Dirac equation. Because of that, a local perturbation which induces a chan ge near the position of the \nvacancy does not affect much the resonance. A perturbation of strength \n0V localized within a length d of the vacancy \nchanges the position of the resonance by \n)/ log()/ log( )(02\n00 aD ad V rr Vd\n . Hence, for \nad\n and \na D\n, the shift in the resonance away from the neutrality point is small. As the extended states have a density of states \n2v )(F D\n, the broadening of the resonance is also small. \n \n \n A) B) \nFig, S2 . A) Density of states at a bulk site. Blue: \n site. Red: \n site. B) As in A), for a surface site. \n \n0\n 3160 \n1\n 390 meV \n2\n -20 meV \n3\n 315 meV \n4\n 44 meV \n5\n 38 meV \n-8 meV 7 \n2. Interaction effects. \n The interactions are described by the Hamiltonian: \n..21\n21)(21\n21\n; ;;;; ;\n;,; int 0\nch n nrV n n Ucc cct H HH\nj\nii ij i\niisisisii sj\nsjisi\n\n (SI 1) \nWhere we have separated the short range (Hubbard) and long range parts of the interaction. The second term in eq.(SI 1) \nincludes the disorder effects introduced by the vacancy. We diagonalize \n0H using the exact one particle eigenvalues: \nskskskk\nsvsvsvv cc cc H\nii i i\n;; ;\n;; ; 0\n (SI 2) \nThe on site interaction term can be written as: \nii isn U H ) (2 2\nint\n (SI 3) \nWith: \n; ; ; ;; ;; ;2; ; ; ;\n) (21\ni i i iz\nii i ii i iz\niz\ni ii ii ii i i i i\ncc ccscc scc sss ssss scc ccn\n\n (SI 4) \nThe local operators \nsic; can be expressed in terms of the eigenvectors of \n0H : \ni\nsk\nki\nk sv\nvi\nv si c c c\nj\njj ; ;\n (SI 5) \nInteractions suppress double occupancy of the resonance. We assume that in graphite, or in graphene near the neutrality \npoint, the resonance is half filled on the average. Then, it can be considered a spin one half interacting with the extended \nstates. For one resonance, the Hamiltonian which describes the interaction can be written as: \njij\nvi\nvj\nki\nk ij\nkkkkv\nii\nvi\nki\nk\nkkkkv kv rV ss ssUH\n,'\n',',2\n'\n',', , )(4\n \n (SI 6) \nWhere the operators \nvs and \n',kks are generaliza tions of those in eq. (SI 4). The coupling is given by the exchange \ninteractions, and it is proportional to the overlap between the wavefunctions. This Hamiltonian describes a ferromagnetic \nKondo system. As the resonance is spread over many sites, i, the localized spin interacts with many conduction band \nchannels, defined in terms of their angular momentum around the position of the impurity. The coupling is ferromagnetic, \nand the moment is not quenched at low temperatures. The effective Kondo coupling can be large, despite the low density of \nstates of graphite near the Fermi level, because the phaseshift induced by a spin flip proce ss in the presence can be large \n(S2). This enhancement of the coupling arises from the strong scalar potential induced by the vacancy. The coupling to \nmagnetic impurities outside the graphene layers is determined by virtual transitions into states with different occupancies, \nleading to an antiferromagnetic coupling suppressed by the low density of states of clean graphite near th e Fermi level (S3). \nThe extension of the orbital associated to the vacancy leads to a long range coupling between magnetic moments, when the \nwavefunctions of resonances around different vacancies overlap. This is the case if the vacancies belong to the sam e \nsublattice. We find a ferromagnetic coupling, given by: \n'2\n22\n,''2\n'2\n' )(vv\njij\nvi\nvj\nvi\nv iji\nv\nii\nv vv vv ssde\ndaU rV ssU H \n (SI 7) \nWhere d is the distance between vacancies, and we write for the long range part as \nij ij re rV 2)(\n . The coupling between \na magnetic moment and the average magnetization induced by the rest is \nv v neanU J2 2\n where \nvn is the \ndensity of vacancies. For low densities, the coupling is determined by the contribution of the long range exchange 8 \ninteraction. For \n2 11cm103\nvn and\n02AeV51\ne , we find a Curie temperature \n25050\nJ TC K. This \nestimate depends strongly on the density of vacancies. \n \nS1. Guinea, F., Tejedor, C., Flores, F., and Louis, E., Phys. Rev. B 28, 4397 -4402 (1983). \nS2. Hentschel, M., and Guinea, F., Phys. Rev. B 76, 115407 2007). \nS3. Uchoa, B., Kotov, V. I., Peres, N. M. R., Castro Neto, A. H., Phys. Rev. Lett. 101, 026805 (2008). " }, { "title": "2005.14478v3.Direction_sensitive_magnetophotonic_surface_crystal.pdf", "content": "Direction-sensitive magnetophotonic surface crystal\nRichard M. Rowan-Robinson* J\u0013 erome Hurst Agne Ciuciulkaite Ioan-Augustin Chioar Merlin Pohlit\nMario Zapata Paolo Vavassori Alexandre Dmitriev* Peter M. Oppeneer Vassilios Kapaklis*\nR. M. Rowan-Robinson\nDepartment of Material Science and Engineering, University of She\u000eeld, She\u000eeld, United Kingdom\nEmail: r.rowan-robinson@she\u000eeld.ac.uk\nJ. Hurst\nUniv. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-Spintec, F-38000 Grenoble, France\nR. M. Rowan-Robinson, J. Hurst, A. Ciuciulkaite, I-A. Chioar, M. Pohlit, Prof. P. M. Oppeneer, Prof.\nV. Kapaklis\nDepartment of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden\nEmail: vassilios.kapaklis@physics.uu.se\nI-A. Chioar\nDepartment of Applied Physics, Yale University, New Haven 06511, CT, USA\nM. Zapata, Prof. P. Vavassori\nCIC nanoGUNE BRTA, E-20018 Donostia-San Sebastian, Spain\nM. Zapata, Prof. P. Vavassori\nIKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain\nProf. A. Dmitriev\nDepartment of Physics, University of Gothenburg, SE-412 96 G oteborg, Sweden\nEmail: alexd@physics.gu.se\nKeywords: magnetoplasmonics, rare-earth{transition-metal ferrimagnets, all-optical switching, magne-\ntophotonic crystal, Fano resonance\nNanometer-thin rare-earth-transition metal (RE-TM) alloys with precisely controlled compositions and out-of-plane magnetic anisotropy\nare currently in the focus for ultrafast magnetophotonic applications. However, achieving lateral nanoscale dimensions, crucial for\npotential device downscaling, while maintaining designable optomagnetic functionality and out-of-plane magnetic anisotropy is ex-\ntremely challenging. Here we integrate nanosized Tb 18Co82ferrimagnetic alloys, having strong out-of-plane magnetic anisotropy,\nwithin a gold plasmonic nanoantenna array to design micrometer-scale a magnetophotonic crystal that exhibit abrupt and narrow\nmagneto-optical spectral features that are both magnetic \feld and light incidence direction controlled. The narrow Fano-type reso-\nnance arises through the interference of the individual nanoantenna's surface plasmons and a Rayleigh anomaly of the whole nanoan-\ntenna array, in both optical and magneto-optical spectra, which we demonstrate and explain using Maxwell-theory simulations. This\nrobust magnetophotonic crystal opens the way for conceptually new high-resolution light incidence direction sensors, as well as for\nbuilding blocks for plasmon-assisted all-optical magnetization switching in ferrimagnetic RE-TM alloys.\n1 Introduction\nNanoscale magnetophotonics merges magnetism with nanophotonics [1], combining seamlessly magneto-\noptical (MO) e\u000bects with surface plasmons, thus being capable of delivering ultra-high performance bi-\nological and chemical sensors [2, 3], active tunability in nano-optics by external magnetic \felds [1, 4, 5,\n6, 7, 8, 9, 10, 11, 12], and setting a platform for ultrafast opto-magnetism and spintronic [13] devices on\nthe nanoscale. Pure ferromagnetic plasmonic systems were earlier considered unfeasible for these pur-\nposes due to the high ohmic losses associated with the transition-metal ferromagnets. However, to a large\nextent, these can be overcome through nanopatterning [14, 15], materials engineering and fabrication of\nhybrid noble metal-ferromagnetic nanostructures [16, 17, 18, 19, 20]. The enhancement of various MO\ne\u000bects is typically achieved in these systems through near-\feld light concentration at the nanoscale, boost-\ning light-magnetism interactions that relate to the MO Voigt parameter of the ferromagnet [21, 22]. Im-\nportantly, by exploiting magnetic anisotropy control, the magnetization can be stabilized in a desired di-\nrection and MO e\u000bects can be recorded at zero external magnetic \feld. Linewidth engineering [17, 23,\n24] wherein high Q-factor resonances are achieved, can furthermore be employed in ordered arrays of\nmagnetoplasmonic nanoantennas with surface lattice resonances.\n1arXiv:2005.14478v3 [physics.optics] 18 May 2021The use of rare-earth{transition-metal alloys is of paramount interest for future nanoscale magnetopho-\ntonic and magnetoplasmonic systems for several key reasons. Firstly, they are known to exhibit very\nlarge MO e\u000bects [25, 26] potentially permitting very high real-time active tunability of light polariza-\ntion. Secondly, they can exhibit strong perpendicular magnetic anisotropy, yet with an amorphous tex-\nture [27, 28, 29, 30, 31]. For instance, carefully engineered Co/Pt multilayered nanodots having large\ninterfacial spin-orbit coupling with perpendicular magnetic anisotropy, demonstrate tenfold enhance-\nments in MO activity and demonstrate the great potential of out-of-plane magnetic anisotropy materi-\nals for magnetoplasmonics [32]. The amorphous texture of RE-TM alloys greatly simpli\fes the otherwise\nstringent requirements on material microstructure for obtaining these highly desired magnetic proper-\nties. As such, they can be grown on noble metals like Au with minimal residual stresses, and with highly\nsmooth interfaces, thereby maintaining much of their original magnetic properties even after patterning\n[27]. Importantly, with perpendicular magnetic anisotropy the remanent magnetization state of the mag-\nnetic nanostructures can be designed to be parallel to the light propagation direction for normal light\nincidence, greatly simplifying potential practical applications of magnetoplasmonic crystals. This allows\none to explore their MO functionality (such as, tunable Faraday e\u000bect) directly, i.e., without the need\nof external magnetic \felds in order to stabilize the magnetization along the out-of-plane axis. Thirdly,\nferrimagnetic alloys such as Tb 18Co82, have recently experienced extensive interest due to the demon-\nstration of enhanced spin-orbit torques [33, 34, 35] and all-optical switching [35, 36, 37, 27], allowing for\nzero-\feld magnetic switching, on picosecond timescales, with the use of pulsed lasers. Thus, demonstrat-\ning the compatibility of these materials with nanoantennas is essential for the development of nanoscale\n(i.e., sub-di\u000braction) all-optical switching technologies [13].\nHere we devise a magnetophotonic crystal composed of nanocone Au plasmonic nanoantenna arrays in-\ncorporating an amorphous RE-TM ferrimagnetic alloy, Tb 18Co82, with perpendicular magnetic anisotropy\n[27]. We show that this hybrid Au/Tb 18Co82system provides high-Q MO resonances, overcoming the\nlosses associated with ferrimagnetic alloys. By Maxwell-theory modelling we show that this is achieved\nthrough the resonant collective excitation of surface lattice modes that exhibit a particularly strong an-\ngular dispersion. This is a result of the interference of a Rayleigh anomaly with the individual nanoan-\ntennas' plasmons, giving rise to surface lattice resonance resonances with characteristic Fano-type asym-\nmetric lineshape in both the optical and MO spectra. We demonstrate an exceptionally strong tunability\nof the spectral position of such resonances by varying the angle of incidence (incident direction) of the\nincoming light, exemplifying the potential of magnetophotonic crystals for high-resolution mechanical\ntilt-angle sensors and, more broadly, for actively-controlled optical systems [38, 39].\n2 Results and Discussion\nNanocone antennas were previously shown to exhibit a very strong \feld enhancement [40], with the elec-\ntromagnetic \feld concentrated at the tip [40, 41]. We build large rectangular lattice arrays of Au/Tb 18Co82\ntruncated nanocone antennas (Fig. 1a) [27] with two selected base diameters (179 \u00065 nm and 227\u0006\n4 nm (see SEM insets in Fig. 1f and h respectively). The light incidence angle ( \u000bi) is varied with respect\nto the lattice plane, directed along either one or the other of the array periodicity (Fig. 1b). We \frst\nuse \fnite-element Maxwell-theory simulations (COMSOL Multiphysics, see Supporting Information) to\npinpoint the emerging resonances' linewidth narrowing and high incidence-angle sensitivity. The mag-\nnetophotonic crytstal is built of Au(80 nm)/Tb 18Co82(15 nm) truncated nanocones (base diameter, DB\n= 179 nm), arranged in a rectangular array with 340 nm \u0002425 nm periodicity (Fig. 1c). The light in-\ncidence direction angle (using the optical convention) de\fnes a scattering plane which is parallel to one\n(340 nm) or the other (425 nm) of the array periodicity axes with azimuthal angles 'i= 0 or'i= 90\u000e,\nrespectively.\n2Figure 1: Magnetophotonic crystals composed of arrays of truncated nanocone hybrid antennas, with tunable optical\ntransmission response. (a) Schematic of a single Au-TbCo nanoantenna featuring PMA (left) and scanning electron micro-\ngraph view of a magnetophotonic crystal (right). (b) Magnetophotonic crystal illumination with resulting Faraday rotation\n(\u0012F), ellipticity ( \u0011F) of the transmitted light, and the Rayleigh anomaly associated with the passing-o\u000b of the di\u000braction\norder. (c) Magnetophotonic crystal illumination with two azimuthal orientations ( 'i= 0 and 90\u000e) with respect to the\nincident light polarisation ( Ei) and scattering plane, with pEdenoting the orientation of the electric dipolar plasmon in\nthe nanoantennas. The reciprocal lattice vectors [1 ;0] and [0;1] are shown to illustrate the 90\u000erotation of the reciprocal\nlattice vectors with respect the real space lattice. (d, e) Calculated transmission spectra for incidence angles \u000bibetween 0\nand 20 degrees, for the 'i= 0 (d) and the 'i= 90\u000e(e) con\fgurations, respectively. (f, g) Measured transmission spectra\nfor incidence \u000biangles 0 - 20 degrees for the magnetophotonic crystal built on DB= 179 nm nanoantennas for the 'i=\n0 (f, inset { SEM of nanoantennas in this magnetophotonic crystal) and the 'i= 90\u000e(g). (h, i) Same as (f, g) but for the\nmagnetophotonic crystal with DB= 227 nm nanoantennas (inset in (h) { nanoantennas SEM).\nSurface lattice resonances are the result of the coupling between a broad lossy resonance, in this case\nthe localised plasmon resonances of individual nanoantennas, and di\u000bracted waves in the plane of the\nnanoantenna array (a detailed description is provided in the supporting information). This condition is\n3generally observed close to a Rayleigh anomaly, where for a given \u000biand lattice periodicity, a Rayleigh\nanomaly exists where a di\u000bracted wave is directed parallel to the grating [42]. This Rayleigh anomaly\nrepresents the passing-o\u000b of a di\u000braction order through a laterally excited beam. There can exist a large\nnumber of these di\u000braction orders, which are labeled by two integers nandm. The allowed waves are\nobtained by imposing the condition that the component of the light wave-vector normal to the lattice\nsurface is real, through the expression\nk?=q\nk2\ns\u0000\u0000\nkk+mG1+nG2\u000122<: (1)\nIn the above formula, ks= 2\u0019nsub=\u0015corresponds to the light wave-vector in the substrate, where nsubis\nthe refractive index of the fused silica substrate ( nsub= 1.45),\u0015the light wavelength,\nkk=k0[sin(\u000bi) cos('i)ux+ sin(\u000bi) sin('i)uy] corresponds to the wave-vector component of the in-\ncident radiation (in air/vacuum) parallel to the lattice surface, k0= 2\u0019=\u0015is the light wave-vector in\nair and G1= (2\u0019=a)ux,G2= (2\u0019=b)uyare the reciprocal lattice vectors, with ux,uybeing the\nreciprocal lattice unit vectors and a= 340 nm, b= 425 nm, being the lattice parameters. The number\nof di\u000bracted waves depends on the lattice parameters, the angle of incidence, the refractive index of the\nsubstrate and the light wavelength. For wavelengths greater than 600 nm, Equation (1) indicates that\nonly the di\u000bracted waves ( n= 0 ;m=\u00001) for'i= 0 and (n=\u00001 ;m= 0) for'i= 90\u000ecan be\nobtained by varying the incidence angle between 0 - 20\u000e(see Supporting Information).\nWe use reciprocal vector notation, such that the Rayleigh anomaly occurs at wavelengths \u0015[n;m]\nRwith\nwave-vector orientated along the reciprocal lattice vectors [ n;m]. Fig. 1c demonstrates how the recip-\nrocal lattice vectors are orientated with respect to the real-space lattice. The analytical expressions for\nthe two allowed substrate waves ([0 ;\u00001];[\u00001;0]) from Equation (1), are given by\n\u0015[0;\u00001]\nR =a[nsub+nairsin (\u000bi)] for'i= 0; (2)\n\u0015[\u00001;0]\nR =b[nsub+nairsin (\u000bi)] for'i= 90\u000e; (3)\nwherenair= 1 is the refractive index of air.\nWe \frst calculate the spectral transmission through the array for p-polarised light (i.e., incident electric\n\feld is in the scattering plane) (Fig. 1d, e). Individual nanoantenna dipole-type plasmons are excited in\nthe respective scattering planes at 690 nm at normal incidence ( \u000bi= 0). For the 'i= 0 con\fguration\n(scattering plane along 340 nm array periodicity, Fig. 1d) the surface lattice resonances from Eq. (2) are\nat\u0015[0;\u00001]\nR = 493 nm, 552 nm, 581 nm and 609 nm for \u000bi= 0, 10, 15, and 20 degrees, respectively, and\ntherefore not spectrally overlapping with the nanoantennas' individual plasmons. For 'i= 90\u000e(scatter-\ning plane along 425 nm array periodicity, Fig. 1e), Eq. (3) gives \u0015[\u00001;0]\nR = 616 nm, 690 nm, 726 nm and\n762 nm, strongly overlapping with the nanoantennas' plasmon, resulting in a very substantial tuning of\nthe spectrally abrupt transmission spectrum by changing \u000bi(see Fig. 1e).\nIn the Fano-type resonance description [43, 44], the nanoantennas plasmon represents a continuum of\nstates, whereas the Rayleigh anomaly is a narrow line-width di\u000bracted wave, which, upon interfering\nwith the continuum, results in the characteristic asymmetric lineshape of the surface lattice resonances.\nA similar behaviour has been seen previously with magnetoplasmonic Ni nanoantennas arrays [24, 23],\nwhere the overlap between \u0015[n;m]\nRand the nanoantenna plasmon was tuned by varying the lattice period-\nicity of the magnetoplasmonic crystal. However, a much simpler alternative method of tuning the sur-\nface lattice resonance spectral position can be obtained using the angular dispersion of \u0015[n;m]\nR. This tun-\ning of the spectral position of the surface lattice resonance opens up applications as mechanical tilt-angle\ntransducers/sensors, and in contrast with previously observed transmission/re\rectance angular depen-\ndence in pure plasmonic arrays [45], this magnetoplasmonic crystal allows one to fully explore angular\nMO tunability.\nThe dipolar radiation \feld is strongest transverse to the dipolar plasmon oscillation given by pE(Fig.\n1c). In our simulations we used p-polarised light and hence the electric dipole excitation within individ-\nual nanocone antennas is orientated within the scattering plane and parallel to the di\u000braction anomaly.\n4This dipole cannot radiate along the oscillation direction, hence there must exist an additional mech-\nanism for light to be scattered along the other periodicity direction for the excitation of the Rayleigh\nanomaly. We show this to be the result of an out-of-plane component to the electric dipole due to the il-\nlumination at oblique incidence (see Supporting Information), which would radiate in all directions within\nthe plane of the lattice [46] providing the excitation for all \u0015[n;m]\nR, e.g. [-1, 0], [0, -1], [-1, -1] waves for p-\npolarised light.\nThe measured transmission spectra are shown in Fig. 1f-i. In agreement with the electromagnetic simu-\nlations above, for the 'i= 0 con\fguration (Fig. 1f, h) the transmission spectra show very little depen-\ndence on\u000bi. The nanoantenna plasmon is red-shifted and spectrally broadened as compared to the sim-\nulations though, which is likely a result of the thin Al 2O3isolation layer (see Methods) and oxidation of\nthe exposed Tb 18Co82side-walls on the fabricated nanocones and also the size and shape distribution of\nthe nanoantenna ensemble. There is a spectral feature between 500 - 600 nm (Fig. 1f, h) that migrates\nto longer wavelengths as \u000biincreases that is most likely due to \u0015[0;\u00001]\nR, since it occurs at the same spec-\ntral positions for both the DB= 179 nm (Fig. 1f) and 227 nm (Fig. 1h) nanoantennas, suggesting its\norigin relates to the lattice and not the individual nanoantenna plasmon resonance.\nWhen rotated into the 'i= 90\u000econ\fguration (Fig. 1g and i), the strong variations in the transmission\nspectra are observed, in excellent agreement with the simulations, in both spectral position and line-\nshape, albeit with reduced amplitude. For both DB= 179 nm (Fig. 1g) and 227 nm (Fig. 1i) nanoanten-\nnas, the\u000bi= 0 incidence shows a small blue shift of the plasmon for the 'i= 90\u000econ\fguration relative\nto the'i= 0 con\fguration. As shown in the inset scanning electron microscopy images, the nanocones\nare not perfectly circular and this discrepancy is likely a result of this asymmetry. Markedly, the broad\nspectral distribution with the DB= 227 nm nanocone antennas allows for a larger tuning bandwidth,\nsuch that there exists a larger range of \u000bifor which\u0015[\u00001;0]\nR overlaps with the nanoantenna plasmon.\nWhile we readily earn high incidence direction tunability of optical transmission with the designed mag-\nnetophotonic crystals, resonances in MO spectra can yield much larger Q-factors [47]. Maccaferri et al.\n[48] showed that an out-of-plane magnetization in the presence of the electric dipolar plasmon gives rise\nto an in-plane MO dipolar plasmon ( pMO) which is orientated orthogonal to pEand is induced in the fer-\nromagnetic layer. The magnitude of pMOis proportional to the magnitude of pE. Given that a material's\noptical constants are typically much larger than their MO constants, even lossy broad localised plas-\nmon resonances can give rise to large enhancements in MO activity as compared to ferrimagnets without\nplasmonic integration. This transverse oscillation is induced via spin-orbit coupling, generating an oscil-\nlation of conduction electrons in-the-plane but orthogonal to pE. With the use of p-polarised light, the\npure optical dipole is orientated along pEand the transverse MO dipole is aligned along pMO(Fig. 2a).\nHence, the use of p-polarised light results in the MO dipole induced in the Tb 18Co82layer which radiates\nstrongly in the scattering plane, and is therefore expected to be most sensitive to the angular dispersion\nof the surface lattice resonances as the crystal is tilted by \u000bi.\nIn Fig. 2b-j the calculated and experimental Faraday rotation ( \u0012F), Faraday ellipticity ( \u0011F) and Faraday\nangle (\u0002 F=p\n\u00122\nF+\u00112\nF) are presented. The calculated Faraday e\u000bect using the experimental permit-\ntivity for a Tb 18Co82thin \flm is shown in Fig. 2b, e, h for the DB= 179 nm nanocone antennas array\n(see Supporting Information for details). The 'i= 0 con\fguration shows no angular dependence for the\nFaraday e\u000bect (see Supporting information) and through \ftting a Lorentzian to the \u000bi= 0 transmission\nand \u0002 Fspectra for the 'i= 0 con\fguration we estimate that the MO resonance exhibits a two-fold re-\nduction in linewidth relative to the pure optical resonance. While for 'i= 0 (no overlap of nanoanten-\nnas plasmon with Rayleigh anomaly) a reasonable spectral feature narrowing is achieved without angular\ndependence, in the 'i= 90\u000econ\fguration the experimental Faraday spectra show strong angular depen-\ndence and suggest that sizeable Faraday angles of up to 0.3\u000eare readily available. The simulated spectra\nprompts that extremely sharp features exist that coincide with \u0015[\u00001;0]\nR (Fig. 2b, e, h).\nThe Rayleigh anomaly is strongest through the substrate and the observation of strong di\u000bractive e\u000bects\nin the Faraday spectra indicates that the MO dipole induced in the Tb 18Co82layer is transferred to the\nrest of the nanoantenna [19]. The experimental MO spectra measured for the nanoantennas with DB\n= 179 nm (Fig. 2c, f, i) compare well to the calculations. The excellent match of the measured spectra\n5Figure 2: Angle-of-incidence spectral dependence of the Faraday e\u000bect in the magnetophotonic crystals. (a) Illumination\ncon\fguration as in Fig, 1c, with added MO plasmon dipole of nanoantenna ( pMO, green). (b, e, h) Calculated spectral \u0012F\n(b),\u0011F(e) and \u0002 F(h) for incidence angles \u000bibetween 0 and 20 degrees. Measured \u0012F(c, d),\u0011F(f, g) and \u0002 F(i, j) for the\nDB= 179 nm and 227 nm nanoantenna arrays respectively. A quadratic polynomial has been \ftted to the \u0012Fmeasure-\nments and subtracted to remove the background contribution which arises from the Faraday rotation of the fused-silica\nsubstrate, which is strongest for short wavelengths and approaches zero with increasing wavelength.\nwith simulations demonstrates the suitability of combining \fnite-element methods with experimentally\nmeasured thin-\flm permittivity for the calculated design of magnetophotonic devices. For the nanoan-\ntennas with DB= 227 nm (Fig. 2d, g, j) there is a stronger Faraday e\u000bect, but with broader spectral\nfeatures, demonstrating the trade-o\u000b between adding more magnetic material in the nanoantenna whilst\nmaintaining small dimensions for narrow plasmon resonances.\nFrom the above it is clear that it is not possible to measure the MO response of the Au-Tb 18Co82nanoan-\ntennas o\u000b-resonance, where \u0012Fand\u0011Fquickly drop to values comparable to the measurement uncer-\ntainty. In e\u000bect, the nanoantennas plasmons strongly amplify the minute magnetic signals that ordi-\nnarily wouldn't be resolved. It is possible to estimate the Tb 18Co82amount in each nanoantenna, corre-\nsponding to a nanodisk with 86 \u000610 nm diameter and 15 nm height for nanoantennas with base diam-\neter of 179 nm. This yields Tb 18Co82e\u000bective \flm thickness (i.e. the thickness of a \flm made with the\n6same amount of material) of approximately 0.6 nm, of the order of an atomic monolayer, demonstrating\nthe MO ampli\fcation obtained through the nanoantenna's plasmons.\nThe experimental \u0002 F,\u0011Fand\u0012Fcurves all show abrupt features that onset with the excitation of the\nsurface lattice resonance associated with \u0015[\u00001;0]\nR in the'i= 90\u000econ\fguration. However, spectrally just\nprior to this resonance there is the greatest change in MO activity for the smallest change in wavelength.\nSince this feature is dependent on the spectral position of \u0015[\u00001;0]\nR, it can be e\u000bectively tuned by varying\n\u000bi, indicating the potential use of such magnetophotonic crystals as light incidence direction/angular\nsensors. This is explored in Fig. 3a, where hysteresis loops are recorded through measurements of the\ntransmitted light ellipticity at a wavelength of 730 nm for the nanoantennas with base diameter 227 nm\nfor di\u000berent \u000bi. The nanoantenna's Tb 18Co82tops maintain perpendicular magnetic anisotropy even af-\nter the lithography process, which is clear from the large remanent magnetization observed in the hys-\nteresis loops in Fig. 3a, reducing the magnetic \feld strength required to saturate the sample along the\nout-of-plane direction. The dynamic tunability of the MO activity by varying \u000biis remarkable in this\ncase, resulting in a dramatic change in the magnitude of \u0011F, where extraordinarily at \u000bi= 15\u000ethe loop\nis even inverted (see a view of \u0011Ffor the spectral region around the surface lattice resonance in Support-\ning information; it is clear that this sign change in \u0011Fis associated with the migration of the surface lat-\ntice resonance to the measurement wavelength of 730 nm).\nThis is explored further in Fig. 3b where the change in Faraday ellipticity ( \u000e\u0011F) between successive wave-\nlength increments ( \u000e\u0015= 5 nm) is plotted. Since the gradient of this feature is positive when it coincides\nwith\u0015[\u00001;0]\nR (see inset of Fig. 3a), the \u000e\u0011F<0 data has been excluded from the \fts. It is evident that \u0011F\nundergoes a sign change, which in turn is tunable by varying \u000bi. This active tuning modality was pre-\nviously envisioned for refractive index biochemosensing, where the spectral region of maximum sensi-\ntivity can be tuned by varying the angle of incidence, thereby allowing to operate in a spectral region\nwhere the analyte solution is minimally absorbing [49]. Here we foresee that the deviations from a set\nangle, i.e., a mechanical tilt, could be employed in high-precision tilt-control systems and detected with\nhigh accuracy, simply as reduced MO activity in transmittance. The latter feature starkly di\u000berentiates\nthis approach from the currently employed optical systems where re\rection is captured by a complex\nsystem of mirrors/detectors often with the need of a microelectromechanical (MEMS) array of actua-\ntors. Lorentzian functions have been \ftted to the \u000e\u0011Fdata, in order to estimate the spectral width of\nthe abrupt transition in \u0011F. Due to the limited number of data points on this abrupt spectral transition,\na full estimate of the full-width at half-maximum (FWHM) is di\u000ecult to obtain from these \fts. How-\never, all values are within the 5 \u000010 nm range (which is comparable to the wavelength resolution of the\nsetup) with the exception of the \u000bi= 10\u000ewhere a FWHM of 24 \u000610 nm is obtained due to the anoma-\nlously large error on this particular measurement.\nCrucially, the perpendicular magnetic anisotropy in this magnetophotonic crystal allows for the measure-\nment of the magnetic di\u000berential absorption of circularly polarised light, underpinning \u0011F, without the\nneed for an out-of-plane magnetic \feld to stabilize the magnetization along the propagation direction of\nlight. When circularly polarized light beam, with a time-varying helicity is incident on the sample, we\ncan measure the ratio, Cq\n!=Cq\n\u000ewhich is proportional to the di\u000berential absorption of circularly polarised\nlight (see Methods) for the two opposite polar magnetization states. Here, Cq\n!is the amplitude of the\n!=2\u0019= 50 kHz signal from the modulation of the light circular polarization (see Methods), for a \fxed\npolar magnetization q=\u0006Mz, whileCq\n\u000eis the DC signal intensity, which contains the helicity indepen-\ndent absorption contribution. Fig. 4a shows several spectra for the nanoantennas with base diameter of\n227 nm for di\u000berent values of \u000bi, in the'i= 90\u000econ\fguration and in zero external magnetic \feld. The\nspectral minima strongly depend on \u000bi. If we include an external \feld, the amplitude of Cq\n!=Cq\n\u000ecan be\nfurther modulated by reversing the magnetization ( q= +Mz!\u0000Mz, and vice versa), as indicated by\nthe variation between the dashed and solid curves. The magnetophotonic crystal then exhibits active\ntransmission tunability, whereby absolute transmission can be enhanced or attenuated with the use of\na magnetic \feld. Similar active magnetic transmission tunability has been devised with magnetoplas-\nmonic chiral nanoantennas [16], however, an external \feld was required to orient the magnetization out-\nof-plane throughout the measurement, whereas here the external \feld is only required to set the mag-\n7-400-2000200400Magnetic Field [mT]-0.2-0.10.00.10.2Ellipticity [deg]0 degs10 degs15degs700750Wavelength [nm]0.00.1F [deg]660700740780Wavelength [nm]0.000.020.040.060.08F [deg]0 degs10 degs15degs20 degsa)b)Figure 3: Dynamic Faraday ellipticity in the magnetophotonic crystal. (a) Hysteresis loops under externally-applied mag-\nnetic \feld recorded from the magnetophotonic crystal with DB= 227 nm, at a wavelength of 730 nm demonstrating how\nthe magnitude and sign of the Faraday ellipticity ( \u0011F) can be controlled through the illumination incidence angle ( \u000bi),\nvarying between 0 and 20 degs. (b) The change in Faraday ellipticity ( \u000e\u0011F) measured at di\u000berent wavelengths. Follow-\ning the onset of the surface lattice resonance, there is an abrupt change in light ellipticity at various illumination angles\n(0-20 degrees), which is associated with a maximum in \u000e\u0011F. The peaks at di\u000berent incidence angles have been \ftted with\nLorentzians.\nnetic state. An additional tuning knob is implemented through the light incidence direction/angle \u000bi,\nwhereby the spectral location of this maximum for magnetic modulation can be tuned with the surface\nlattice resonance.\nWe de\fne a magnetic asymmetry ratio ( C\u0000Mz!\u0000C+Mz!)=(C\u0000Mz\u000e+C+Mz\u000e), which represents the avail-\nable helicity-dependent transmission modulation between the two antiparallel magnetization states (see\nMethods) which is plotted in Fig. 4b. The dispersion of the surface lattice resonance calculated from\nequation (3) is given by the dashed lines. Here, it is clear that the latter dictates the onset wavelength\nfor the magnetic modulation of the di\u000berential circular transmission, meaning that the peak sensitiv-\nity can be tuned to arbitrary wavelength between 650 nm - 800 nm. This tunability range is governed\nby the FWHM of the magnetophotonic crystal transmission spectra. A maximum magnetic asymme-\ntry ratio of around 0.5% can be obtained, however, we believe there is enormous scope for improvement\nthrough composition optimisation of the RE-TMs and the noble metal thicknesses in the nanoantennas,\nincluding exploring new geometries sustaining plasmon optically dark modes, which result in a stronger\nplasmonic enhancement of the MO activity than achieved with the here-used strongly scattering dipo-\nlar plasmons [50]. The essential operation of a simple mechanical tilt-control/light incidence optical sen-\nsor can be further envisioned as in Fig. 4c. The di\u000berential chiral transmission ( Cq\n!=Cq\n\u000e) reports the me-\nchanical tilt/change of light incidence direction angle on the pre-magnetized magnetophotonic crystal by\nhaving sharp spectral dips at various wavelengths. We can also envision that by using materials exhibit-\ning all-optical magnetization switching [27] such as the TbCo family of alloys employed here, the need\nfor the external magnetic \feld to setup the magnetic state of the magnetophotonic crystal or for mag-\nnetic transmission modulation can be entirely removed, whereby the transmission would be modulated\npurely optically at the ultrafast (fs) timescale and allowing for sub-wavelength (nanoscale) miniaturiza-\ntion [35, 51].\n8Figure 4: Angle-controlled chiral transmittance and mechanical tilt-angle sensing. (a) Spectral dependence of the Cq\n!=C\u000e\nsignals, where q= +Mzor -Mzfor the solid and dashed curves respectively. Cq\n!is related to the total circular dichroism\nfor a particular magnetization state, containing both magnetic and non-magnetic contributions. (b) The amplitude of\nthe magnetic modulation of the helicity dependent transmission as function of both wavelength and \u000bi, which relates to\nthe di\u000berence between the solid and dashed curves in (a). The dashed white line indicates the expected location of the\nRayleigh anomaly calculated from Eq. (3). (c) Schematics of the tilt-angle sensing device, where the di\u000berence in left- and\nright- circularly-polarized light, passing through the pre-magnetized magnetophotonic crystal, is detected as spectrally-\nresolved di\u000berential chiral transmission, having sharp spectral dips at distinct wavelengths, depending on the light's angle\nof incidence.\n3 Conclusion\nIn conclusion, our work demonstrates the seamless integration of a rare-earth{transition-metal into mag-\nnetophotonic crystals. A strong angular dispersion is engineered through the interference of the Rayleigh\nanomaly and the nanoantenna's plasmons, producing a sharp surface lattice resonance in both the op-\ntical and MO responses. We showcase dynamic tunability of magnetophotonic crystals using the light's\nincidence direction angle, which strongly modi\fes the MO response, as rationalized using \fnite-element\nmethod electromagnetic simulations. Further, we have shown the magnetic modulation of the di\u000berential\ncircular transmission of a magnetophotonic crystal, with measurements performed in zero external mag-\nnetic \feld, exploiting the perpendicular magnetic anisotropy of the magnetic nanoantennas. More gen-\nerally, the integration of rare-earth{transition-metals within plasmonic nanoantennas o\u000bers an exciting\nplatform for highly tunable, ultrafast all-optical switching active magnetophotonic devices [51, 1]. Such\narchitectures could also \fnd further application scope where the optical response from magnetophotonic\ncrystals can be tuned by the angle of incidence [38, 39] in combination with the recon\fgurable magnetic\nstructure [52] steered by all-optical ultrafast magnetization switching or by external magnetic \felds.\n4 Experimental Section\nSample Fabrication :\nThe plasmonic nanoantennas are fabricated using a top-down approach, based on the method outlined\nby Horrer et al. [40]. Au(80 nm) \flms were deposited using electron-beam evaporation onto fused-silica\nsubstrates. Later, Al 2O3(3.5 nm)/Tb 18Co82(15 nm)/Al 2O3(2 nm) \flms were sputter deposited onto these\n\flms. The Tb 18Co82layer was deposited through co-sputtering, with the complete structure being Au(80\nnm)/Al 2O3(3.5 nm)/Tb 18Co82(15 nm)/Al 2O3(2 nm). The additional thin Al 2O3layers were used as cap-\nping and isolating layers for the Tb 18Co82. Here, the composition of the \flm can be varied by adjusted\n9the relative power of the Co and Tb magnetrons. Calibration \flms were made with di\u000berent power ra-\ntios on the two magnetrons and compositions were veri\fed using Rutherford back scattering. Electron\nbeam lithography was used to de\fne disk shaped apertures in a MicroChem 496PMMA A4 electron-\nbeam resist. Electron-beam evaporation was used to deposit an Al mask through the resist followed by\nremoval of the PMMA mask with Acetone. The resulting structure was then milled at a 5 deg incidence\nangle with sample rotation, removing all material unprotected by the Al mask. Any remaining Al mask\nwas then removed with the photoresist developer Microdeposit 351, which in this case was used as a se-\nlective etcher to target the Al. A conical pro\fle is induced through a combination of the small lateral\ncomponent of the milling which depends to some extent on the small milling incidence angle [53]. In our\nsamples, this results in a constant slope pro\fle of approximately 62 degrees for all nanoantenna arrays.\nTherefore, by varying the diameter of the Al mask, the resulting structures can be tuned from truncated\nto conical pro\fles.\nMagneto-optical characterisation :\nThe experimental values of \u0012F,\u0011Fand \u0002 Fwere measured using the photoelastic modulator methodology\nwith an applied \feld of 450 mT along the light propagation direction, which is described in the Support-\ning Information. A quadratic polynomial was \ftted to the raw \u0012Fdata in order to subtract the back-\nground contribution which arises from the Faraday rotation of the fused-silica substrate, which is strongest\nfor short wavelengths and decreases for longer wavelengths [54]. For the di\u000berential absorption of circu-\nlarly polarised light measurement, a time varying light polarisation, which alternates between left and\nright circularly polarised light states at 50 kHz was generated using a photoelastic modulator (PEM)\nand directed at the sample at normal incidence. This is achieved by passing linearly polarised light ori-\nentated at 45\u000eto the fast axis of the PEM, with the PEM retardation set to 0.25 wavelengths. Any mech-\nanism in the TNC array which results in a di\u000berence in absorption for opposite helicities (including mag-\nnetic circular dichroism) will contribute to an oscillating light intensity at the detector at the photoelas-\ntic modulator frequency. It is common to express this measurement as the ratio Cq\n!=Cq\n\u000e, whereCq\n!is the\namplitude of the != 50 kHz signal for a \fxed polar magnetization q=\u0006Mz, andCq\n\u000eis the DC signal\nintensity, which contains the helicity independent absorption contribution. Prior to the measurement, a\nsaturating magnetic \feld was used to initialise the magnetization along the light propagation direction\n(q= +Mz) and then removed. For the subsequent measurement, the magnetization was saturated in the\nopposite polar direction ( q=\u0000Mz) and the measurement repeated.\nIt is important to note that the spectra in Fig. 4a contain additional fakeCD contributions, which arise\nfrom leaking-in of the large linear dichroism signal as a result of the rectangular array with which the\nnanostructures are arranged. By observing the di\u000berence between the antiparallel magnetization states,\nthese e\u000bects, which are independent of the magnetization orientation, can be subtracted out, yielding the\navailable magnetic modulation. We de\fne this magnetic modulation of the helicity dependent transmis-\nsion as (C\u0000Mz!\u0000C+Mz!)=(C\u0000Mz\u000e+C+Mz\u000e), and this quantity is plotted in Fig. 4b as a function of both \u000bi\nand wavelength.\nSupporting Information\nSupporting Information is available from the Wiley Online Library or from the author.\nAcknowledgements\nThe authors would like to express their gratitude towards Prof. Bengt Lindgren of Uppsala University,\nSweden, for fruitful discussions and support with the ellipsometric characterization of TbCo thin \flm\nmaterials. The excellent support and infrastructure of the MyFab facility at the \u0017Angstr om Laboratory\nof Uppsala University is also highly appreciated. The authors acknowledge support from the Knut and\nAlice Wallenberg Foundation project \\ Harnessing light and spins through plasmons at the nanoscale \"\n(Project No. 2015.0060), the Swedish Research Council (Project No. 2019-03581), the Swedish Founda-\ntion for International Cooperation in Research and Higher Education (Project No. KO2016-6889), and\nthe Swedish National Infrastructure for Computing (SNIC). This work is part of a project which has re-\nceived funding from the European Union's Horizon 2020 research and innovation programme under grant\nagreement no. 737093, \\ femtoterabyte \". P.V. acknowledges funding from the Spanish Ministry of\n10Science and Innovation under the Maria de Maeztu Units of Excellence Programme (MDM-2016-0618)\nand the project RTI2018-094881-B-I00 (MICINN/FEDER).\nAuthor Contributions\nR.M.R-R. and V.K. designed the material and nanofabrication processing, with input from A.D. con-\ncerning the nanocone design approach. 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Hirao, Journal of Materials Research 1998 ,13, 5 1358.\n13" }, { "title": "0809.2450v1.Kinetics_of_a_mixed_spin_1_2_and_spin_3_2_Ising_ferrimagnetic_model.pdf", "content": "1 \n Kinetics of a mixed spin-1/2 and spin-3/2 Ising ferrimagnetic model \nBayram Devirena, Mustafa Keskinb, *, Osman Cankob \n \na Institute of Science, Erciyes University, 38039 Kayseri, Turkey \nb Department of Physics, Erciyes University, 38039 Kayseri, Turkey \n \nWe present a study, within a mean-field approach, of the kinetics of a mixed \nferrimagnetic model on a square lattice in wh ich two interpenetrating square sublattices \nhave spins that can take two values, σ = ± 1/2, alternated with spins that can take the four \nvalues, S = ± 3/2, ± 1/2. We use the Glauber-type stochastic dynamics to describe the \ntime evolution of the system w ith a crystal-field interaction in the presence of a time-\ndependent oscillating external magnetic field. The nature (continuous a nd discontinuous) \nof transition is characterized by studying the thermal behaviors of average order \nparameters in a period. The dynamic phase transition points are obtained and the phase diagrams are presented in the reduced ma gnetic field amplitude (h) and reduced \ntemperature (T) plane, and in the reduced te mperature and interaction parameter planes, \nnamely in the (h, T) and (d, T) planes, d is the reduced crystal-field interaction. The phase \ndiagrams always exhibit a tricri tical point in (h, T) plane, bu t do not exhibit in the (d, T) \nplane for low values of h. The dynamic multicri tical point or dynamic critical end point \nexist in the (d, T) plane for low values of h. Moreover, phase diagrams contain paramagnetic (p), ferromagnetic (f), ferrimagne tic (i) phases, two co existence or mixed \nphase regions, (f+p) and (i+p), that strongly depend on interaction parameters. \n \n PACS : 05.50.+q; 05.70.Fh; 64.60.Ht; 75.10.Hk \n \nKeywords : Mixed spin-1/2 and spin-3/2 Ising ferrimagnetic model; Glauber-type \nstochastic dynamics; Dynamic phase transition; Phase diagrams \n \n1. Introduction \n \nIn last two decades, mixed spin Ising systems ha ve attracted a great deal of attention. The \nreasons are follows: (i) These problems are ma inly related to the potential technological \napplications in the area of thermomagnetic recording [1]. (ii) The systems have less translational \nsymmetry than their single spin counterparts; hence exhibit many new phenomena that cannot be \nobserved in the single-spin Ising systems. (iii) The study of these systems can be relevant for \nunderstanding of bimetallic molecular systems ba sed magnetic materials [2]. One of the well \nknown mixed spin Ising systems is the mixed sp in-1/2 and spin-3/2 Ising model. Amorphous \nV(TCNE)\nx. y (solvent), where TCNE is tetracyanoet hylene, are organometallic compounds that \nseem to have a 1/2 - 3/2 ferrimagnetic structure and order ferrimagnetically as high as 400K [3, \n4]. An early attempt to study the magnetic properti es of the diluted mixed spin-1/2 and spin-\n3/2 Ising model Hamiltonian with only a bilinear exchange interaction (J) was made with in the \n \n \n* Corresponding author. \nTel: + 90 (352) 4374938#33105; Fax: + 90 (352) 4374931 E-mail address: keskin@erciyes.edu.tr\n (M. Keskin) \n 2 \n framework of the effective-field theory (EFT) by Bobák and Jurčišin [5]. They found that the \ncompensation point which depends not only on the magnitude of spins but also on the lattice \nstructure. Bobák and Jurčišin [6] investigated the diluted mixe d spin-1/2 and spin -3/2 Ising model \nHamiltonian with J and the crystal-field (D) in teractions on the honeycomb lattice within the \nEFT and found that the system exhibit two compen sation points. Benayad et al. [7] studied the \nmixed spin-1/2 and spin-3/2 Ising model Hamiltonian with J and the crystal- field (D) interactions \non the honeycomb lattice by using the EFT, and they found a variety of interesting phenomena in \nphase diagrams due to the influence of the crys tal-field interaction. Magne tic properties of the \nmixed spin-1/2 and spin-3/2 transverse Ising model with a crystal-field interaction were studied within the EFT, extensively [8]. Especially, the thermal behavior of order parameters are \ninvestigated and phase diagrams are presented. Monte Carlo (MC) study of a mixed spin-1/2 and \nspin-3/2 Ising model on a square lattice was done by Buendia a nd Cardona [9], and observed that \nthe compensation temperatures are extremely depe ndent on the interactions in the Hamiltonian. \nMagnetic properties of the mixed spin-1/2 and sp in-3/2 Ising model in a longitudinal magnetic \nfield were investigated, and thermal behaviors of magnetizations, magnetic susceptibilities and \nthe phase diagram are examined in detail [10]. Li et al. [11] studi ed the mixed spin-1/2 and spin-\n3/2 quantum Heisenberg system on a square lattice with the double- time-temperature Green \nfunction method to investigate the effects of the nearest- and next- neares t-neighbor interactions \nbetween spins on the magnetic beha vior of the system, especia lly on the compensation point. \nThe system has also been investigated on the Be the lattice [12] and two- fold Cayley tree [13] \nusing the exact recursion relati ons, on the honeycomb lattice within the framework of an exact \nstar-triangle mapping transformations [14], an d on the extended Kagomé lattice [15] and union \nJack (centered square) lattice [16] by establ ishing a mapping correspo ndence with the eight-\nvertex model. Despite of all these equilibrium studies, as far as we know, the nonequilibrium aspects of \nthis system have not been investigated. Theref ore, the purpose of the present work is to \ninvestigate dynamical aspect of the mixed spin-1/2 and spin-3/2 Ising fe rrimagnetic model with a \ncrystal-field interaction in the presence of a time- dependent oscillating exte rnal magnetic field. \nWe use the Glauber-type stochastic dynamics [17] to describe the time evolution of the system. \nThe nature (continuous and discontinuous) of tran sition is characterized by studying the thermal \nbehaviors of average order parameters in a period. The dynamic phase transition (DPT) points \nare obtained and the dynamic phase diagra ms are presented in different planes. \nThe organization of the remaining part of this paper is as follows. In Section 2, the model \nand its formulations, namely the derivation of th e set of mean-field dynamic equations, are given \nby using Glauber-type stochastic dynamics in the presence of a time-dependent oscillating \nexternal magnetic field. In Section 3, we solve the coupled set of dynamic equations and present \nthe behaviors of time variations of order para meters and the behavior of the average order \nparameters in a period, which are also called th e dynamic order parameters, as functions of the \nreduced temperature and as a re sult, the DPT points are calculated. Section 4 contains the \npresentation and the discussion of the dyna mic phase diagrams. Finally, summary and \nconclusion are given in Section 5. \n2. Model and formulations \n \nThe mixed spin-1/2 and spin-3 /2 Ising model is described as a two-sublattice system, \nwith spin variables σ\ni = ±1/2 and Sj = ±3/2, ±1/2 on the sites of s ublattices A and B, respectively. \nThe system has two long-range order parameters, namely the average magnetizations < σ > and \n for the A and B sublattices, respectively, which are the excess of one orientation over the \nother, also called the dipole moments. 3 \n The Hamiltonian of the mixed spin-1/2 a nd spin-3/2 Ising mode l with the bilinear ( J) \nnearest-neighbor pair in teraction and a single-ion potential or crystal-field interaction ( D) in the \npresence of a time-dependent oscill ating external ma gnetic field is \n \n ,AB B 2 A B\nij j i j\nij j i j= J σSD( S ) - 5 / 4 H σ+S H⎛⎞⎡⎤ −− − ⎜⎟ ⎣⎦⎝⎠∑∑ ∑ ∑ (1) \n \nwhere < ij> indicates a summation over all pairs of nearest-neighboring sites, and H is an \noscillating magnetic field of the form \n0 H(t)=H cos(wt), (2) \n \nwhere H0 and w = 2πν are the amplitude and the angular frequency of the oscillating field, \nrespectively. The system is in contact with an isothermal heat bath at absolute temperature. \nNow, we apply Glauber-type stochastic dynamics [17] to obt ain the mean-field \ndynamic equation of motion. Thus, the system evol ves according to a Glauber-type stochastic \nprocess at a rate of 1/ τ transitions per unit time. Leaving the S spins fixed, we define \nA\n12 N P( , , , ; t )σσ σ… as the probability that the system has the σ-spin configuration, \n12 N,,,σσ σ… , at time t, also, by leaving the σ spins fixed, we define B\n12 N P( S , S , , S; t ) … as the \nprobability that the system ha s the S-spin configuration, 12 NS, S, , S… , at time t. Then, we \ncalculate A\niiW( )σ and B\njj jW( S S) ′→ , the probabilities pe r unit time that the ith σ spin \nchanges from σi to – σi ( while the spins on B sublatti ce momentarily fixed) and the jth S spin \nchanges from S j to jS′ (while the spins on A sublattice mome ntarily fixed), respectively. Thus, \nif the spins on the sublattice B momentarily fixe d, the master equation for the sublattice A can \nbe written as \n \nAA A\n12 N i i 12 i N\ni\nAA\nii 1 2 i N\nidP ( , ,..., ;t) W ( ) P ( , ,..., ,... ;t)dt\nW ( ) P ( , ,..., ,... ;t).σσ σ = − − σ σσ σ σ\n+σ σ σ − σ σ∑\n∑ ( 3 ) \n \nSince the system is in contact with a heat bath at absolute temperature T A, each spin σ can flip \nwith the probability per unit time; \n()\n()\niA\ni A\nii A\niexp E ( )1W( )\nexp E ( )\nσ−βΔ σ\nσ=τ−βΔ σ∑ , ( 4 ) \n \nwhere BA1/k T ,β= Bk is the Boltzmann factor, \niσ∑is the sum over the tw o possible values of \nA\niσ, 12± , and 4 \n \nA\nii j\njE( ) 2 ( H J S )Δσ = σ+ ∑ , ( 5 ) \ngives the change in the energy of the system when the σi-spin changes. The probabilities satisfy \nthe detailed balance condition \nA A\n12 i N ii\nAA\nii 1 2 i NP ( , ,..., ,... ) W( )\nW ( ) P ( , ,..., ,... )σσ− σ σ −σ=σσ σ σ σ, ( 6 ) \n \nand substituting the possible values of σi, we get \n \nA\ni1 1 exp( x 2)W( ) ,22 c o s h ( x 2 )−β−=τβ (7a) \n \nA\ni1 1 exp( x 2)W() ,22 c o s h ( x 2 )β=τβ (7b) \n \n \nwhere j\njx=H+J S∑ . From the master equation associated wi th the stochastic process, it follows \nthat the average < σk > satisfies the following equation [18] \n \nkk j\njd1tanh H+J Sdt 2 2⎡ ⎤ ⎛⎞βτσ = − σ + ⎢ ⎥ ⎜⎟⎢ ⎥ ⎝⎠⎣ ⎦∑ . ( 8 ) \n \nThis dynamic equation can be written in terms of a mean-field approach and hence the first \nmean-field dynamical equation of the system in the presence of a time-varying field is: \n()() AA Bd1 1mm t a n h m h c o sd2 2 T⎡ ⎤Ω= − + + ξ ⎢ ⎥ξ ⎣ ⎦, ( 9 ) \n \nwhere Am=σ, BmS= , wtξ= , 1T( z J )−=β , 0h=H /zJ and Ω = wτ. \n \nNow assuming that the spins on sublattice A remain momentarily fixed and the spins \non the sublattice B change, we obtain the mean-field dynamical equation of Bm f o r t h e B \nsublattice. Since S j =32 , 12±± , the master equation for the s ublattice B can be written as \n 5 \n jj\njjBB B\n12 N j j j 12 j N\njS S\nBB\njj j 1 2 j N\njS SdP (S ,S ,...,S ;t) W (S S ) P (S ,S ,...,S ,...,S ;t)dt\nW (S S )P (S ,S ,...,S ,...,S ;t) ,′≠\n′≠⎛⎞′ =− →⎜⎟⎜⎟⎝⎠\n⎛⎞′′ +→⎜⎟⎜⎟⎝⎠∑∑\n∑∑ (10) \n \nwhere B\njj jW( S S) ′→ is the probability per unit time that the jth spin changes from the value jS \nto jS′, and in this sense the Glauber model is stocha stic. Since the system is in contact with a \nheat bath at absolute temperature T A, each spin can change from the value jS to jS′ with the \nprobability per unit time; \n \n()\n()\n'\njB\njj B\njj j B\njj\nSexp E (S S )1W( S S )\nexp E (S S )′ −βΔ →′→=τ ′ −βΔ →∑, ( 1 1 ) \n \nwhere \njS′∑is the sum over the four possible values of jS′, 32 , 12±± , and \n \nB2 2\njj j j i j j\niE( S S) ( S S ) ( H J ) ( S ) ( S ) D ′′ ′⎡ ⎤ Δ→ = − −+ σ −−⎣ ⎦ ∑ , ( 1 2 ) \n \ngives the change in the energy of the system when the S j-spin changes. Using the detailed \nbalance condition and substituting the possible values of jS, w e g e t \n \nBBB\njjj33 13 13W( ) W( ) W( )22 22 22\n1 exp( D)exp( 3 y 2), (13a)2 exp( D)cosh(3 y 2) exp( D)cosh( y 2)→− = →− = − →−\n−β − β=τβ β+− β β \n \nBBB\njjj31 11 31W( ) W( ) W( )22 22 22\n1 exp( D)exp( y 2), (13b)2 exp( D)cosh(3 y 2) exp( D)cosh( y 2)→− = →− = − →−\n−β −β=τβ β+− β β \n \nBB B\njj j31 11 31W( ) W( ) W( )22 22 22\n1 exp( D)exp( y 2), (13c)2 exp( D)cosh(3 y 2) exp( D)cosh( y 2)→= − →= − →\n−β β=τβ β+− β β \n 6 \n BB B\njj j13 13 33W( ) W( ) W( )22 22 22\n1 exp( D)exp(3 y 2), (13d)2 exp( D)cosh(3 y 2) exp( D)cosh( y 2)→= − →= − →\nββ=τβ β+− β β \n \nwhere i\niy=H+Jσ∑ . Notice that, since B\njj jW( S S ) ′→ does not depend on the value jS. W e c a n \ntherefore write BB\njj j jjW( S S ) W( S ) ′′→= , then the master equation becomes \n \njj\njjBB B\n12 N j j 12 j N\njS S\nBB\njj 1 2 j N\njS SdP (S ,S ,...,S ;t) W (S ) P (S ,S ,...,S ,...,S ;t)dt\nW (S ) P (S ,S ,...,S ,...,S ;t) ,′≠\n′≠⎛⎞′ =−⎜⎟⎜⎟⎝⎠\n⎛⎞′ + ⎜⎟⎜⎟⎝⎠∑∑\n∑∑ (14) \n \nSince the sum of probabilities is normalized to one, by multiplying both sides of Eq. (14) by S j \nfor m B and taking the average, we obtain \n \n \njjd 3exp( D)sinh(3 y / 2) exp( D)sinh( y / 2)SS ,dt 2exp( D)cosh(3 y / 2) 2exp( D)cosh( y / 2)ββ + − β βτ= − +ββ + − β β (15) \n \n \nThis dynamic equation can be written in terms of a mean-field approach; hence the second mean-\nfield dynamical equation of the system in the presence of a time-varying field is: \n \n[] [ ]\n[][ ]BB\nAA\nAAdmmd\n3exp(d / T)sinh 3(m h cos ) / 2T exp( d / T)sinh (m h cos ) / 2T,2exp(d / T)cosh 3(m h cos ) / 2T 2exp( d / T)cosh (m h cos ) / 2TΩ= −ξ\n+ξ +− +ξ++ξ + − +ξ (16) \n \n \nwhere d = D/zJ. Thus, the set of the mean-field dy namical equations for the average \nmagnetizations are obtained, namely Eq s. (9) and (16). We fixed z=4 and Ω=2π. In the next \nsection, we will give the solution and discussi ons of the set of coupled mean-field dynamical \nequations. \n3. Thermal behaviors of dynamic order para meters and dynamic phase transition \n \nIn this section, we first investigate the beha viors of time variations of magnetizations and \nthen the thermal variation of the average magnetizations in a period, which are also called the dynamic magnetizations, as functions of the reduced temperature and as a result the nature of \ntransition is found and the DPT points are calculate d. We also investigate the behavior of the \ndynamic magnetizations as a functio n of the reduced crystal-field interaction. For these purposes, \nfirst we have to study the st ationary solutions of the set of coupled mean-field dynamical 7 \n equations, given in Eqs. (9) and (16), when the pa rameters T, d and h are varied. The stationary \nsolutions of these equations will be periodic functions of ξ with period 2 π; that is, \n()( )AAm2 mξ+ π = ξ and () ()BBm2 m .ξ+ π = ξ Moreover, they can be one of third types \naccording to whether they have or do not have the property \n \n() ()AAmmξ+π =− ξ and ()()BBmmξ+π =− ξ. (17) \n \nThe first type of solution satisfies both Eq. ( 17) is called a symmetric solution which corresponds \nto a paramagnetic (p) solution. In this solution, the submagnetizations Am and Bm are equal to \neach other (ABmm= ) and Am()ξ and Bm()ξ oscillate around zero and are delayed with respect \nto the external magnetic field. The second type of solution which does not satisfy Eq. (17), is \ncalled a nonsymmetric solution that corresponds to a ferromagnetic solution. In this solution, the \nsubmagnetizations Am and Bm are equal each other (ABmm= ). In this case the magnetizations \ndo not follow the external magnetic field any more, but instead of oscillating around zero; they \noscillate around a nonzero value, namely ±1/2; he nce, we have the ferromagnetic ±1/2 (f) phase. \nThe third type of solution, which does not satisf y Eq. (17), is also called a nonsymmetric solution \nbut this solution corresponds to a ferrimagnetic (i) solution because the submagnetizations Am \nand Bm are not equal to each other, and Am()ξ and Bm()ξ oscillate around ±1/2 and ±3/2, \nrespectively. These facts are seen explicitly by solving Eqs. (9) and (16) within the Adams-\nMoulton predictor-corrector method for a given set of parameters a nd initial values and \npresented in Fig. 1. From Fig. 1, one can see fo llowing five different solu tions or phases, namely \nthe p, f and i fundamental phases or solutions, and two coexistence phases or solutions, namely \nthe f + p in which f and p soluti ons coexist; the i + p in which i and p solutions coexist, have \nbeen found. In Fig. 1(a) only the symmetric solution is always obtained, in this case ABmm= \noscillate around zero value AB(m ( ) m ( ) 0)ξ= ξ= . Hence, we have a paramagnetic (p) solution or \nphase. On the other hand in Fig. 1 (b) and (c) only the nonsymmetric solutions are found; \ntherefore, we have the f and i solu tions, respectively. In Fig. 1(b), Am()ξ and Bm()ξ oscillate \naround ±1/2; hence we have the ferromagnetic ±1/2 ( f) phase. In Fig. 1(c), Am()ξ oscillates \naround ±1/2 and Bm()ξ oscillates around ±3/2, this soluti on corresponds to the ferrimagnetic (i) \nphase AB(m ( ) m ( ) 0)ξ≠ ξ≠ . In Fig. 1(d), Am()ξ and Bm()ξ oscillate around either ±1/2, that \ncorresponds to the f phase, or zer o values which corresponds to th e p phase; hence we have the \ncoexistence solution (f + p), as explained above. In Fig. 1(e), Am()ξ oscillates around ±1/2 and \nBm()ξ oscillates around ±3/2, which corr esponds to the i phase, and also Am()ξ and Bm()ξ are \nequal to each other and they oscillate around zero value, this solution corresponds to the p phase; \nhence we have the coexistence solution (i + p). A symmetric solution does not depend on the \ninitial values, but the other solu tions depend on the init ial values. Finally we should also mention \nthat the ferromagnetic phase has been defined as ABmm0≠≠ in general [19], but in a few work, \nit was defined as ABmm 0≠− ≠ [20]. \nIn order to see the dynamic boundaries among these phases, we have to calculate DPT \npoints and then we can present the phase diagrams of the system. DPT points will be obtained by \ninvestigating the behavior of the averag e magnetizations in a period or the dynamic \nmagnetizations as a function of the reduced temperature. The dynamic order parameters, namely \ndynamic sublattice magnetizations (AM, BM ) are defined as 8 \n \n2\nAA\n01Mm ( ) d2π\n=ξ ξπ∫ and 2\nBB\n01Mm ( ) d .2π\n= ξξπ∫\n (18) \n \nThe behaviors of AM and BM as a function of the reduced temp erature for several values of d \nand h are obtained by combining the numerical methods of Adams-Moulton predictor corrector \nwith the Romberg integration. A few interesting re sults are plotted in Figs . 2(a)-(d) in order to \nillustrate the calculation of the DPT points a nd the dynamic phase boundaries among the phases. \nTC and T C' are the second-order phase transition temperature from the i phase to the p phase, and \nfrom the f phase to the p phase, respectively. T t represents the first- order phase transition \ntemperature. Fig. 2(a) shows the behavior of AM and BM as a function of the reduced \ntemperature for d = 0.125 and h = 0.125. In this figure, AM =1 2 and BM= 3 2 a t z e r o \ntemperature, and they decrease to zero continuously as the re duced temperature increases, \ntherefore a second-order phase transition occurs at T C = 0.555. In this case the dynamic phase \ntransition is from the i phase (AM≠BM≠0) to the p phase (AM= BM = 0) and the solution \ndoes not depend on initial values of AM and BM . Fig. 2(b) presents the thermal variations of \nAM and BM for d = -0.5 and h = 0.125. In Fig. 2(b), ABM= M 1 2= at zero temperature, and \nthey decrease to zero continuous ly as the reduced temperatur e increases, therefore a second-\norder phase transition occurs at T C' = 0.265 from the f phase to the p phase. This solution does \nnot also depend on in itial values of AM and BM . Figs. 2(c) and (d) illustrate the thermal \nvariations of AM and BM for d = 0.125 and h = 0.575 for two different initial values; i.e., the \ninitial values of Am =1 2 and Bm =3 2 for Fig. 2(c), and ABm = m = 1/2 or zero for Fig. 2(d). The \nbehavior of Fig. 2(c) is similar to Fig. 2( a), hence the system undergoes a second-order phase \ntransition from the i phase to the p phase at T C = 0.2875. In Fig. 2(d), ABMM0== at zero \ntemperature, the system undergoes two successive phase transition as the temperature increases: \nThe first one is a first-order phase transition, because discontinuity occurs for the dynamic \nmagnetizations, and the transition is fr om the p phase to the i phase at T t = 0.2125. The second \none is a second-order phase transition from the i phase to the p phase at T C = 0.2875 as similar to \nFigs. 2(a) and (c). From Figs. 2(c) and (d), one can see that the i + p coexistence region also \nexists in the system and this fact is seen in the phase diagram of Fig. 5(a), explicitly. \n It is worth mentioning that if the single Ising [21] or mixed Ising [22] systems are in the \nstatic magnetic field, the systems do not under go any phase transition w ithin the mean-field \napproach. This fact is also correct for our calcu lation in this work that has been shown in our \nprevious paper of the single spin-1 Blume-Capel (BC) model [23]. Now, we have also checked \nthis fact for the mixed spin-1/2 and spin-3/2 Ising ferrimagnetic model, namely we have investigated the behavior of the dynamic order para meters in the static ex ternal magnetic field. \nFig. 3 shows the ther mal variations of \nAM and BM for several values of static h and d = − 0.125; \nhence this figure indicates that the system does not undergo any phase transition. These \nbehaviors are similar to Fig. 6 (a) of Ref. 23, compare Fig. 3 with Fig. 6 (a) of Ref. 23. \nThe behaviors of dynamic magnetizations as a function of the re duced crystal-field \ninteraction or single -ion anisotropy (d ) are also investigated and pr esented four representative \ngraphs, seen in Fig. 4. Fig. 4(a) is obtained for h = 0.375 and T = 0.25, and the system undergoes \na second-order phase transition at d C = − 0.3825, because AM and BM become zero \ncontinuously. Figs. 4(b) and (c) ar e calculated for h = 0.625 and T = 0.1 for two different initial 9 \n values; i.e., the initial values of Am =1 2 and Bm = 3 2 for Fig. 4(b) and ABm = m = 1/2 or zero \nfor Fig. 4(c). In Fig. 4(b), th e system undergoes two successive pha se transitions; the first one is \na first-order phase transition and the transition is from the p phase to the i phase at d t1 = 0.00, and \nthe second one is a second-order phase transi tion from the i phase to the p phase at d C = − 0.285. \nThe behavior of Fig. 4(c) is similar to Fig. 4( b), but the first-order phase transition occurs at d t2 = \n− 0.2075. From Figs. 4(b) and 4(c) one can see that the p phase until d t1 = 0.00; the i + p \ncoexistence phase between d t1 = 0.00 and d t2 = − 0.2075; the i phase between d t2 = − 0.2075 and \ndC = - 0.285; after d t2 = - 0.2075 the p phase, exist in the system and this f act is seen in the phase \ndiagram of Fig. 6(c) explicitly [compare in Figs. 4( b) and 4(c) with Fig. 6( c)]. Fig. 4(d) displays \nthe behaviors of magnetizations for h = 0.125 and T = 0.05. At the high values of a reduced \ncrystal-field interaction, AM =1/2 and BM = 3 2 ; hence we have the ferrimagnetic (i) phase, and \nas the reduced crystal-field decreases the i phase becomes the ferromagnetic (f) phase \n(ABMM 1 / 2== ) with the second-order phase transition d C′ = − 0.3125. \n \n \n4. Dynamic phase diagrams \n \nSince we have obtained the DPT points in Section 3, we can now present the phase \ndiagrams of the system. The calculated phase di agrams in the (h, T) and (d, T) planes are \npresented in Figs. 5 and 6, resp ectively for various values of in teraction parameters. In these \nphase diagrams, the solid and dashed lines repres ent the second- and first- order phase transition \nlines, respectively, and the dynamic tricritical points are also denoted by a solid circle. The \ndotted line is an ordered line smoothly mediating, with no phase transition, between the different \nordered phases. \nIn Fig. 5, only one dynamic tricritical point exists and two different topological types of \nphase diagrams are found. (i) Fig. 5(a) represents the phase di agram in the (h, T) plane for d = \n0.125. In this phase diagram, at high reduced temperature (T) and high reduced external \nmagnetic field (h), the solutions are paramagnetic (p); and at low values of T and h, are ferrimagnetic (i). The dynamic phase boundary between these regions, i → p, is the second-order \nphase transition line. At low reduced temperatures, there is a range of values of h in which the p \nand i phases or regions coexist, called the coexis tence or mixed region, i + p. The i + p region is \nseparated from the i and the p phases by the first-order phase transition lines. The system also exhibits only one dynamic tricritical point where the both first-order phase transition lines merge \nand signals the change from the first- to the second-order phase transition. Finally, we should \nalso mention that very similar phase diagrams we re also obtained in kinetics of the mixed spin-\n1/2 and spin-1 Ising ferrimagnetic system [24], the kinetic spin-1 Ising systems [23, 25] and the \nkinetic spin-3/2 Ising systems [26], but the phases other than the p phases are different.\n (ii) Fig. 5 \n(b) calculated for d = - 0.5 and it is similar to Fig. 5(a), except that the i + p phase becomes f + p \nphase and the i phase turns to the f phase. \nThe calculated phase diagrams of the system in the (d, T) are seen in Figs. 6 (a)-(c). As \nseen in Fig.6, we have obtained th ree different phase diagram topologies. (i) For h = 0.125, we \nare performed the phase diagram, seen in Fig. 6(a). The system always undergoes a second-order \nphase transition. Besides one dynamic multicritical point (A), the p, f and i phases exist in the \nphase diagram. The dynamic phase boundaries among the p, f and i are the second-order phase \ntransition lines. For high values of T, the p pha se always exists, but low values of T and large \nnegative values of d, the f phase exists and for lo w values of T and high values of d, the i phase \noccurs. We have found a similar dynamic phase di agram to the one obtained in the kinetic spin-\n3/2 BC model [27], except the following diffe rences: (1) The i phase becomes the f 3/2 phase, (2) 10 \n For very low values of T and d, the f 3/2 + f 1/2 coexistence phase exis ts and the dynamic phase \nboundary between the f 3/2 + f1/2 and f 3/2, and between the f 3/2 + f1/2 and f 1/2 phase are first-order \nphase lines. Moreover, we have also found the similar phase di agram, except the second-order \nphase transition line between the f and i phases beco mes a first-order line, to the one obtained by \nmethods in the equilibrium statis tical physics in spin-3/2 Ising sy stems, namely the mean-field \napproximation and the Monte Carlo simulation [ 28], a renormalization-group transformation in \nposition-space based on the Migdal-Kadanoff recursion relations [29], the cluster expansion method [30] and in the exact solution of th e model on the Bethe la ttice by using the exact \nrecursion equa tions [31]. \n(ii) For h = 0.375, the phase diagram is constructed in Fig. 6(b) and is \nsimilar to the phase diagram of Fig. 6(a) bu t following differences have been found: (1) The \nsecond-order phase line and the f phase occur at low temperatures disappear. (2) Two more \ncoexistence phases, namely the f + p, i + p phases, occur for ve ry low values of T, and the \ndynamic phase boundary between these two mixe d phases is a second-order line. (3) The \ndynamic phase boundaries between the f + p and p pha ses, and between the i + p and the i phases \nare the first-order phase lines. (4) The dynamic critical end poi nt (E) appears instead of the \ndynamic multicritical point (A). (5) The dynamic tricritical points, where the both first-order \nphase transition lines merge and si gnals the change from the firs t- to the second-order phase \ntransitions, occurs. (iii) For h = 0.625, the phase diagram is given in Fig. 6(c). This phase \ndiagram exhibits the p, i and i + p phases besi des the two dynamic tricritical points. The dynamic \nphase boundary between the i and p phase is a sec ond-order line that occurs for negative values \nof d, and all other phase lines among the other phases are first-order lines. \n \n5. Summary and Conclusion \n \nWe have analyzed, within a mean-field appr oach, the stationary states of the kinetic \nmixed spin-1/2 and spin-3/2 Ising ferrimagnetic model with a crystal-fi eld interaction under the \npresence of a time varying (si nusoidal) magnetic field. We use a Glauber-type stochastic \ndynamics to describe the time evol ution of the system. First we ha ve studied time variations of \nthe average magnetizations in orde r to find the phases in the syst em. Then, the behavior of the \ndynamic magnetizations as a function of the redu ced temperature and a cr ystal-field interaction \nis investigated to find the nature of phase transi tions and as well as to calculate DPT points. The \ndynamic phase diagrams are presen ted in the (h, T) and (d, T) planes. We have found that the \nbehavior of the system strongly depends on the values of the interac tion parameters and two \ndifferent phase diagram topologies are obtained in the (h, T) pl ane and three fundamental phase \ndiagrams are found in the (d, T) plane. The phase diagrams exhibit the p, f, i, f+p and/or i+p \ncoexistence regions depending on the interac tion parameter values and the dynamic phase \nboundaries among these phases are first-order lines for most cases and second-order lines for a \nfew cases. Therefore, the phase diagrams always e xhibits dynamic tricritical points in the (h, T) \nplane, but does not exhibit in the (d, T) plane fo r low values of h, seen in Fig. 6(a). Moreover, \nthe dynamic critical end point (E) and dynamic multicri tical point (A) exist in the (d, T) plane for \nlow values of h, seen in Fig. 6 (a) and (b), respectively. \nFinally, it should be mentioned that this mean-field dynamic study, in spite of its \nlimitations such as the correlation of spin fluctuations have not been considered, suggests that the kinetic mixed spin-1/2 and spin-3/2 Ising fe rrimagnetic model with crystal field has an \ninteresting dynamic behavior. Hen ce, we hope that our detailed theoretical investigation may \nstimulate further works to study the nonequilibrium or the dynamic phase transition (DPT) in the \nmixed Ising model by using the dynamic Monte Carlo (MC) simulations in which our results will \nbe instructive for the time cons uming process searching critical be havior of this system while \nusing the dynamic MC simulations. We also menti on that some of the first-order lines and as \nwell as tricritical points might be artifact of th e mean-field calculation, this fact has been 11 \n discussed extensively in the kinetic spin-1/2 Is ing model in the recent works [32-34]; hence this \nsystem should be studied by non-perturbati ve methods, such as MC simulations and \nrenormalization-group (RG) calculations in order to find the artifact first-order phase line as well \nas the tricritical point. \n \n \n \nAcknowledgements \n \n This work was supported by the Scientif ic and Technological Research Council of \nTurkey (TÜB İTAK), Grant No: 107T533 and Erciyes Un iversity Research Funds, Grant No: \nFBA-06-01. One of us (B.D.) would like to express his gratitude to the TÜB İTAK for the Ph.D \nscholarship. \n \nReferences \n [1] M. Monsuripur, J. Appl. Phys. 61 (1987) 1580. [2] O. Kahn, in: E. Coronado, et al., (Eds.), Fr om Molecular Assemblies to the Devices, Kluwer \n Academic Publishers, Dordrecht, 1996. 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E 66 (2002) 056127. \n[34] M. Acharyya, A. B. Acharyya, Commun. Comput. Phys. 3 (2008) 397. \n \n \nList of the Figure Captions \n \nFig. 1. Time variations of the average magnetizations (m A, mB): \n \na) Exhibiting a paramagnetic (p) phase: d = - 0.5, h = 0.25 and T = 0.375. \nb) Exhibiting a ferromagnetic-1/2 (f) phase: d = - 0.5, h = 0.15 and T = 0.10. \nc) Exhibiting a ferrimagnetic (i) phase: d = 0.125, h = 0.20 and T = 0.50. \nd) Exhibiting a coexistence region (f+p): d = - 0.5, h = 0.40 and T = 0.05. \ne) Exhibiting a coexistence region (i+p): d = 0.125, h = 0.60 and T = 0.025. \n \nFig. 2. The reduced temperature dependence of the dynamic magnetizations, M A and M B. The T C \nis the second-order phase transi tion temperature from the i phase to the p phase; T C' is from the f \nphase to the p phase; T t represents the first-order phase transition temperature from the i phase to \nthe p phase. \na) Exhibiting a second-order phase transition from the i phase to the p phase for d = 0.125 \nand h = 0.125; 0.555 is found T C. \nb) Exhibiting a second-order phase transition from the f phase to the p phase for d = - 0.5 \nand h = 0.125; 0.265 is found T C'. \nc) Exhibiting a second-order phase transition from the i phase to the p phase for d = 0.125, \nh = 0.575 and the initial values of AM= 1 2 a n d BM = 3 2 ; 0.2875 is found T C. \nd) Exhibiting two successive phase transition, th e first one is a first-order phase phase \ntransition from the p phase to the i phase and the second one is a second-order phase \ntransition from the i phase to the p phase for d = 0.125, h = 0.575 and the initial values of \nABM = M = 1/2 or zero; 0.2125 and 0.2875 are found T t and T C, respectively. \nFig. 3. Thermal variations of the dynamic order para meters for several values of the static \nexternal magnetic fields h and d = - 0.125. 13 \n Fig. 4. The behavior of dynamic magnetizations as a function of the reduced crystal-field \ninteraction or singl e-ion anisotropy. \na) Exhibiting a second-order phase transition from the i phase to the p phase for h = 0.375 \nand T = 0.25; - 0.3825 is found d C. \nb) Exhibiting two successive phase transitions, the first one is a first-order phase transition \nfrom the p phase to the i phase and the sec ond one is a second-order phase transition from \nthe i phase to the p phase for h = 0.625 and T = 0.1 and the initial values of AM= 1 2 a n d \nBM = 3 2 ; 0.00 and - 0.285 are found dt1 and dC, respectively. \nc) Same as (b) but the initial values of ABM = M = 1/2 or zero; - 0.2075 and - 0.285 are \nfound dt2 and dC, respectively. \nd) Exhibiting a second-order phase transition from the f phase to the i phase for h = 0.125 \nand T = 0.05; - 0.3125 is found d C′. \n \nFig. 5. Phase diagrams of the mixed spin-1/2 and spin -3/2 Ising ferrimagnetic model in the (h, T) \nplane. The paramagnetic (p), ferromagnetic (f), fe rrimagnetic (i) and two different coexistence or \nmixed phases, namely the i+p and f+p phases, ar e found. Dashed and solid lines represent the \nfirst- and second-order phase tran sitions, respectively, and dynamic tricritical point is indicated \nwith a filled circle. a) d = 0.125, b) d = - 0.50. \nFig. 6. Same as Fig. 5, but in the (d, T) plane. a) h = 0.125, b) h = 0.375, c) h = 0.625. ξ0 50 100 150mA(ξ), mB(ξ)\n-2-1012(a)\nmA = mB = 0\nξ05 0 1 0 0 1 5 0mA(ξ), mB(ξ)\n-2-1012(b)\nmA=mB=-1/2mA=mB=1/2\nCol 1 vs Col 2 Col 1 vs Col 2 \nξ0 100 200 300mA(ξ), mB(ξ)\n-2-1012\n(c)\nmA=-1/2mB=3/2\nmA=1/2\nmB=-3/2\nξ0 50 100 150 200mA(ξ), mB(ξ)\n-2-1012\n(d)\nmA=mB=-1/2mA=mB=1/2\nmA=mB=0\nξ0 2 55 07 5 1 0 0mA(ξ), mB(ξ)\n-2-1012(e)\nmB=-3/2mA=1/2\nmA=-1/2mB=3/2\nmA=mB=0\nFig. 1MA, MB(a)\n0.00 0.15 0.30 0.45 0.600.00.51.01.5\nTcMB\nMA(b)\n0.0 0.1 0.2 0.30.00.51.01.5\nMA = MB\nTc'\n(c)\n0.0 0.1 0.2 0.30.00.51.01.5(d)\n0.0 0.1 0.2 0.30.00.51.01.5MA, MB\nTtMB\nMA\nMA = MB\nT T\nFig. 2TcTcMB\nMAMA, MB\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.5MB\nMAh = 0.05\n0.1\n0.20.30.4\nT0.5\nFig. 3-0.75 -0.50 -0.25 0.00 0.25MA, MB\n0.00.20.40.60.81.01.21.41.6\nMA=MBMB\nMA\ndC(a)\n-0.4 -0.3 -0.2 -0.1 0.0 0.1MA, MB\n0.00.20.40.60.81.0\nMA=MBMB\nMA\ndC(b)\nd-0.75 -0.50 -0.25 0.00 0.25MA, MB\n0.00.51.01.5\nMA=MB= f1/2MB= f3/2\nMA= f1/2\ndC'(d)dt1\nFig. 4MA=MB\n-0.4 -0.3 -0.2 -0.1 0.0 0.10.00.20.40.60.81.0\nMA=MBMB\nMA\ndC(c)\ndt2MA=MBT0.000 0.125 0.250 0.375 0.500 0.625h\n0.0000.1750.3500.5250.700\nipi + p(a)\n(b)\nT0.0 0.1 0.2 0.3h\n0.0000.1250.2500.3750.500\nfpf + p\nFig. 5(a)\n-1.00 -0.75 -0.50 -0.25 0.00 0.25T\n0.000.250.500.75p\nfi\n-0.75 -0.50 -0.25 0.00 0.25T\n0.0000.1250.2500.3750.5000.625p\nf + pi\ni + p\nd-0.375 -0.250 -0.125 0.000 0.125 0.250T\n0.000.050.100.150.20\ni\ni + p\ni + pp(b)\n(c)\nFig. 6EA" }, { "title": "1910.14263v1.Time_resolving_magnetic_scattering_on_rare_earth_ferrimagnets_with_a_bright_soft_X_ray_high_harmonic_source.pdf", "content": "1\t\tTime-resolving magnetic scattering on rare-earth ferrimagnets with a bright soft-X-ray high-harmonic source G. Fan1,2, K. Légaré2, V. Cardin2, X. Xie1,9, E. Kaksis1, G. Andriukaitis1, A. Pugžlys1, B. E. Schmidt3, J.P. Wolf4, M. Hehn7, G. Malinowski7, B. Vodungbo5, E. Jal5, J. Lüning5, N. Jaouen6, Z. Tao8*, A. Baltuška1, F. Légaré2 and T. Balčiūnas1,4 1 Institute of Photonics, TU Wien, Gusshausstrasse 27/387, Vienna, Austria 2 Institut National de la Recherche Scientifique, Varennes, Quebec J3X1S2, Canada 3 few-cycle, Inc., 2890 Rue de Beaurivage, Montreal, Quebec H1L 5W5, Canada 4 GAP-Biophotonics, Université de Genève, 1205 Geneva, Switzerland 5 Sorbonne Université, CNRS, Laboratoire de Chimie Physique - Matière et Rayonnement, LCPMR, 75005 Paris, France 6 Synchrotron SOLEIL, L’Orme des Merisiers, 91192 Gif-sur-Yvette, France 7 Institut Jean Lamour, CNRS UMR 7198, Université de Lorraine, 54000 Nancy, France. 8 State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, People’ s Republic of China 9 SwissFEL, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland * Email: Zhenshengtao@fudan.edu.cn Abstract We demonstrate the first time-resolved X-ray resonant magnetic scattering (tr-XRMS) experiment at the N edge of Tb at 155 eV performed using a tabletop high-brightness high-harmonic generation (HHG) source. In contrast to static X-ray imaging applications, such optical-pump—X-ray-probe studies pose a different set of challenges for the ultrafast driver laser because a high photon flux of X-rays resonant with the N edge must be attained at a low repetition rate to avoid thermal damage of the sample. This laboratory-scale X-ray magnetic diffractometer is enabled by directly driving HHG in helium with terawatt-level 1 µm laser fields, which are obtained through pulse compression after a high-energy kHz-repetition-rate Yb:CaF2 amplifier. The high peak power of the driving fields allows us to reach the fully phase-matching conditions in helium, which yields the highest photon flux (>2x109 photons/s/1% bandwidth) in the 2\t\t100-220 eV spectral range, to the best of our knowledge. Our proof-of-concept tr-XRMS measurements clearly resolve the spatio-temporal evolution of magnetic domains in Co/Tb ferrimagnetic alloys with femtosecond and nanometer resolution. In addition to the ultrafast demagnetization, we observe magnetic domain expansion with a domain wall velocity similar to that induced by spin transfer torque. The demonstrated method opens up new opportunities for time-space-resolved magnetic scattering with elemental specificity on various magnetic, orbital and electronic orderings in condensed matter systems. Spontaneous emergence of magnetic orders in nanoscale and mesoscale structures has been widely observed and plays an important role in a variety of macroscopic phenomena in magnetic materials. Such heterogeneity of magnetic states in an otherwise spatially homogeneous material is a result of a complex interplay between electron spins and other degrees of freedom (electron orbitals and lattice). Driven by optical excitation, spin angular momentum can be transferred between neighboring magnetic nanoregions with different constituent magnetic elements, facilitating magnetization reversal on sub-picosecond timescales in rare-earth-transition-metal ferrimagnetic alloys and multilayers1–3. Such ultrafast spin transfer promises potential applications in future data storage and spintronic devices functioning on the picosecond timescale, but its underlying mechanisms are still under debate4–6. To further advance the field, it is essential to develop microscopic methods that can map optically-induced magnetization reordering processes with elemental specificity, and at their characteristic sub-picosecond temporal and nanometer spatial scales. High-brightness HHG7–9 light sources enable tr-XRMS as an excellent laboratory-scale tool for studying nanoscale magnetic dynamics. Owing to its high spatial coherence, HHG radiation has been widely used to provide high-quality images of nanostructures with a large-scale view10–12. In addition to the outstanding spatial properties, HHG pulses enable femto-to-3\t\tattosecond temporal resolution13,14. The broad spectral bandwidth of HHG spans the characteristic M and N absorption edges of the transition-metal (TM) and rare-earth (RE) elements that exhibit magneto-optical activity. X-ray magnetic circular dichroism (XMCD) at these edges15 provides access to an element-specific16,17 mapping of magnetic states and allows following their evolution in time. Compact tabletop HHG sources addressed in this Letter favorably differ in size and complexity from facility-scale sources such as free-electron lasers (FEL) 3,18 and synchrotrons with femtosecond slicing technology19. Compared to other spectroscopic methods, resonant magnetic scattering requires orders of magnitude higher photon flux because of the extremely low, ~10-5 scattering cross-section of magnetic structures in the XUV spectral range20, which imposes a major challenge for the applications of HHG radiation in scattering experiments. To date, tr-XRMS based on tabletop HHG sources has only been carried out on 3d TM ferromagnets (iron, cobalt and nickel) and a few multilayer structures20–22, by covering the easily accessible M absorption edges of the TM elements at ~60 eV. To image the laser-induced nanoscale spin transfer in ferrimagnets, such as TbFeCo23 and GdFeCo1,3, it is thus crucial to further extend HHG photon energy and cover the N edges of 4f RE ferromagnets (Gd (~148 eV), Tb (~155 eV) and Dy (~153 eV)). One the one hand, it is possible to extend the phase-matched HHG cutoff with longwave mid-IR driver pulses from optical parametric amplifiers (OPA)24–28. On the other hand, scaling the wavelength of the driver pulse reduces the HHG efficiency29 as lL-5.5±0.5. Moreover, the low efficiency of optical frequency conversion in an OPA (10-20%) further limits the HHG flux. In contrast, high-brightness HHG can be achieved using 1030 nm pulses from high-repetition-rate Yb fiber lasers30 by applying a tight-focusing geometry and a high-ionization-potential target gas. Nevertheless, because the phase-matching pressure of the target gas scales inversely with the square of the beam diameter31, it is very challenging to fulfill phase-matching conditions, precluding applications of such sources in the >150 eV region. 4\t\tIn this work, we demonstrate the first tr-XRMS on Co0.88Tb0.12 ferrimagnetic alloys with high-brightness HHG radiation covering the N edge of Tb at 155 eV. This ultrafast magnetic diffractometer is enabled by HHG up to 220 eV with the highest brightness (>2´109 photons/s/1% bandwidth (BW)) ever reported in the literature. This is accomplished by directly driving the HHG process in helium with 1030-nm 0.3-TW-peak-power femtosecond laser pulses resulting from hollow-core fiber (HCF) pulse compression of a high-energy Yb:CaF2 laser system. The combination of optimum driver wavelength and high peak power boosts the HHG flux at 155 eV for tr-XRMS. In this work, we prove this assertion both experimentally and theoretically by comparing various generation schemes employing different wavelengths, gases and pressures. Our 220 eV, high-brightness HHG source with a-few-kHz repetition rate is especially suitable for time-resolved soft X-ray spectroscopy and imaging experiments on solids, for which high-repetition-rate sources could cause irreversible thermal damage induced by the optical pump. From the tr-XRMS measurements, we observed the laser-induced evolution of magnetic domains in a Co0.88Tb0.12 ferrimagnetic alloys on the femtosecond time scale with domain changes on the nanometer scale, revealing different dynamics compared to previous work18,21. The experimental setup is illustrated in Fig. 1. The fundamental ~220 fs, 1030 nm pulses from a Yb:CaF2 amplifier were first compressed by a post-pulse-compressor, consisting of a HCF and a set of chirped mirrors. The implemented HCF is 3-m-long and has a large core diameter (1mm), enabling a compression ratio of ~10 for the high-energy (~11 mJ) pulses. The post-compressed pulse has a peak power of 0.3 TW (~25 fs, 8 mJ), which is the key to achieving efficient laser-like HHG (Fig. 2c) up to 220 eV in a 20 mm helium-filled gas cell with adjustable backing pressure. The harmonic spectrum is characterized with a soft X-ray spectrometer (see Methods). As shown in Fig. 2a and e, the change of the pulse duration from 5\t\t220 fs to 25 fs extends the HHG cutoff from 150 eV to 220 eV. This result can be explained by the suppression of ionization for a shorter driver pulse32. To generate high-brightness HHG radiation, it is essential to reach the optimal phase matching conditions. In Fig. 2b, we plot the total spectrum intensity for photon energy >100 eV as a function of the backing pressure of helium, at a laser peak intensity of 6.5´1014 W cm-2. The HHG intensity grows quadratically at low backing pressure, followed by a saturation of the signal at a backing pressure of 200 mbar. By further increasing the pressure, the spectral intensity decreases due to the absorption of the generated harmonics within and after the generation volume. This observation is reproduced with high fidelity in our ab-initio simulation based on the strong-field approximation (see Fig. 2c and Methods) and demonstrates that the HHG generated in helium using our approach is fully phase-matched and absorption-limited in intensity (see Supplementary). Compared to previous experiments30,31, the high peak intensity of the driving laser here allows us to use a loose focusing geometry and relax the requirements on the phase-matching pressure. We also experimentally compare the conversion efficiency of our approach (i) 1 µm in helium with two other generation schemes based on the use of OPAs: (ii) 1.5 µm in neon and (iii) 2.4 µm in argon. Since our target is to optimize the HHG brightness at 220 eV, we adjust the peak power, pulse duration and focusing geometry of the driving laser in every case to reach the same cutoff energy. For gases with a lower ionization potential, it is essential to use a longwave driver pulse to stay below the critical ionization level33. The backing pressure of the gas medium is also optimized to obtain the phase-matched and absorption-limited HHG. The spectrometer efficiency, filter transmission and the input pulse energies at different conditions are taken into account to extract the conversion efficiency right after the gas cell (see Methods). The experimentally measured conversion efficiencies for the three different approaches are plotted in Fig. 3a, showing that our method ((i) 1µm in helium) yields the 6\t\thighest conversion efficiency throughout the 100~200 eV range of interest, which is supported by our simulations shown in Fig. 3b. For the phase-matched HHG, the absorption-limited conversion efficiency can be described as 𝜉!=𝜆\"#$$%%&$' 34, where is the amplitude of the single-atom recombination cross-section at the harmonic frequency , represents the wavelength scaling due to the electron wave packet diffusion during its free-space excursion, with n =5.5 ±0.529, and is the X-ray absorption cross-section. As shown in Fig. 3c, the longest electron wave packet excursion occurs for argon with lL=2.4 µm, which leads to significant reduction of the recombination probability for the case (iii)9. In contrast, the single-atom responses for (i), 1 µm in helium, and (ii), 1.5 µm in neon, are similar since the stronger wave packet diffusion for the longer lL is compensated by the larger recombination cross-section associated with the larger ionic core of neon in case (ii). Summarizing the above discussion of microscopic single-atom response, the expected HHG efficiency is similar for both helium and neon. Nevertheless, macroscopic propagation changes the situation in favor of helium because its X-ray absorption cross-section () is one order of magnitude lower than neon at the energy of 200 eV (Fig. 3a and b). For a fixed driver pulse duration, this simple model suggests a straightforward recipe for reaching the highest flux at a target X-ray photon energy located in a resonance-free plateau region of the harmonic spectrum in the vicinity of the cutoff. The highest efficiency is achieved using helium driven by the shortest laser wavelength capable of reaching the corresponding semi-classical cutoff, given that the laser pulse intensity is sufficient to sustain a fully phase-matched HHG regime. In Fig. 3d, we show that, owing to the high conversion efficiency for 1 µm driving HHG in helium, we achieve the highest flux of 2´109 photons/s/1%BW at 200 eV. Unlike most of qAqwnLl-s\ns7\t\tthe previous experiments summarized in Fig. 3d, in which lL-dependent cutoff extension was studied, our method is free of additional energy loss due to the absence of parametric frequency conversion. Reviewing HHG results with direct laser driving at different wavelengths, it must be noted that with 800 nm laser pulses from a Ti:Sapphire amplifier, it was possible to extend the cutoff beyond 200 eV by employing sub-10-fs driver pulses35 or using quasi-phase matching techniques36. However, in these situations, ionization-induced-phase-mismatch quickly outruns the dispersion contribution of the neutral atoms, making the macroscopic phase matching very challenging37. By contrast, in our experiments with 1-µm driver pulses, we significantly suppressed the ionization of helium thus facilitating the phase matching. The estimated ionization fraction was below 0.38%, while the critical ionization of helium is 0.4%. The high-brightness HHG up to 220 eV generated using our new method allows us, for the first time, to carry out the tr-XRMS measurements on the ferrimagnetic alloy Co0.88Tb0.12. The sample was a 50 nm film grown on a Si3N4 membrane, exhibiting an out-of-plane magnetic anisotropy with a stripe domain structure (Fig. 1). A concave multilayer mirror focuses a portion of the harmonic beam on the ferrimagnetic sample and selects a 5-eV-wide spectral bandwidth covering the N edge of Tb at 155 eV (see Fig. 2d). Due to the XMCD effects, the alternating oppositely-magnetized domains serve as a diffraction grating for the incident linearly polarized soft X-ray beam, giving rise to the ±1st-order diffraction peaks in the far field, as illustrated in Fig. 4a. The magnetic domain structure resolved using magnetic-force microscopy (MFM) of the same sample is plotted in Fig. 4b. Correspondingly, the Fourier transform (FT) of the real-space stripe-like structure yields a diffraction pattern consisting of two well-defined diffraction spots, the momentum transfer (k) of which is consistent with that obtained through XRMS using the bright harmonic beam, as shown in Fig. 4c. 8\t\tThe ultrafast dynamics in the magnetic material is induced by 1.5 µm, 80 fs laser pulses obtained from an OPA driven by the same Yb driver laser, and is probed by soft X-ray pulses arriving at the sample with a time delay td. The repetition rate of the laser was intentionally reduced from 2 kHz to 500 Hz in order to prevent thermal damage of the sample by the accumulated heat from the pump excitation. This demonstrates the need for high pulse energy laser systems operating at low repetition rate for driving tabletop X-ray sources for applications in solid-state physics, e.g. thin films with low heat dissipation and materials with slow recovery. The domain magnetization amplitude M can be measured by the square root of the diffraction intensity, while the spatial evolution of the magnetic domains is revealed by the change of the momentum transfer ()21. As shown in Fig. 4d, with a pump fluence of 8 mJ/cm2, the intensity of the diffraction peaks is suppressed by ~70%, which corresponds to demagnetization up to ~50%. The demagnetization of the sample exhibits two timescales: an ultrafast demagnetization process quickly suppresses ~10% of the sample magnetization in the first 680 fs, followed by a slow demagnetization process in ~18 ps. The two-step process is consistent with the “Type II” demagnetization dynamics previously observed in pure Tb and Gd1-xTbx alloys6,38. More interestingly, we find that the momentum transfer (k) is reduced by ~3% in ~10 ps after pump laser excitation, indicating an expansion of the periodicity of the magnetic domains perpendicular to the stripe-structures (Fig. 4e, x direction). As shown in Fig. 4e, the decay of is much slower than the sample demagnetization () and can be approximated as a linear decrease as a function of time. The slope of the change yields a velocity for the domain expansion of ~750 m/s. Very interestingly, this velocity coincides with the domain-wall velocity under the current-induced spin-transfer torque, with a current density >3´1012 A/m2 39. Indeed, it has been shown that such a high current density can possibly be created with similar pump fluences40 as in our experiment, leading to kD\nkkD()0Mt M9\t\trearrangement of the domain pattern on picosecond timescales. We note that the domain wall velocity here is much smaller compared to the velocity observed in the CoPt multilayers18, indicating a very different driving mechanism in 4f RE ferrimagnets materials compared to 3d metals. In conclusion, we demonstrated the first time-resolved X-ray magnetic diffractometer based on tabletop high-brightness HHG source reaching the N edge of Tb at 155 eV. The measured time-dependent diffraction patterns allow us to extract the ultrafast demagnetization as well as the temporal evolution of magnetic domains with nanometer spatial resolution. Seemingly counter-intuitively, the contrast of pump-probe measurements was improved by decreasing the repetition rate, thus allowing us to increase the pump fluence to quench efficiently the magnetization while keeping the average pump power low enough to prevent thermal damage. This indicates the urgent necessity to develop high-energy sub-kHz-repetition-rate laser sources delivering ultrashort pulse duration. These future technologies will enable high flux harmonic sources, despite lower repetition rate, to perform high-resolution spatio-temporal imaging of various magnetic, orbital and charge orderings of condensed matter. 10\t\t Figures 1-4 \n Fig. 1 The schematics of the Tr-XRMS experiment setup. High-brightness high-order harmonics are generated in a gas cell filled with helium and driven by the compressed pulses in the full phase matching conditions (inset a). The harmonic beam is then focused by a multilayer mirror, which selects the harmonic spectrum around 155 eV, corresponding to the N edge of Tb (inset b). Tr-XRMS experiments are carried out in a pump-probe geometry, by exciting the sample with 1550 nm IR pulses before HHG probe pulses arrive. The sample is a 50 nm Co0.88Tb0.12 thin film deposited on a Si3N4 membrane with an out-of-plane magnetic anisotropy (inset c). The spatio-temporal evolution of the magnetic domains is measured with the time-resolved diffraction patterns from the harmonic beam, recorded with a charge-coupled-device (CCD) camera as a function of the delay time td. \n11\t\t Fig. 2 Driver pulse compression and HHG characterization. (a) Temporal characterization of uncompressed and compressed driver pulses with the second-harmonic-generation frequency-resolved optical gating (SHG-FROG). (b) Measured and simulated HHG signals as a function of the helium backing pressure with a peak power of ~0.4 TW at the wavelength of 1030 nm. (c) The beam profile of the narrow band harmonic beam after the multilayer mirror with the central photon energy of 155 eV. (d) Absorption cross section of Tb near its N edge and the reflectance of the X-ray multilayer mirror. (e) High-order harmonic spectra driven by the uncompressed (200 fs) and compressed (~25 fs) pulses, respectively 0.10.2Tb 4d-4f absorption coefficientμ (nm-1)Multilayer mirror reflectance−200−10001002000100200300\nTime (fs)Peak power (GW)−202−20200.51\nX divergence (mrad)\nY divergence (mrad)\nIntensity (a.u.)\n0.20.4Reflectance\n10012014016018020022000.20.40.60.81\nPhoton energy (eV)Intensity (a.u)ac\neb\n020040060000.20.40.60.81\nPressure (mBar)Norm. HHG yield (a.u)ExperimentSimulationParabolic scalingb\n0d25 fs 200 fsDriven by:~25 fs, FWHMCompressedInput12\t\t Fig 3. High brightness of HHG directly driven by compressed 1µm laser pulses. (a) Measured and (b) simulated conversion efficiencies of HHG in the 100~200 eV spectral range generated in (i) helium with 1 µm, in (ii) neon with 1.5 µm and in (iii) argon with 2.4 µm driving wavelengths. The peak power, pulse duration and focusing geometry are adjusted to yield the same cut-off energy of the spectra. The conversion efficiencies in (a) and (b) are all normalized to the conversion efficiency in case (i) at ~200 eV. c) The simulated cutoff trajectories for the three cases shown in (a) and (b). The radii of the trajectories linearly increase with the excursion time, symbolizing wave packet spreading. The more efficient single-atom response for 1 µm in helium is due to reduced electron wave packet (EWP) spread caused by the shorter wavelength driving field. (d) Overview of the experimentally generated HHG flux in photons per shot per 1%BW above 100 eV in helium (red)24,28,30,32, neon (green)25,26,28,30,36 and argon (blue)43. The driving field wavelength (λL) is illustrated by marker colors. \n13\t\t \n Fig. 4 tr-XRMS measurements of CoTb sample at the N edge of Tb. (a) Illustration of the physical mechanism of XRMS using linearly polarized XUV light. (b) Magnetic domain structure of the same CoTb sample measured using a magnetic force microscope (MFM). (c) The diffraction pattern corresponding to the magnetic domain. The upper panel plots the experimentally measured diffraction pattern using XRMS as a function of momentum transfers kx and ky. The lower panel shows the Fourier transform (FT) of MFM image in (b), which yields the diffraction peaks with the momentum transfer consistent with the XRMS measurement. (d) The variation of averaged magnetization (M(t)) as a function of the pump-probe delay time, extracted from the time-dependent variation of the XRMS diffraction patterns. The magnetization is normalized to the ground-state magnetization M0. (e) The change of momentum transfer in the x direction in percentage (Δkx/kx) as a function of the delay time. \n−150\n−30−1501530−15015\nd\n4d\n-1001020300.50.60.70.80.91.01.1\nDelay time (ps)Delay (ps) \t\n0MtM\nax xk\nRELERE-10010-4-20e4f e− LE\nRXMSMFM FTcDelay time (ps) \t\n%xxkk∆b1µm kx (μm-1) ky (μm-1)14\t\tReferences: 1. Stanciu, C. D. et al. 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High flux coherent super-continuum soft X-ray source driven by a single-stage , 10mJ , Ti: sapphire amplifier-pumped OPA. Opt. Express 22, 6194–6202 (2014). 43. Hong, K.-H. et al. Multi-mJ, kHz, 2.1 µm optical parametric chirped-pulse amplifier and high-flux soft x-ray high-harmonic generation. Opt. Lett. 39, 3145–3148 (2014). 44. Fan, G. et al. Hollow-core-waveguide compression of multi-millijoule CEP-stable 32μm pulses. Optica 3, 2–5 (2016). 45. Xie, X. et al. Subcycle dynamics in the laser ionization of molecules. Phys. Rev. A 76, 23426 (2007). 46. Hellwig, O., Denbeaux, G. P., Kortright, J. B. & Fullerton, E. E. X-ray studies of aligned magnetic stripe domains in perpendicular multilayers. Phys. B 336, 136–144 (2003). Methods Setup for XRMS measurement HHG generation and characterization High-brightness HHG beam by 1030 nm femtosecond pulses with TW-level peak power, which was directly obtained through pulse compression of ~200 fs, 11 mJ, 1030 nm pulses from a Yb:CaF2 amplifier. The pulse compression was achieved in a 3-meter long Ar-filled HCF and the spectrally broadened pulses were compressed by a set of chirped mirrors down to ~25 fs pulse duration with an efficiency of >75%. To achieve a high compression factor (~10), we employed a stretched waveguide technique44. The specially designed flexible HCF has a 1mm inner diameter with ~300-µm-thick fused silica cladding surrounded by a polymer layer. 18\t\tIn order to find the optimal conditions for generating HHG in the 100-220 eV spectral region, we compare the HHG conversion efficiency of HHG generated in helium with 1µm driving fields with generation schemes using OPAs (1.5 µm in neon and 2.4 µm in argon). The 2.4 µm light was generated by a 3-stage OPA with a pulse energy of ~0.8 mJ, while 1.5 µm light was generated by a 4-stage OPA with a pulse energy of 3.7 mJ. The phase matching conditions in argon and neon were optimized correspondingly by adjusting the focusing geometry as well as the gas-cell length to achieve fully phase-matching conditions, respectively. HHG spectrum was measured with an XUV spectrometer, which is based on a flat-field grating with a nominal groove number of 1200 lines/mm and an X-ray CCD camera (Andor Newton 920). The image of the 50-µm-wide slit is imaged direct onto the camera through the concave grating. The absolute photon flux of HHG generated in helium with 1µm fields was estimated, on the other hand, in the diffraction experimental geometry by considering the quantum efficiency of the CCD camera, the transmittance of the filter and the reflectance of the multilayer mirror in the diffraction experimental geometry. The fluxes of HHG generated by the other two generation schemes were then calculated relatively (see Supplementary). TDSE Simulation In order to obtain simulated harmonic spectra from the interaction of noble gases with a strong laser field, we numerically solved three-dimensional time-dependent Schrödinger equation in velocity gauge with single active electron approximation 45, , where V(r) is the potential and is the vector potential of the laser field with a Gaussian envelope, a peak intensity of I0, a center frequency of ω0 and a pulse duration of τ. In these calculations, we employed the pseudospectral method with Tong-Lin models45 of the helium, neon and argon atom. Afterwards, single-atom harmonic spectra were calculated from the Fourier transform of the dipole acceleration with the simulated electron wave functions. The laser wavelengths and the pulse durations for the simulations were the same as those in the corresponding measurements, while the pulse intensities were chosen to generate the corresponding cutoff energies. Sample magnetization Probing of magnetization state via small-angle X-ray magnetic scattering using linearly polarized X-ray pulses rely on preparation of periodically magnetized magnetic domain structure of the sample. The aligned magnetic stripe domains can be achieved using a magnetization procedure described in 46. A thin-layer ferromagnetic sample with saturated out-of-plane magnetic moment forms a “labyrinth” domain state with a typical domain size that is dependent on the thickness of the sample. In order to ()()()()()2,,2tit V ttéù-¶êúY= - Y¶êúëûpArr r()()221.3800 0costtIe ttww-=A19\t\tachieve the aligned “stripe” domain state, the sample is de-magnetized using a strong magnetic field along an in-plane axis. Acknowledgements: Z. Tao gratefully acknowledge support from the National Natural Science Foundation of China (grant no. 11874121) and the Youth Thousand Talent program of China. T.B. acknowledges funding from the EU H2020 resarch and innovation programme under the Marie Sklodowska-Curie grant agreement No 798176. X. Xie acknowledges funding from Austrian Science Fund (FWF) P30465-N27. " }, { "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 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. 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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": "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. 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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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Ono, Correlation \nbetween Compensation Temperatures of Magnetization and Angular \nMomentum in GdFeCo Ferrimagnets, arXiv:1710.07779, (2017).\n \n \n" }, { "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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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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Palstra, Physical Review B 75 , 174435 (2007). \n \n " }, { "title": "2402.03734v1.Magnon_mediated_spin_pumping_by_coupled_ferrimagnetic_garnets_heterostructure.pdf", "content": "Magnon mediated spin pumping by coupled ferrimagnetic garnets\nheterostructure\nAnupama Swain*,1Kshitij Singh Rathore*,1Pushpendra Gupta,1Abhisek Mishra,1Gary Lee,2Jinho Lim,3Axel\nHoffmann,3Ramanathan Mahendiran,2and Subhankar Bedanta1, 4,a)\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), An OCC of Homi Bhabha National Institute (HBNI),\nJatni-752050, Odisha, India\n2)Department of Physics, 2 Science Drive 3, National University of Singapore, 117551,\nRepublic of Singapore\n3)Department of Materials Science and Engineering and Materials Research Laboratory,\nUniversity of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA\n4)Center for Interdisciplinary Sciences (CIS), NISER, An OCC of Homi Bhabha National Institute (HBNI),\nJatni-752050, Odisha, India\n*equal contribution\nABSTRACT\nSpin pumping has significant implications for spintronics, providing a mechanism to manipulate and trans-\nport spins for information processing. Understanding and harnessing spin currents through spin pumping is\ncritical for the development of efficient spintronic devices. The use of a magnetic insulator with low damping,\nenhances the signal-to-noise ratio in crucial experiments such as spin-torque ferromagnetic resonance (FMR)\nand spin pumping. A magnetic insulator coupled with a heavy metal or quantum material offers a more\nstraight forward model system, especially when investigating spin-charge interconversion processes to greater\naccuracy. This simplicity arises from the absence of unwanted effects caused by conduction electrons unlike\nin ferromagnetic metals. Here, we investigate the spin pumping in coupled ferrimagnetic (FiM) Y 3Fe5O12\n(YIG)/Tm 3Fe5O12(TmIG)bilayerscombinedwithheavy-metal(Pt)usingtheinversespinHalleffect(ISHE).\nIt is observed that magnon transmission occurs at both of the FiMs FMR positions. The enhancement of\nspin pumping voltage ( Vsp) in the FiM garnet heterostructures is attributed to the strong interfacial exchange\ncoupling between FiMs. The modulation of Vspis achieved by tuning the bilayer structure. Further, the spin\nmixing conductance for these coupled systems is found to be ≈1018m−2. Our findings describe a novel\ncoupled FiM system for the investigation of magnon coupling providing new prospects for magnonic devices.\nKeywords: Ferrimagnet, Spin Pumping, Magnons, Coupling, Thin Films.\nThe need for ultra-low power consumption devices has\ngiven rise to the field of spintronics. This field uses\nthe spin degree of freedom of electrons1,2. Spintronic-\nbased applications rely on the generation of spin cur-\nrents and their conversion into charge currents in mag-\nnetic heterostructures3–6. However, in this process, the\nscattering of conduction electrons in the magnetic layer\nresults in Joule heating. In this context, moving to-\nwards the magnon-based information processing in in-\nsulators will solve this issue as there is no physical move-\nment of the electrons7. Magnons are the quanta of spin\nwaves, defined as the collective excitations of magnetic\nmoments in magnetically ordered materials. Magnons\ncan propagate over distances ranging from micrometers\nin low damping metallic thin films to around cm in high\nquality magnetic insulators8,9. The magnonic spin cur-\nrent can be employed to carry, transport, and process\ninformation10,11, as well as generate a spin torque act-\ning on the local magnetic moment that can be exploited\nto drive magnetization dynamics12,13and magnetic do-\nmain walls14–16. A promising technique for detecting\nmagnonic spin currents is the spin pumping induced in-\nverse spin Hall effect (ISHE)10,17. Spin pumping refers\nto the transfer of spin angular momentum via magne-tization precession from the ferromagnetic material to\nthe adjacent spin sink layer18. These pure spin currents\nare transformed into conventional charge currents by the\nISHE, which allows for a convenient electrical detection\nof spin-wave based spin currents11. After the discovery\nof the spin-pumping effect and the ways for enhancement\nof spin current in ferrimagnetic insulator (yttrium iron\ngarnet, Y 3Fe5O12, YIG)/non-magnetic metal (platinum,\nPt), there was rapidly emerging interest in the investi-\ngation of these structures19,20. In these magnetic insu-\nlators movement of individual electron gets restricted,\nwhich helps to avoid Joule heating dissipation and there-\nfore benefits modern upcoming spintronic devices21.\nYIG is often considered as the best medium for spin\nwavepropagationbecauseofitsverysmallGilbertdamp-\ning coefficient ( 2×10−5for bulk YIG)22. Being an elec-\ntrical insulator, electron-mediated angular momentum\ntransfer can only occur at the interface between YIG and\na metallic layer. In that context, metals with large spin\norbit coupling (SOC) like Pt where a pure spin current\ncan be generated through spin Hall effect (SHE) have\nbeen used to excite or amplify propagating spin waves\nthrough loss compensation in YIG23–25. Recently, other\ninsulator garnets like Tm 3Fe5O12(TmIG), Gd 3Fe5O12arXiv:2402.03734v1 [cond-mat.mtrl-sci] 6 Feb 20242\n(GdIG) etc. have been investigated with a focus on low\nGilbert damping, interlayer coupling and spintronics ap-\nplications. Here we investigate the manipulation, genera-\ntionanddetectionofmagnon-basedspincurrentsinferri-\nmagnetic insulators capped with heavy metal Pt. These\ninsulators with Pt have been explored a lot for magnon\ntransport physics in effects like ISHE26, Spin Hall mag-\nnetoresistance (SMR)27and spin-Seebeck effect (SSE)28.\nThere are few results published on spin waves in coupled\nferrimagnetic layers29,30, however, the effect of coupling\non spin pumping in such FiM system is scarce in liter-\nature. In order to study the interface and the growth\neffects on spin pumping, this work presents a systematic\nstudy by considering YIG/Pt, TmIG/Pt and bilayers of\nYIG/TmIG with Pt. This study reveals the increase in\nthe spin pumping voltage in the bilayers which can be\nattributed to the interfacial exchange coupling between\nYIG and TmIG.\nHigh-quality garnet films were prepared on (111)\noriented GGG (GdGa 5O12)single crystal substrate by\npulsed laser deposition (PLD) technique using an ex-\ncimer laser (λ= 248 nm ). The garnet targets were com-\nmercially purchased from M/s. Testbourne, UK. The\nbase pressure of the chamber was 8×10−7mbar. The\nsubstrate temperature was maintained at 560◦Cin a\n9×10−4mbar of oxygen partial pressure during YIG\nlayer deposition. The laser fluence and repetition rate\nwere 1.8 J/cm2and8 Hz, respectively. After deposition,\nthe sample was annealed for 2 hat 800◦Cin 300 mbar\nof ambient oxygen environment and cooled at 10◦C/min\nrate. TheTmIGlayerwasgrownbymaintainingthesub-\nstrate temperature at 750◦C, laser fluence at 1 J/cm2,\nwith a repetition rate of 6 Hzin 0.26 mbar of oxygen par-\ntial pressure. Post deposition, the prepared TmIG film\nwas in-situ annealed at same growth temperature for 25\nmins at 100 mbar oxygen pressure followed by cooling\nat 5◦C/min. During deposition the substrate to target\ndistance was kept 5 cmfor both the films. The bilayer\nsamples were prepared by following the same growth con-\ndition as the individual single layers. For the bilayer\nsamples, each FiM layer was first deposited and then an-\nnealed in similar conditions as of its corresponding sin-\ngle reference layer samples and then the subsequent layer\nwas deposited followed by its post-annealing. The details\nof prepared sample structure are mentioned in Table I.\nThe thickness of the corresponding layer is mentioned\nin the brackets. The Pt layer has been prepared via dc\nmagnetron sputtering in a high vacuum multi-deposition\nchamber manufactured by Mantis Deposition Ltd., UK.\nThe growth quality and thickness of the prepared\nfilmswereinvestigatedby X-raydiffraction(XRD).High-\nresolution transmission electron microscopy (HR-TEM)\nhas been performed in this study to verify the epitaxy of\ndeposited films. The saturation magnetization value has\nbeen taken from the superconducting quantum interfer-\nencedevice(SQUID)magnetometrydata. Theferromag-\nnetic resonance (FMR) and spin pumping induced ISHE\nmeasurements were performed using a coplanar waveg-TABLE I. Details of the sample structure studied in this work\nSample Sample\nname structure\nS1 GGG(111)/YIG(100 nm)/Pt( (5 nm)\nS2 GGG(111)/TmIG(30 nm)/Pt(5 nm)\nS3GGG(111)/YIG(100 nm)/TmIG(30 nm)/Pt(5 nm)\nS4GGG(111)/TmIG(30nm)/YIG(100 nm)/Pt(5 nm)\nuide (CPW) based setup. The sample was placed up-\nside down on the CPW. FMR measurements were car-\nried out in the 3-12 GHz of frequency range with 25 mW\nmicrowave power. The ISHE measurements were carried\noutat7GHz rffrequencybyconnectingananovoltmeter\nat the opposite edges of the sample. The measurement\ndetails are mentioned in our previous work31.\nFIG. 1. XRD patterns of (a) S3 and (b) S4 samples (the\ncorresponding insets show the XRD pattern for the 2 θrange\n20◦-80◦). HRTEM image and SAED pattern for sample S3\nare shown in (c) and (d), respectively. The inset shows the\nmagnified part of YIG and TmIG interface.\nThe prepared samples were structurally characterized\nby XRD to confirm the phase and growth quality. The\nXRDpatternofS3andS4samplesareshowninFig. 1(a-\nb). The XRD pattern of S1 and S2 is given in the supple-\nmentary file. The YIG and TmIG lattice parameters are\nveryclosetothatofthesubstrateGGGwhichensuresthe\nepitaxial growth of the structure. The observed diffrac-\ntion peaks were indexed with the corresponding crystal\nplane (h k l) values, and it is evident from the analysis\nthatthegrowthofpreparedsamplesisalongthe(111)di-\nrection. The absence of any additional peaks other than\nthe peaks corresponding to YIG, TmIG, and GGG in\nthe patterns ensures the phase purity of the grown struc-\ntures (shown in the corresponding insets). In Fig. 1(a),\nthe peak corresponding to YIG layer is dominating over\nthe TmIG peak as the YIG thickness is high. By ana-\nlyzing the XRD peak at (444) reflection yields a cubic\nlattice parameter of YIG in S1 is 12.46 Å, which is com-3\nparable to 12.38 Åfor the bulk YIG32. Moreover, the\nlattice parameters of YIG and TmIG are estimated for\nall the samples. The YIG lattice parameter in S3 and\nS4 is 12.43 Åand 12.48 Å, respectively. Likewise, the lat-\ntice parameters of TmIG in S2, S3, and S4 are 12.45 Å,\n12.54 Å, and 12.60 Å, respectively. It is to be noted that,\nthe lattice parameter has changed in S3 and S4 samples\nfor both YIG and TmIG with respective to their single\nlayer i.e., S1 and S2.This indicates the presence of strain\nin the films which may have the impact on the physical\nproperties of the samples.\nThe interface is further explored by cross-sectional\nHRTEM for the sample S3. The zoomed-in image [shown\nin the inset of Fig. 1(c)] reveals the epitaxial growth\nof YIG and TmIG. These high-quality images provide\nclearevidenceofthewell-definedinterfaceandcrystalline\nstructure of the film showing in-plane lattice matching\nwith the substrate. The thicknesses of YIG and TmIG\nwere found to be 100 nm and 30 nm respectively. No-\ntably, no defects or misalignment in lattice planes were\nobserved in the HRTEM image shown in Fig. 1(c). Fig.\n1 (d) shows the selected area electron diffraction (SAED)\npattern of sample S3, which confirms the single crys-\ntalline nature. The validation provided by these HRTEM\nimages and SAED pattern is crucial evidence, confirm-\ning the successful fabrication of the thin films with sharp\ninterface.\nThe spin dynamics properties were investigated by\nFMR measurements at room temperature. Fig. 2 (a-\nd) show the FMR spectra of the prepared samples at\ndifferent frequencies ( f) ranging from 3- 12 GHz. Dis-\ntinct FMR peaks were observed for YIG and TmIG in\nall samples. This confirms the room temperature mag-\nnetic phase of the prepared samples. A shift in Hres\ncorresponding to YIG and TmIG in the S3 and S4 sam-\nples compared to S1 and S2 samples is observed. This\ncould be attributed to the interfacial exchange coupling\nbetween YIG and TmIG, similar to other multilayered\nsystems33.\nFurther, the FMR signals were fitted by Lorentzian\nfunction to obtain line width (∆H)and resonance field\n(Hres) at each frequency. Later, the damping analy-\nsis is carried out for all the samples by plotting fvs\nHresand∆Hvsf(shown in Fig. 3 ). In case of S1,\nthe damping value is estimated considering the uniform\n(n=0) mode. Here, for the bilayer samples, the damping\nanalysis is done for the respective resonance fields of YIG\nand TmIG.\nThe plotted data in Fig 3 (a) were fitted by the Kittel\nequation34,\nf=γ\n2πq\n(Hres+HK)(Hres+ 4πMeff+HK)(1)\nwhere, γ\u0000\n=gµB\nℏ) is gyromagnetic ratio ( gis Lande g-\nfactor, µBis Bohr magneton), HKis in-plane anisotropic\nfield), 4πMeff\u0010\n= 4πMs+2KS\nMstFM\u0011\nis the effective de-\nmagnetizing field (KS, Ms, and tFMare perpendicu-\nFIG. 2. FMR spectra of (a) S1 (b) S2 (c) S3 and (d) S4\nsamples at different frequencies.\nlarsurfaceanisotropyconstant, saturationmagnetization\nand thickness of the magnetic layer, respectively). Af-\nterwards, the Gilbert damping constant ( α) values were\nestimated by fitting the plot in Fig 3(b) and (c) by the\nequation\n∆H= ∆H0+4παf\nγ. (2)\nTheobtainedvaluesof g,Msandα(tabulatedinTable\nII)areingoodagreementwiththeexistingliterature26,35.\nAn increase in αvalue is observed for both YIG and\nTmIG in S3 and S4 samples compared to the S1andS2\nsamples.\nThe ISHE measurements were carried out by a FMR\nbased setup. The observed voltage signal corresponding\nto the FMR resonance is shown in Fig 4 (a-d) for the\nprepared samples. In general case, for ISHE two layers\nare required, one is the source for spin current i.e., the\nmagnetic layer and the other one is the spin sink i.e., the\nhigh spin orbit coupling (HS) material. In this study, the\nHS layer is the Pt layer where the spin current source is\nthe YIG/TmIG bilayer in the samples S3 and S4. In-\nterestingly, here we have observed ISHE voltage at two\ndistinct FMR resonance field positions corresponding to\nthe resonances and concomitant spin currents mainly ex-\ncited from different layers, i.e., YIG and TmIG layers in\nboth S3 and S4 samples as shown in Fig. 4.\nIn order to extract the spin pumping voltage from the\nrectifications such as anomalous Hall effect (AHE) and\nanisotropic magneto resistance (AMR), the voltage was\nmeasured at different value of the angle ( ϕ) at a step of\n5◦in the range of 0◦to360◦at a constant frequency of 7\nGHz. We deliberately used higher frequency to avoid the\n3 magnon splitting phenomenon which in general takes4\nFIG. 3. (a) fvsHresplot for all the samples. ∆Hvsfplot for (b) YIG and (c) TmIG for respective resonance in FMR signals\nwith corresponding fittings.\nplace at lower frequencies36. Here ϕis defined as the\nangle between the direction of Hand the perpendicular\ndirection to contacts for voltage measurement. The mea-\nsured voltage is fitted by the Lorentzian equation. The\nsymmetric ( Vsym) and antisymmetric ( Vasym) contribu-\ntions of the voltage were extracted. The obtained Vsym\nandVasymvalues were plotted as a function of angle ϕ\n(shown in Fig. 5). It is to be noted that the antisym-\nmetric component is almost negligible compared to the\nsymmetric component. This clearly indicates the domi-\nnance of the spin pumping induced ISHE in the samples.\nMoreover, to quantify the spin pumping voltage and\nother rectification effects the estimated VsymandVasym\nvalues in Fig. 5 were fitted by the given equations (3)\nand (4), respectively37.\nVsym=Vspcos3(ϕ) +VAHE cos(θ)cos(ϕ)\n+VAMR⊥\nsym cos(2ϕ)cos(ϕ)\n+VAMR||\nsym sin(2ϕ)cos(ϕ)(3)\nVasym =VAHE sin(θ)cos(ϕ)+\nVAMR⊥\nasym cos(2ϕ)cos(ϕ)+\nVAMR||\nasym sin(2ϕ)cos(ϕ)(4)\nwhere VspandVAHEare voltages due to spin pumping\nand the anomalous Hall effect. Furthermore, VAMR∥\nasym,sym\nandVAMR⊥\nasym,sym are the parallel and perpendicular com-\nponents of the AMR voltage, respectively. θis the angle\nbetween the electric and magnetic fields of the microwave\nwhich is 90◦.\nThe extracted value from fitting is tabulated in Table\nII. The V∥,⊥\nAMRcomponent is calculated by the following\nformula38\nV∥,⊥\nAMR =r\u0010\nVAMR∥,⊥\nsym\u00112\n+\u0010\nVAMR∥,⊥\nasym\u00112(5)\nFrom Table II, it is further confirmed quantitatively\nthat the spin pumping voltage Vspis the dominating\nFIG. 4. (a)FMR and ISHE spectra for (a) S1, (b) S2, (c) S3\nand S4 samples at 7 GHz.\ncontribution to the measured ISHE voltage. The power-\ndependent data (shown in supplementary file Fig. 2) fur-\nther ensure the spin pumping dominance in all the sam-\nples. It is observed that the Vsphas decreased by one\norder for YIG in S3 sample as compared to the S1 sam-\nple. This is expected as in S3 sample, the spin current\ngenerated at YIG layer has to pass through the TmIG\nlayer before converting to charge current at Pt layer. In\nthis process, there may be dissipation of the spin angular\nmomentumwhichledtothedecreaseinthespinpumping\nvoltage. The Vspisalmost20timeslargerforTmIGinS3\nas compared to the S2 sample. The observed enhanced\nVspfor TmIG in S3 and S4 as compared to the S2 sam-\nple can be ascribed to the interfacial exchange coupling.\nThe presence of interlayer exchange coupling in magnetic\nbilayersystemshasalreadybeenshowntochangetheam-\nplitudes of different ferromagnetic resonance modes due\nto the dynamic exchange field at the interface39,40. As\nobservedfromtheFMRdata, thecouplingofthemagnon\ncurrent at the interface of YIG and TmIG led to the en-\nhancement of the Vspfor TmIG. Interestingly, the Vspis\nmaximum for YIG in S4 sample. This may be ascribed\nto the interfacial exchange coupling between YIG and\nTmIG.5\nTABLE II. Fitted parameters\nSample S1(GGG/YIG/Pt) S2(GGG/TmIG/Pt) S3(GGG/YIG/TmIG/Pt) S4(GGG/TmIG/YIG/Pt)\nYIG TmIG YIG TmIG\nα×10−30.51±0.07 17.00±0.03 1.10±0.0317.00±0.04 2.60±0.01 30.00±0.38\nVsp(µV) 180.0±7.8 0.06±0.01 18.0±0.71.00±0.01220.00±0.52 0.16±0.12\nVasym\nAHE (µV) 17.0±1.5 0.010±0.004 0.94±0.070.16±0.07 3.20±0.82(0.40±0.02)×10−2\nV⊥\nAMR(µV) 12.0±1.0 0.04±0.01 11.0±0.80.58±0.01 1.10±0.16 0.17±0.15\nV∥\nAMR(µV) 0.62±0.04 (0.40±0.04)×10−21.40±0.380.110±0.002 1.40±0.14 0.050±0.006\ng↑↓\neff 4.78×10182.39×10171.56×10171.12×10171.24×10181.76×1018\nFIG. 5. Angular dependence of VsymandVasymwith corre-\nsponding fits for (a) S1, (b) YIG in S3, (c) TmIG in S3, and\n(d) YIG in S4.\nIn this context, in order to quantify the spin current\npropagation, effective spin mixing conductance (g↑↓\neff) is\nevaluated by the following expression using the obtained\ndamping constant value6.\ng↑↓\neff=4π∆αMstFM\ngµB(6)\nwhere MS,tFM, and ∆αare the saturation magneti-\nzation, thickness of magnetic layer and change in Gilbert\ndamping from bilayer to single layer films, respectively.\nThese g↑↓\neffvalues (of the order of 1018m−2) are well\nmatched with the existing literature26. Hence, the inter-\nfacial exchange coupling led to the enhancement of spin\npumping for YIG and TmIG layer41.\nWehavestudiedspinpumpingandISHEforgarnet/Pt\nsystems. The damping analysis exhibits an enhancement\nofαin the bilayer garnet sample. The measured ISHE\nvoltage for YIG/Pt layer is around 180 µV where the ma-\njor contribution is from spin pumping. A decrease in the\nISHE voltage is observed by one order i.e., 17 µV in S3\nwhich is attributed to the presence of TmIG layer play-\ning as a hindrance to the transfer of angular momentumfrom the YIG layer. Whereas, an increase in spin pump-\ning voltage for YIG in S4 and TmIG in S3 is observed as\ncompared to their respective single layers. This may be\nattributed to the interfacial exchange coupling between\nYIG and TmIG. Moreover, further study can be carried\nout to have an insight to interface exchange coupling and\nthe effects of spin pumping on magnon-magnon interac-\ntions.\nACKNOWLEDGEMENT\nS.B., A.S., K. S. R., P.G., and A.M. thank the De-\npartment of Atomic Energy, Department of Science and\nTechnology, Science and Engineering Research Board\n(Grant No. CRG/2021/001245), Government of In-\ndia, Chanakya Post-doctoral fellowship, i-Hub quan-\ntum technology foundation (Sanction Order No. I-\nHUB/PDF/2022-23/04) for providing financial support.\nWork from J. L. and A. H. with respect to the data\nanalysis and discussion, as well as manuscript prepara-\ntion has been supported by the U.S. Department of En-\nergy, Office of Science, Basic Energy Sciences, Materials\nSciences and Engineering Division through Contract No.\nDE-SC0022060.\nDATA AVAILABILITY\nThe data that support the findings of this study are\navailable from the corresponding author upon request.\nREFERENCES\naElectronic mail: sbedanta@niser.ac.in\n1S. D. Bader and S. Parkin, Annu. Rev. Condens. Matter Phys.\n1, 71 (2010).\n2I. Žutić, J. Fabian, and S. D. Sarma, Reviews of modern physics\n76, 323 (2004).\n3A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands,\nNature physics 11, 453 (2015).\n4B. B. Singh, K. Roy, P. Gupta, T. Seki, K. Takanashi, and S. Be-\ndanta, NPG Asia Materials 13, 9 (2021).\n5P. Gupta, B. B. Singh, K. Roy, A. Sarkar, M. Waschk,\nT. Brueckel, and S. 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Chen, et al., arXiv preprint\narXiv:2309.03116 (2023)." }, { "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\nSz\ntSz\nt′−J22/summationdisplay\nSz\nmSz\nm′\n−J33/summationdisplay\nSz\nbSz\nb′−J12/summationdisplay\nSz\ntSz\nm−J23/summationdisplay\nSz\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\nSz\ntSz\nt′\n+/summationdisplay\nSz\nbSz\nb′)+JBB/summationdisplay\nSz\nmSz\nm′\n+ JAB(/summationdisplay\nSz\ntSz\nm+/summationdisplay\nSz\nmSz\nb)] (8)\nand\n/an}bracketle{tE(T)/an}bracketri}htAAB=−1\n3L2δ/integraldisplayt0+δ\nt0dt[JAA(/summationdisplay\nSz\ntSz\nt′\n+/summationdisplay\nSz\nmSz\nm′/summationdisplay\nSz\ntSz\nm)\n+ JBB/summationdisplay\nSz\nbSz\nb′+JAB/summationdisplay\nSz\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\n��=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 ( ��B)\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": "2202.04700v1.Coexistence_of_antiferromagnetism_and_ferrimagnetism_in_adjacent_honeycomb_layers.pdf", "content": "Coexistence of antiferromagnetism and ferrimagnetism in adjacent honeycomb layers\nD. Szallery,1,\u0003L. Prodan,2, 3,\u0003K. Geirhos,2V. Felea,2, 3, 4Y. Skourski,4D. Gorbunov,4T. F orster,4T. Helm,4T.\nNomura,4, 5A. Miyata,4S. Zherlitsyn,4J. Wosnitza,4, 6A. A. Tsirlin,7V. Tsurkan,2, 3and I. K\u0013 ezsm\u0013 arki2\n1Institute of Solid State Physics, TU Wien, 1040 Vienna, Austriay\n2Experimental Physics 5, Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, D-86159, Augsburg, Germany\n3Institute of Applied Physics, MD 2028, Chisinau, R. Moldova\n4Hochfeld-Magnetlabor Dresden (HLD-EMFL) and W urzburg-Dresden Cluster of Excellence ct.qmat,\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany\n5Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan\n6Institut f ur Festk orper- und Materialphysik, Technische Universit at Dresden, 01062 Dresden, Germany\n7Experimental Physics 6, Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, D-86159, Augsburg, Germany\n(Dated: February 11, 2022)\nAntiferromagnetic and ferro/ferrimagnetic orders are typically exclusive in nature, thus, their co-\nexistence in atomic-scale proximity is expected only in heterostructures. Breaking this paradigm and\nbroadening the range of unconventional magnetic states, we report here on an atomic-scale hybrid\nspin state, which is stabilized in three-dimensional crystals of the polar antiferromagnet Co 2Mo3O8\nby magnetic \felds applied perpendicular to the Cohoneycomb layers and possesses a spontaneous in-\nplane ferromagnetic moment. Our microscopic spin model, capturing the observed \feld dependence\nof the longitudinal and transverse magnetization as well as the magnetoelectric/elastic properties,\nreveals that this novel spin state is composed of an alternating stacking of antiferromagnetic and\nferrimagnetic honeycomb layers. The strong intra-layer and the weak inter-layer exchange couplings\ntogether with competing anisotropies at octahedral and tetrahedral Cosites are identi\fed as the\nkey ingredients to stabilize antiferromagnetic and ferrimagnetic layers in such a close proximity. We\nshow that the proper balance of magnetic interactions can extend the stability range of this hybrid\nphase down to zero magnetic \feld. The possibility to realize a layer-by-layer stacking of such distinct\nspin orders via suitable combinations of microscopic interactions opens a new dimension towards\nthe nanoscale engineering of magnetic states.\nExploring new magnetic phases with peculiar spin or-\nders that can lead to unique material functionalities is\na fundamental goal of spintronics1{3. While a great\nfraction of the spintronic devices is still based on ferro-\nmagnets and their intefaces/heterostructures with heavy\nmetals4{6, antiferromagnets progressively gain ground in\nsuch applications, as witnessed by the exponential grow-\ning \feld of antiferromagnetic (AFM) spintronics1,7,8. Al-\nthough ferromagnetism and antiferromagnetism are two\nmutually exclusive phases, the possibility to combine\ntheir best qualities triggered extensive research, leading\nto the realization of bilayers and heterostructures fea-\nturing a large exchange bias9,10. These arti\fcial struc-\ntures are typically composed of thin \flms of soft ferro-\nmagnets deposited on an AFM layer. Since the mag-\nnetization of the ferromagnetic layer is often unstable,\nbeing susceptible even to weak \ructuations of magnetic\n\feld and/or spin current, the exchange coupling to the\nanisotropic antiferromagnet is required to make the ferro-\nmagnetic (FM) state more robust for reliable device ap-\nplications6,11. In fact, the exchange-bias e\u000bect is essential\nfor current and future spin valve and MRAM technolo-\ngies12{16.\nIn addition to metallic ferro- and antiferromagnets,\nrecently magnetic insulators have also been considered\nas active elements in spintronic applications17{22. Their\nmain potential advantage, as compared to the metallic\nsystems, is that their operation is based on electric-\feldcontrol and not on current control, thus heat dissipation\nis considerably reduced. Furthermore, as opposed to met-\nals, they are characterized by short-range magnetic inter-\nactions, i.e. lack the non-local Ruderman-Kittel-Kasuya-\nYosida interactions23,24. Short-range interactions in in-\nsulating crystals could, in principle, lead to the real-\nization of distinct magnetic orders even in atomic-scale\nproximity. This concept naturally requires materials\nthat are composed of structural subunits, where the ex-\nchange interactions within these subunits are consider-\nably stronger than between them.\nAs a proof of principle, we demonstrate the coexistence\nof antiferromagnetism and ferrimagnetism in adjacent\nhoneycomb layers of Co 2Mo3O8. While this compound\nfeatures a strict hierarchy of intra- and inter-layer ex-\nchange interactions, covalent and ionic chemical bondings\nbetween the neighbouring layers are still preserved, dis-\ntinguishing this system from van der Waals magnets and\nheterostructures25. This unconventional hybrid phase\nemerges in external magnetic \felds applied perpendic-\nular to the layers as an intermediate state between the\nlow-\feld easy-axis collinear AFM state and the high-\feld\nspin-\rop phase. The latter two represent conventional\nmagnetic phases with the same types of spin patterns\nin each honeycomb layer. As a further peculiar aspect\nof this intermediate phase, the ferromagnetic moment\nemerging at every second layer has a transverse compo-\nnent, that is even larger than the component parallel to\nTypeset by REVT EXarXiv:2202.04700v1 [cond-mat.str-el] 9 Feb 20222\nthe \feld. We show that competing magnetic anisotropies\nof chemically di\u000berent Cosites are indispensable for the\nlayer-by-layer alternation of the magnetic state, in addi-\ntion to the weak inter-layer coupling. In a simplistic spin\nmodel, we specify the general microscopic conditions, i.e.\nthe ratio of exchange couplings and anisotropies, required\nto stabilize this hybrid phase without an external \feld.\nTernary transition-metal molybdenum oxides,\nM2Mo3O8with M= Mn, Fe, Co, and Ni, crystal-\nlize in a hexagonal polar structure (space group P63mc).\nTheir structure is a stack of honeycomb layers of\nmagnetic M2+ions along the caxis, where adjacent\nMsites have tetrahedral and octahedral coordinations\nin an alternating fashion both within the honeycomb\nlayers and along the caxis26, as shown in Fig. 1.\nThese materials exhibit fascinating phenomena, such as\naxion-type electrodynamics27, high-temperature optical\ndiode e\u000bect28, vibronic excitations29, doping induced\nferrimagnetism30, and precursor short-range magnetic\norder within the honeycomb layers31. Furthermore, these\ncompounds were predicted to have topological spin-wave\nexcitations32. According to their symmetry they are\ncandidates to host N\u0013 eel-type skyrmion lattices33, in case\nferro- or ferrimagnetic order can be stabilized, like in\nMn2Mo3O834and Zn-doped Fe 2Mo3O830.\nThese compounds undergo an AFM transition at tem-\nperatures between TN=6-60 K26,30,32,34{36. Not onlyTN\nbut also the spin pattern in the ground state shows a\nstrong variation with the transition metal ion. Without\nan external magnetic \feld, Co 2Mo3O8and Fe 2Mo3O8\nshare a common easy-axis collinear AFM structure37,38,\nas depicted in Fig. 1, where adjacent spins are arranged\nantiferromagnetically in the honeycomb layers and form\nferromagnetic chains along the caxis. This is a fully com-\npensated AFM state, though in Fe 2Mo3O8the individ-\nual honeycomb layers may possess a \fnite magnetization\ndue to slightly di\u000berent magnetic moments of the tetra-\nand octahedral Fesites35,37. In contrast, the net mag-\nnetization of all layers point along the same direction\nin Mn 2Mo3O8, forming an easy-axis ferrimagnet36. In\nNi2Mo3O8, a more complicated zig-zag AFM order has\nbeen reported32. Such versatility of ground-state spin\npatterns, realized in the same crystal structure, implies\na complex network of exchange paths and signi\fcant de-\npendence of magnetic interactions on the orbital occupa-\ntion, spin-orbit e\u000bects and spin-lattice coupling.\nAt \frst, we verify by susceptibility measurements that\nthe layered crystal structure of Co 2Mo3O8leads to strong\nmagnetic anisotropies and con\fnes the dominant ex-\nchange paths to individual honeycomb layers, as implied\nby the lack of dispersion of the magnon branches along\nthecaxis39. Indeed, we \fnd a large di\u000berence between\nthe susceptibility values obtained for \felds parallel and\nperpendicular to the layers. Based on the temperature-\ndependent susceptibility curves, shown in Fig. S1 of the\nsupplement, we re\fned the basic set of magnetic inter-\nactions using quantum Monte Carlo simulations40. The\nimportant aspects of the magnetic interactions revealed\nba\nKOJin\nterJinter-OTJinter-TT\nKTJi\nntra\nJinter-OO\nCo\nO\na\nbFIG. 1.jCrystal structure of Co 2Mo 3O8.(a) Co sites\n(blue spheres) and O positions (red spheres) in the abplane\nand (b) in the acplane including the spin pattern of the\ncollinear AFM ground state37. Inter-spacer Mo ions are omit-\nted.Jintra stands for the intra-layer interaction between the\noctahedral and tetrahedral sites in the same honeycomb layer,\nwhile Jinter\u0000OO,Jinter\u0000TT, and Jinter\u0000OTare inter-plane\ninteractions between octahedral sites, tetrahedral sites, and\noctahedral-to-tetrahedral sites, respectively. Jinter is an ef-\nfective inter-layer coupling along the caxis, capturing the\ne\u000bect of the three inter-layer exchange paths in a simpli\fed\nscheme. Values of the magnetic interactions are listed up in\nTable I.\nhere are the dominance of intra-layer AFM exchange,\nJintra= 14 K, the weak inter-layer FM exchange, Jinter=-\n1.4 K, the strong easy-axis anisotropy, KO=14 K, and\nthe weak easy-plane anisotropy, KT=-1.4K, of Co sites\nwith octahedral and tetrahedral coordination, respec-\ntively.Jinter refers to an overall e\u000bective inter-layer\ncoupling, which has contributions from exchanges be-\ntween octahedral sites ( Jinter\u0000OO) and tetrahedral sites\n(Jinter\u0000TT) on adjacent layers according to Jinter=-\n(Jinter\u0000TT+Jinter\u0000OO)/4. The di\u000berent exchange paths\nand the single-ion anisotropies are depicted in Fig. 1.\nThis hierarchy of intra- and inter-layer exchanges as well\nas the presence of strong single-ion anisotropies, that are\ncomparable to the dominant exchange coupling, are fur-\nther supported by our DFT results. (The list of magnetic\nparameters is given in Table I; for details of the Monte\nCarlo and DFT calculations see the supplement).\nTo explore possible magnetic orders emerging from the\nmagnetic interactions identi\fed above, we constructed\na simpli\fed spin model34,41. We describe the four-spin\nunit cell of Co 2Mo3O8, by introducing Ising spins on the\ntwo octahedral Cosites ( S1andS3), while spins at the\ntetrahedral sites ( S2andS4) are characterized by easy-\nplane anisotropy, KT<0. The pairs in the same layers,3\nTABLE I. Comparison of exchange constants and single-\nion anisotropies as determined by three di\u000berent approaches:\nQuantum Monte Carlo simulation of temperature-dependent\nsusceptibility, ab initio calculations (DFT) and mean-\feld\nspin model. All values are in kelvin units.\nMonte Carlo Jintra=14Jinter=-1.4 KO=14 KT=-1.4\nDFT Jintra=18Jinter=-2.2 KO=7.7 KT=2.3\nMean \feld Jintra=12Jinter=-1.5 KO=26.8 KT=-1.73\nFIG. 2.jMagnetic phase diagram of the simplistic\nfour-spin model. Ground-state spin orders on the mag-\nnetic anisotropy ( KT) versus Zeeman energy plane for mag-\nnetic \felds applied parallel to the vertical Ising spins (black\narrows). Spins indicated by gray arrows experience KT<0\neasy-plane anisotropy, favouring horizontal spin alignment.\nJintra >0 is the AFM exchange between spins in the same\nhorizontal \"layer\", while Jinter =\u0000xJintra is the ferromag-\nnetic coupling between spins on adjacent \"layers\", as depicted\nin Fig. 1. For these calculations the value of x= 0:1 was\ntaken. Dashed horizontal and vertical lines indicate special\nvalues of ( KT) and the magnetic \feld ( H), respectivley. The\nanisotropy value corresponding to the case of Co 2Mo3O8is\nalso indicated by horizontal black line. Purple, red, yellow,\nblue and green regions correspond to the tilted antiferromag-\nnetic (TAF), symmetric spin-\rop (SF), asymmetrically canted\n(ASC), collinear antiferromagnetic (AF), and \feld-polarized\nferromagnetic (FM) phases, respectively.\nS1\u0000S2andS3\u0000S4, are coupled by a strong antifer-\nromagnetic exchange Jintra>0, while the S1\u0000S4and\nS2\u0000S3pairs are connected by the weaker Jinter<0 fer-\nromagnetic coupling. We studied the ground state of this\nmodel by varying KTand the strength of the magnetic\n\feld, applied along the Ising spins. The phase diagram\nin Fig. 2, obtained for Jinter =\u0000Jintra=10, reveals \fve\ndistinct phases. At most four of them can be realized for\na given set of parameters by varying the magnetic \feld.\nWhile this choice of the inter- and intra-layer exchanges\nis realistic for Co 2Mo3O8according to our Monte Carlo\nand DFT calculations (see Table I and Fig. 2), we ap-proximate the magnetic state of octahedral sites by Ising\nspins, by formally setting KOto in\fnity. In each mag-\nnetic phase, depicted in Fig. 2 and discussed below, we\nillustrate the magnetic order schematically. Black and\ngray arrows represent the vertical Ising spins and the\nspins with horizontal easy-plane anisotropy, respectively.\nIfKTis su\u000eciently weak, 2 jKTj\n(Jintra +jJinterj)=2, a tilted antiferromagnetic phase\n(purple TAF region) becomes the zero-\feld ground state,\nwhere the tetrahedral spins stay antiparallel with each\nother yet they are not co-aligned with the Ising spins\nanymore, but tilted away to their easy plane instead. In\n\fnite \felds, the magnetization of this phase has \fnite\ncomponents both along and perpendicular to the \feld.\nIn the following, we test the predictions of our spin\nmodel on Co 2Mo3O8, which is expected to o\u000ber an ideal\nlaboratory for exploring a layer-by-layer alternation of\nAFM and ferrimagnetic states. First, we study the evo-\nlution of its ground state by high-\feld magnetometry.\nFig. 3(b) shows the magnetization measured in pulsed\n\felds applied parallel and perpendicular to the caxis at\n1.5 K. ForHkc, the magnetization shows only a weak\nincrease up to the \frst critical \feld, Hc1=25 T, above\nwhich it grows rapidly, leading to a kink-like feature at\nHc1. With further increasing the \feld, a sudden jump of\nthe magnetization to nearly half of the saturation value\nis observed at the second critical \feld, Hc2=30 T. Then,\nvia a linear increase, the saturation to the spin-only mo-4\n-2-100\n24-\n1.5-1.0-0.50.00\n2 04 06 00240264\nH\n II cH II c Mlong (µB/f.u.)Hc1H sHc2(\na)H\n ^ cE\nxp. M\nodelExp. M\nodelHc1Hc2H s(\nb)(\nc)(d)(\ne)(\nf)(\ng)\nAFMS FF MA SC0\n2 04 06 00.00.5 Mtrans (µB/f.u.) E\nxp. (a.u.)M\nodel Δ\nP (mC/m2)E\nxp.M\nodel \nH || cΔ\nL/L (10-4) E\nxp.M\nodelΔ\nv/v (10-2)T (K) µ\n0H (T)µ 0H (T)\nFIG. 3.jSequence of magnetic phases in Co 2Mo 3O8.(a) Spin con\fgurations in the collinear antiferromagnetic (AFM),\nasymmetric canted (ASC), symmetric spin \rop (SF) and spin-polarized ferromagnetic (FM) phases. Color coding for the\noctahedral and tetrahedral sites is the same as in Fig. 1. (b) Magnetization measured in magnetic \felds parallel (red line) and\nperpendicular (green line) to the caxis at 1.5 K, together with the corresponding curves (blue and brown) obtained from the\nmean-\feld model. (c) Transverse magnetization as measured (red line) and as predicted by the mean-\feld model (blue line).\nThe measured data are plotted in arbitrary units, since the calibration of the magnetic torque was not possible in pulsed \felds.\nField dependence of (d) the magnetically induced polarization \u0001 P, (e) the relative change of the sample length \u0001 L=L, (f) the\nrelative change of the sound velocity \u0001 v=v, and (g) the change of the sample temperature (magnetocaloric e\u000bect) at 1.5 K with\n\felds parallel to the caxis. Blue lines in panels (d) and (e) show the results of the mean-\feld model. In each panel, vertical\ndashed lines mark the critical \felds Hc1andHc2and the saturation \feld Hs, while the region of the intermediate ASC phase\nis indicated by a grey area.\nment of 6\u0016B/f.u. is reached at Hs=60 T. For H?c,\nthe magnetization shows a nearly linear increase over the\nwhole \feld range with a slope approximately half as large\nas forHc2JijSiSj\u0000X\niKi(Sz\ni)2(1)\nwere obtained using a mapping procedure50. Here,Jij\nis the exchange interaction between lattice sites iand\nj,Kiis the single-ion anisotropy at site iand S=3/2.In addition to DFT, classical mean-\feld and quantum\nMonte Carlo simulations were used to re\fne the exchange\nand anisotropy parameters of Eq. (1). In the mean-\feld\nmodel, the spin con\fgurations minimizing the energy\nwere numerically determined for the whole experimen-\ntally studied \feld range. While a larger set of magnetic\ninteractions were identi\fed in DFT ( Jintra,Jinter\u0000OO,\nJinter\u0000TT,Jinter\u0000OT,KO,KT), only the minimal set\n(Jintra,Jinter,KO,KT), necessary to reproduce the ob-\nserved temperature- and \feld-dependent magnetic prop-\nerties, were kept in the Monte-Carlo and mean-\feld cal-\nculations.\nThe mean-\feld results were obtained by \fnding the\nminimal energy of the four-spin cluster,\nHMF= 3Jintra (SO1ST1+SO2ST2) + 2Jinter(SO1ST2+ST1SO2)\n\u0000KO\u0000\n(Sz\nO1)2+ (Sz\nO2)2\u0001\n\u0000KT\u0000\n(Sz\nT1)2+ (Sz\nT2)2\u0001\n\u0000g\u00160\u0016BH(SO1+ST1+SO2+ST2);(2)\nwhere SO1andSO2are the classical spin vectors corre-\nsponding to the Co ions in octahedral environment in the\nodd and even ab-plane layers of the crystal, respectively.\nSimilarly, ST1andST2represent the spins of the tetrahe-\ndrally coordinated Co ions. Integer multiplicators of theexchange terms correspond to the coordination numbers.\nThe g-factor, vacuum permeability, Bohr magneton and\nmagnetic \feld are denoted by g,\u0016B,\u00160andH, respec-\ntively.\nThe Hamiltonian of the simplistic four-spin model\n(Fig. 2) was\nHsimpl =J1(S1Sz\n2+S3Sz\n4) +J2(S1Sz\n4+Sz\n2S3)\u0000K2\u0000\n(Sz\n2)2+ (Sz\n4)2\u0001\n\u0000g\u00160\u0016BHz(S1+Sz\n2+S3+Sz\n4);(3)\nwhereS1=\u0006SandS3=\u0006Sare Ising spins.\n\u0003These authors contributed equally to this work\nydavid.szaller@tuwien.ac.at\n1Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J.,\nAntiferromagnetic spintronics, Nat. Nanotechnol. 11, 231\n(2016).\n2Yang, S.H., Naaman, R., Paltiel, Y. & Parkin, S. S. P.,\nChiral spintronics, Nat. Rev. Phys. 3, 328 (2021).\n3Zhou, Y., Magnetic skyrmions: intriguing physics and new\nspintronic device concepts, Natl. Sci. Rev. 6, 210 (2018).\n4Zuti\u0013 c, I., Fabian, J. & Sarma, S.D., Spintronics: Funda-\nmentals and applications, Rev. Mod. 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G., Predicting the spin-lattice order of frustrated\nsystems from \frst principles, Phys. Rev. B 84, 224429\n(2011).\nAcknowledgements We acknowledge S.-W. Cheong,\nS. Bordacs, J. Deisenhofer, O. Zaharko and A. Pimenov\nfor fruitful discussions. This work was supported by the\nDeutsche Forschungsgemeinschaft (DFG) through Tran-\nsregional Research Collaboration TRR 80 (Augsburg,\nMunich, and Stuttgart), SFB 1143, and the excellence\ncluster ct.qmat (EXC 2147, Project ID 39085490), and by\nthe BMBF via DAAD (Project No. 57457940), as well asby the project ANCD 20.80009.5007.19 (Moldova). D.S.\nacknowledges the support of the FWF Austrian Science\nFund grants I 2816-N27 and TAI 334-N. We acknowl-\nedge support by HLD at HZDR, member of the European\nMagnetic Field Laboratory (EMFL).\nAuthor Contributions D.S. and L.P. contributed\nequally to this work. L.P., K.G., V.T. performed the\nmeasurements in static magnetic \felds. V.F., Y.S., D.G.,\nT.F., T.N., A.M., S.Z., J.W. performed the experiments\nin pulsed magnetic \felds. V.T. analysed the data. L.P.,\nV.T. synthesized and characterized the single crystals.\nT.H. fabricated the lamella for the cantilever torque mag-\nnetometry measurements. A.T. performed the Monte\nCarlo and the ab initio calculations. D.S. performed the\nmean-\feld calculations. I.K. wrote the manuscript with\ncontributions from D.S., V.T. and A.T. I.K. planned the\nproject.\nAdditional information The authors declare no\ncompeting \fnancial interests. Supplementary informa-\ntion accompanies this paper. Correspondence and re-\nquests for materials should be addressed to D.S. and I.K." }, { "title": "2012.00576v1.Magnon_hybridization_in_ferrimagnetic_heterostructures.pdf", "content": "Magnon hybridization in ferrimagnetic heterostructures\nSong Li,1, 2Ka Shen,2,\u0003and Ke Xia3\n1School of Science, Tianjin University, Tianjin 300072, China\n2The Center for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, Beijing 100875, China\n3Beijing Computational Science Research Center, Beijing 100193, China\n(Dated: December 2, 2020)\nWe study magnon hybridization in a ferrimagnetic heterostructure consisting of ultrathin gadolin-\nium iron garnet and yttrium iron garnet layers and show the localized and extended spatial pro\fles\nof the magnon modes with di\u000berent polarizations. These modes are expected to have distinct\nthermal excitation properties in the presence of a temperature gradient across the heterostructure.\nFrom a quantitative analysis of their consequences on longitudinal spin Seebeck e\u000bect, we predict\nan observable shift of the sign-changing temperature with respect to the one previously observed\nin gadolinium iron garnet. Moreover, the sign-changing point of spin Seebeck signal is found to be\ntunable by YIG thickness. Our results suggest the necessity of taking into account the temperature\ndi\u000berence between the magnon modes in ferrimagnetic heterostructures.\nI. INTRODUCTION\nMagnons [1, 2], collective excitations in magnetic or-\ndering systems, have been considered as potential in-\nformation carriers in low-power devices. They can be\nactivated by microwave [3, 4], laser [5{7] and thermal\n\ructuation [8{10] and interact with each other through,\ne.g., exchange coupling [11, 12] and magnetic dipolar in-\nteraction [11, 13, 14]. In the past decade, many inter-\nesting magnon-related phenomena, such as spin Seebeck\ne\u000bect (SSE) [8, 15], orbital Nernst e\u000bect of magnon [16]\nand corner states in ferromagnetic breathing Kagome lat-\ntice [17], have been reported. To understand the under-\nlying physics of these phenomena, we need to explore the\nproperties of magnons.\nWhile earlier studies mainly focused on the magnons in\na single magnetic layer, recent experiments revealed at-\ntractive properties in hybrid magnetic structure. For in-\nstance, unexpected enhancements were observed in spin\npumping (SP) [18] and SSE signals [19] when an ultra-\nthin antiferromagnetic NiO was inserted between yttrium\niron garnet (YIG) and Pt layers [20{23]. Such enhance-\nments were attributed to either the increased interfacial\nspin mixing conductance [20, 21] or the interference of\nevanescent waves [22], one type of hybrid spin waves in\nthe magnetic bilayer structure. Recent phase-resolved\nx-ray pump-probe measurements showed the evidence\nof magnon transmission via the evanescent waves [24].\nAnother important feature in hybrid magnetic structure\nis the anticrossing between di\u000berent ferromagnetic reso-\nnances observed in, e.g., YIG-Ni [25], YIG-Co [26] and\nYIG-CoFeB [27] bilayer structures, which reveals the for-\nmation of hybrid spin wave modes around the anticross-\ning. These mode hybridizations can a\u000bect the measur-\nable quantities in practical experiments. For example,\nthe suppression and enhancement of ferromagnetic reso-\n\u0003kashen@bnu.edu.cnnance linewidth were observed for the in-phase and out-\nof-phase coupled modes, respectively, in YIG-permalloy\n(YIG-Py) system [28]. A precise description of these ob-\nservations requires a detailed calculation of the hybrid\nmagnon modes which could play an essential role, espe-\ncially when part of the system is of only a few nanometers\nthick.\nYIG as one of the most important magnetic materi-\nals due to its low-damping coe\u000ecient is usually grown\non gadolinium gallium garnet (GGG) substrate [29]. Re-\ncently, Gomez-Perez et al. showed that near the interface\nbetween YIG and GGG, the Gd atoms from GGG can\ndi\u000buse into the YIG layer and substitute Y atoms in YIG,\nforming a natural YIG-GdIG magnetic bilayer [30], where\nthe thickness of the GdIG layer is around 3 nanometers.\nTheoretically, whereas the magnon spectra in both YIG\nand GdIG have been studied in literatures [31{35], the\nhybrid magnon spectrum in their hybrid system is still\nmissing. On the other hand, although GdIG shares the\nsame structure with YIG, its SSE [36] is found to be\nquite di\u000berent from that in YIG [9, 37]. An interesting\nquestion one may ask is: What are the consequences of\nhybrid magnon modes in YIG-GdIG bilayer in the spin\nSeebeck measurement. Therefore in this work, we calcu-\nlate the hybrid magnon spectrum in YIG-GdIG bilayer\nsystem and analyze its consequences in the longitudinal\nspin Seebeck e\u000bect (LSSE).\nII. HYBRID SPECTRUM AND MODE\nHYBRIDIZATIONS IN THE YIG-GDIG BILAYER\nSYSTEM\nA. Qualitative analysis\nWe consider the situation with the thickness of YIG in\nthe YIG-GdIG bilayer much larger than that of GdIG.\nAs experimentally demonstrated in Ref.[30], the net mag-\nnetic moments of the two parts in such structure align\nantiparallelly under a weak in-plane \feld. Therefore, asarXiv:2012.00576v1 [cond-mat.mes-hall] 1 Dec 20202\nFIG. 1. Schematic of the structure (a), the magnon spectrum\nand hybrid modes (b) in the YIG-GdIG hybrid system, where\nthe red and blue lines stand for the left-handed ( \u000b) and right-\nhanded (\f) modes, respectively.\nsketched in Fig.1(a), when a small magnetic \feld is ap-\nplied along\u0000^z, the magnetic moment of the YIG layer\ndominated by the d-Fe sublattice [31] points to \u0000^zand\nthat of the GdIG layer determined by c-Gd sublattice [31]\npoints to ^z. Notice also that both orientations of a-Fe and\nd-Fe sublattice in the GdIG layer are the same as those\nin the YIG layer.\nIn such a system, the YIG layer contains two types\nof modes, where the one with lower frequency (higher\nfrequency) is dominated by the precession of d-Fe (a-Fe)\nsublattice [32] and the GdIG layer contains three types of\nmagnons, dominated by the precessions of c-Gd sublat-\ntice, d-Fe sublattice and a-Fe sublattice, respectively [33].\nThe magnon dispersions in the two parts of this bilayer\nsystem are sketched in the upper panel of Fig.1(b), where\nthe two bands in the two layers are of opposite chiralities\nwith the gap in YIG larger than that in GdIG [31, 32, 35].Notice that the high energy mode dominated by the pre-\ncession of a-Fe sublattice in GdIG will not a\u000bect our main\nresults, we therefore discard it in the \fgure. Due to the\nantiferromagnetically aligned magnetic moments of Gd\natoms and d-Fe atoms (as seen in Fig.1(a)), the lower\nbranches in the YIG layer and the GdIG layer carry op-\nposite spin angular momentums. By taking into account\nthe hybridization between the two layers, one expects\nfour types of hybrid modes (as plotted in the lower panel\nof Fig.1(b)): left-handed ( \u000b1) modes propagating in the\nGdIG layer but evanescent in the YIG layer and right-\nhanded (\f1) modes propagating in the YIG layer but\nevanescent in the GdIG layer, right-handed ( \f2) and left-\nhanded (\u000b2) modes propagating in both layers.\nB. Heisenberg model\nTo calculate the concrete spectrum, we apply the\natomic spin exchange model to our bilayer structure\nH=\u0000NX\nn=1[naX\ni=1X\njrijj=raaJaa\nijSa(Rin)\u0001Sa(Rin+rij)\n+ndX\ni=1X\njrijj=rddJdd\nijSd(Rin)\u0001Sd(Rin+rij)\n+ncX\ni=1X\njrijj=rccJcc\nijSc(Rin)\u0001Sc(Rin+rij)\n+ 2naX\ni=1X\njrijj=radJad\nijSa(Rin)\u0001Sd(Rin+rij)\n+ 2naX\ni=1X\njrijj=racJac\nijSa(Rin)\u0001Sc(Rin+rij)\n+ 2ndX\ni=1X\njrijj=rdcJdc\nijSd(Rin)\u0001Sc(Rin+rij)]; (1)\nwherena,ncandndare the total numbers of local spins\nat a-Fe, c-Gd and d-Fe sites in one unit cell. rss0andJss0\nij\nare the nearest neighbor distance and the position depen-\ndent exchange coupling between magnetic atoms sand\ns0. From crystal structure of garnet, we extract the set\nof nearest neighbor distances as raa= (p\n3=4)a0,rdd=\n(p\n6=8)a0,rdc= (1=4)a0andrad=rac= (p\n5=8)a0, with\na0= 1:24 nm. We then derive the bosonic Bogoliubov-de3\nFIG. 2. (a) Spin wave dispersion in YIG(7.4nm)-GdIG(2.5nm) [001] bilayer system, where red and blue lines are speci\fed for\n\u000band\fmodes, respectively. Pink dots and green pentagrams are \fbranches in 7.4-nm-thick YIG and 2.5-nm-thick GdIG.\n(b) and (c) Transverse spin orientations of a-Fe atoms along the bilayer at k= 0:05\u0019=a0andk= 0:78\u0019=a0. Black dotted lines\nenclose the interfacial regions between YIG and GdIG. The capital letters, L and R, are the abbreviations for left-handed mode\nand right-handed mode, respectively.\nGennes (BdG) Hamiltonian as [32, 33]\nHk=naX\ni;j=1ay\ni(k)Aij(k)aj(k) +ncX\ni;j=1cy\ni(k)Cij(k)cj(k)\n+ndX\ni;j=1dy\ni(\u0000k)Dij(\u0000k)dj(\u0000k)\n+naX\ni=1ncX\nj=1h\nay\ni(k)Bac\nij(k)cj(k) +h:c:i\n+naX\ni=1ndX\nj=1h\nay\ni(k)Bad\nij(k)dy\nj(\u0000k) +h:c:i\n+ncX\ni=1ndX\nj=1h\ncy\ni(k)Bcd\nij(k)dy\nj(\u0000k) +h:c:i\n: (2)\nThe matrix elements Aij,Cij,DijandBss0\nijare given\nin Appendix A. Operators ai,diandciare de\fned\nby Holstein-Primako\u000b (H-P) transformation [1] of their\natomic spins as\nSz\na;i=Sa;i\u0000ay\niai;S+\na;i=q\n2Sa;i\u0000ay\niaiai;\nSz\nc;i=Sc;i\u0000cy\nici;S+\nc;i=q\n2Sc;i\u0000cy\nicici;\nSz\nd;i=\u0000Sd;i+dy\nidi;S\u0000\nd;i=diq\n2Sd;i\u0000dy\nidi:(3)\nBy diagonalizing the Hamiltonian through paraunitary\ntransformation [38, 39], we can obtain the magnon spec-\ntrum of this hybrid bilayer system. The resultant eigen-\nstates are linear combination of the local spin operators\nat a, c and d sites [32]\n\u000bi0\nk=pi0i\na;kai;k+pi0j\nc;kcj;k+pi0l\nd;\u0000kdy\nl;\u0000k;\n\fj0\nk=pj0i\na;\u0000kay\ni;\u0000k+pj0j\nc;\u0000kcy\nj;\u0000k+pj0l\nd;kdl;k; (4)\nwhere superscripts i0= 1;\u0001\u0001\u0001;na+ncandj0= 1;\u0001\u0001\u0001;nd\nare the indexes of modes \u000band\f.i= 1;\u0001\u0001\u0001;na,j=1;\u0001\u0001\u0001;ncandl= 1;\u0001\u0001\u0001;ndhere are the indexes of local\nspin operators at a, c and d sites. Einstein summation\nconvention is applied for i,j, andl.\nC. Numerical results\nIn this subsection, we present the hybrid magnon\nspectrum and wave functions in a YIG-GdIG [001] bi-\nlayer system with YIG and GdIG layers 6-unit-cell-\nthick (7.4 nm) and 2-unit-cell-thick (2.5 nm), respec-\ntively. We adopt Jaa\nij=\u00000:329meV,Jdd\nij=\u00001:161meV,\nJad\nij=\u00003:449meV and Sa;i=Sd;i= 2:5 for the YIG\nlayer [31] and Jaa\nij=\u00000:081meV,Jdd\nij=\u00000:137meV,\nJad\nij=\u00002:487meV,Jac\nij= 0:032meV,Jcd\nij=\u00000:157meV,\nSa;i= 2:1,Sd;i= 2:05 andSc;i= 3:5 for the GdIG and\ninterfacial regions [35]. Fig.2(a) shows the magnon spec-\ntra of the two separated layers and their hybrid bilayer\nsystem. For a 7.4-nm-thick YIG \flm, the lowest seven\nmagnon branches, corresponding to the right-handed fer-\nromagnetic resonance mode and its subbands with in-\ncreasing nodes are shown in pink dots and \u000bmodes are\nalso found in high-frequency range (not shown). For the\nmagnons in a 2.5-nm-thick GdIG layer, a \fmode (shown\nin green pentagrams) and many \u000bmodes lying within 0-\n0.5 THz are plotted. As we can see in this \fgure, the \f\nmode in the GdIG layer crosses with other three \fmodes\nin the YIG layer. As a result of interlayer coupling in the\nYIG-GdIG structure, gaps are opened at these crossing\npoints in the hybrid magnon spectrum (shown in the red\nand blue lines). The slight deviation between the bands\nof the bare YIG \flm and the hybrid system is due to the\nmissing atomic layer at the interface.\nTo give more details of the hybrid modes in this bi-\nlayer system, we plot the instantaneous orientations of\nmagnetic moments for a-Fe atoms in x-y plane along the\nwhole bilayer system. Fig.2(b) shows the low-frequency\nhybrid wave functions near the center of Brillouin zone\n(k= 0:05\u0019=a 0), including three types of hybrid modes,4\ni.e.,\u000b1modes for L1 and L2, \f1modes for R1, R2 (not\nshown), R3 and \f2modes for R4, R5, \u0001\u0001\u0001, R8. As the\nanticrossings in the hybrid spectrum reveal the forma-\ntion of new hybrid modes, we plot the wave functions at\nk= 0:78\u0019=a 0(marked by the dashed line in Fig.2(a)) in\nFig.2(c) and \fnd that R3 is transformed from \f1mode\nto\f2mode while the types of the other modes remain\nunchanged. Note that \u000b2-type magnons are also found at\nthese two wavevectors in the high-frequency regime (not\nshown here).\nIII. LSSE IN YIG-GDIG-NM TRILAYER\nSYSTEM\nFIG. 3. Relation between the local temperature of phonons\n(green curve) in YIG-GdIG-NM trilayer in LSSE con\fgura-\ntion and that of magnons for \u000b(red curve) and \f(blue curve)\nmodes in GdIG layer.\nWith the hybrid magnon spectra and wave functions,\none can analyze the consequences of hybrid modes in the\ntransport properties, e.g., LSSE, in which the nonequi-\nlibrium between the phonons and magnons near the mag-\nnetic insulator-normal metal (NM) interface is considered\nas the driving force [40{44]. During the measurement of\nLSSE, the magnons accumulated at the GdIG-NM in-\nterface are mainly \u000b1-type and\f2-type while the contri-\nbutions from \f1-type and\u000b2-type modes are far lesser\ndue to the blockage by GdIG and the high excitation fre-\nquency, respectively. Considering the extended and lo-\ncalized features of the \f2-type and\u000b1-type magnons, the\ntemperatures of the \u000band\fmagnons near the GdIG-NM\ninterface in the LSSE should be di\u000berent. Speci\fcally,\nthe di\u000berences between the temperature of \fmagnons,\nT\f\nm, and the temperature of phonons, TF\np, should be\nlarger than that between the temperature of \u000bmagnons,\nT\u000b\nm, and TF\np. Therefore, we introduce a two-temperature\nmodel to describe the LSSE in the YIG-GdIG-NM tri-\nlayer system as sketched in Fig.3, where an interfacial\ntemperature discontinuity between phonons in the GdIGlayer and the NM layer, equal to the temperature of elec-\ntrons, T e, is also introduced. Note that since the thick-\nness of each magnetic layer is smaller than the phonon\nmean free path, we here assume the local temperature\ninside the YIG and GdIG layers are both uniform and\nfocus on the temperature di\u000berence across each interface\ndue to Kapitza heat resistance.\nWithin the linear-response regime, the spin currents\ngenerated in YIG-GdIG-NM trilayer are proportional to\nthe temperature di\u000berences between magnons and elec-\ntrons [36, 41, 45], i.e.,\nIs=AT\u0001T\u000b\nme\u0000BT\u0001T\f\nme; (5)\nwhereATandBTare the spin Seebeck conductances\n(SSC) for\u000bmagnons and \fmagnons. The temperature\ndi\u000berences in Eq.(5) can be read from Fig.3\n\u0001T\u000b\nme= \u0001T\u000b\nmp+ \u0001TFN\npp;\n\u0001T\f\nme= \u0001T\f\nmp+ \u0001TFN\npp; (6)\nwhere \u0001T\u000b\nmp, \u0001T\f\nmpare the temperature di\u000berences be-\ntween\u000bmodes and local phonons, \fmodes and local\nphonons near the GdIG-NM interface. Based on Eq.(5),\nwe can compare the situations in the conventional GdIG-\nNM bilayer and YIG-GdIG-NM trilayer: In the GdIG-\nNM bilayer, where \u0001T\u000b\nmeand \u0001T\f\nmeare equal and \fxed\nby the boundary, changing SSC is the only approach to\ntune the spin Seebeck signals; In contrast, the hybrid\nmodes in the YIG-GdIG-NM trilayer system would cause\na di\u000berence between \u0001T\u000b\nmpand \u0001T\f\nmpand thus provide\nan additional possibility to manipulate LSSE.\nA. SSC in the LSSE\nTo calculate the SSC in Eq.(5), we use the s-d exchange\nmodel at the GdIG-NM interface [41]\nH0=l2\n0NX\nn=1X\ni;j2intJsSin\u0001\u001bj\u000e(rj\u0000Rin); (7)\nwhere\u001bjis the electron spin at position rj,Nis the total\nnumber of unit cells in the 2-dimensional plane, l2\n0is the\narea of cross section, int is the abbreviation of interface\nandJsis the coupling strength between magnetic atoms\nand s electrons. One has the second-quantized Hamilto-\nnian after performing H-P transformation [1] as\nH0=~JaX\nq;kh\n(X\ni2intai;kge;y\n#;qge\n\";q\u0000k+h:c:)\n+ (X\nj2int\u0011cj;kge;y\n#;qge\n\";q\u0000k+h:c:)\n+ (X\nl2intdy\nl;\u0000kge;y\n#;qge\n\";q\u0000k+h:c:)i\n; (8)\nwherege\n\"(#)is the annihilation operator of spin-up (down)\nelectrons and ~Js=1\n2Jsp2SsN. We also de\fne \u0011=5\n~Jc=~Ja. After substituting the inverse transformation of\nEq.(4),\nai;k=X\ni0Tii0\n\u000b;k\u000bi0\nk+X\nj0Tij0\n\f;k(\fj0\n\u0000k)y;\ncj;k=X\ni0Tji0\n\u000b;k\u000bi0\nk+X\nj0Tjj0\n\f;k(\fj0\n\u0000k)y;\ndl;k=X\ni0Tli0\n\u000b;k(\u000bi0\n\u0000k)y+X\nj0Tlj0\n\f;k\fj0\nk; (9)\ninto Hamiltonian (8), we obtain a perturbation\nHamiltonian[12]\nH0=~JaX\nq;kn\n(X\ni0Ti0\n\u000b;k\u000bi0\nkge;y\n#;qge\n\";q\u0000k+h:c:)\n+ [X\nj0Tj0\n\f;k(\fj0\n\u0000k)yge;y\n#;qge\n\";q\u0000k+h:c:]o\n: (10)\nThe coe\u000ecients are de\fned as\nTi0\n\u000b;k=X\ni2intTii0\n\u000b;k+X\nj2int\u0011Tji0\n\u000b;k+X\nl2int(Tli0\n\u000b;\u0000k)\u0003;\nTj0\n\f;k=X\ni2intTij0\n\f;k+X\nj2int\u0011Tjj0\n\f;k+X\nl2int(Tlj0\n\f;\u0000k)\u0003:(11)\nWe then follow the procedures presented in the Ap-\npendix B and obtain the expressions for SSCs in Eq.(5)\nAT=D~X\nkX\ni0(@TnjT=Teq)jTi0\n\u000b;kj2!k\ni0;\nBT=D~X\nkX\nj0(@TnjT=Teq)jTj0;y\n\f;kj2!\u0000k\nj0; (12)\nwhereD,nand T eqare dimensionless coe\u000ecient de\fned\nin Appendix B, magnon distribution function and equi-\nlibrium temperature, respectively. From Eq.(12), we \fnd\nthat the magnon occupation, the dispersion of hybrid\nmodes and the rescaled numbers of magnons accumu-\nlated at the magnetic insultor-NM interface (the square\nof the coe\u000ecients in Eq.(11)) together determine these\nSSCs.\nB. Numerical results in the YIG-GdIG-NM system\nIn Fig.4(a), we project jTi0\n\u000b;kj2andjTj0\n\f;kj2at GdIG sur-\nface to the hybrid spectrum in Fig.2(a) with the ratio\nbetween the interfacial couplings \u0011= 0:14 [36]. Since\nthe amplitudes of \u000b1-type,\u000b2-type and\f2-type modes\nat GdIG surface are sizable while those of \f1-type modes\nare rather small according to Fig.2(b) and (c), the projec-\ntions of\u000b1-type,\u000b2-type and\f2-type modes in Fig.4(a)\nare much more visable than those of \f1-type modes.\nFor similar reason, as shown in Fig.4(b), the projec-\ntions of\f1-type,\f2-type and\u000b2-type magnons are rel-\natively large at the YIG surface. Considering the neg-\nligible magnon occupation of \u000b2-type magnons at low\nFIG. 4. Magnon spectrum weighted by the projection of wave\nfunctions at (a) the GdIG end and (b) the YIG end in YIG\n(7.4 nm)-GdIG (2.5 nm) [001] system with \u0011= 0:14. (c)\nWeighted spectrum on the surface of 9.9-nm-thick YIG with\nthe outmost atomic layer removed. (d) The same as (a) with\n\u0011= 1.\ntemperature, the SSCs in the YIG-GdIG-NM trilayer are\nmainly determined by \u000b1-type and\f2-type modes while\nthose in the GdIG-YIG-NM system are determined by\n\f1-type and\f2-type modes. Notice that the uniform\nmode in Fig.4(b) does not contribute to the magnons\non the YIG end. This is because we use an antiferro-\nmagnetic terminal plane in our calculation. In realistic\nsituation, imperfect interface, di\u000berent crystal orienta-\ntions or the di\u000berent coupling strengths, ~Jaand ~Jd, will\nchange the contribution of uniform mode and cause the\nmeasurable spin pumping signals [18]. To check the e\u000bect\nof imperfect interface on the uniform mode, we inspect\na [001] orientated 8-unit-cell-thick (9.9 nm) YIG layer\nstructure and \fnd a ferromagnetic atomic plane under\nthe outmost antiferromagnetic plane. We thus remove\nthe topmost antiferromagnetic layer and see the acoustic\nmode has nonzero contribution as seen in Fig.4(c). As\n\u0011is a free but crucial parameter, we increase \u0011to 1 in\nFig.4(d) and \fnd the nearly dispersionless \u000b1-type modes\nare enhanced more greatly than the others. This is be-\ncause these low-frequency \u000b1-type modes are dominated\nby the precession of Gd sublattice.\nOne of the most intriguing phenomena of LSSE in\nGdIG is the two sign-changing points (SCP) found in\nGdIG-NM bilayer system [36], where the higher and lower\nones were attributed to the magnetic compensation and\nthe competition between modes of opposite chiralites at\nthe interface. As YIG-GdIG-NM trilayer owns the same\ninterface as GdIG-NM bilayer, the lower SCP is also ex-\npected in YIG-GdIG-NM trilayer.\nBy solving the equation\nIs= \u0001T\f\nme(\rAT\u0000BT) = 0; (13)\none can obtain the sign-changing temperature, which de-\npends on two elements, i.e., the parameter \r= (\u0001T\u000b\nmp+6\n\u0001TFN\npp)=(\u0001T\f\nmp+ \u0001TFN\npp) and the SSC. In general, both\n\rand SSC could be function of YIG thickness: \ris\napproximately 1 when YIG is very thin (just like the\ncase in the GdIG-NM bilayer) and converge to a certain\nvalue when YIG is thick enough; SSC relies on the in-\ncrease of YIG thickness due to the increase of subbands.\nTherefore, we study the relation between SCP and the\nYIG thickness with these two factors. From the discus-\nsion above, we see while the SSCs of hybrid structures\nwith di\u000berent YIG thicknesses can be calculated from\nthe properties of magnons, but the value of \rremains\nunclear. Here, we use a hypothetic function to describe \r\nvarying with the YIG thickness. Considering the smaller\nmagnitude of \u0001T\u000b\nmpcompared to \u0001T\f\nmpaccording to the\ndiscussion at the beginning of this section, we assume\nFIG. 5. (a) SCPs and \r(inset) as function of YIG thickness\nin LSSE in YIG-GdIG-NM structure with di\u000berent values of\n\u0012c. (b) The relation between the ratio of the two SSCs and\nambient temperature for tYIG= 0, 12.4 and 22.3 nm.\n\u0001T\u000b\nmp= 0: (14)\nOn the other hand, \u0001T\f\nmpapproximately equals to \u0001T\u000b\nmp\nif the YIG layer is very thin and signi\fcantly deviatesfrom \u0001T\u000b\nmpwhen YIG becomes thick. We therefore use\nan asymptotic expression\n\u0001T\f\nmp=\u0012c\u0001TFN\nppftanh[(tYIG\u0000dh)=\u0015T] + 1g;(15)\nwhere 2\u0012c\u0001TFN\nppand\u0015Tare the converged value and char-\nacteristic length of \u0001T\f\nmp.dhis the thickness where\n\u0001T\f\nmpreaches a half of converged value. When dhand\n\u0015Tare set to be 10 and 2.5 unit cells respectively, the\nfunction in Eq.(15) at tYIG= 0 nm and 22 nm gives\n\u0001T\f\nmp= 0 and 2\u0012c\u0001TFN\npp, respectively. Following this\nestimation, \rcan be expressed as\n\r=1\n\u0012cftanh[(tYIG\u0000dh)=\u0015T] + 1g+ 1: (16)\nFor the value of \u0012c, we refer to the case in the YIG due\nto the lack of parameters in the GdIG. Ref.[44] showed\nthat the ratio between the magnon-phonon temperature\ndi\u000berence and \u0001TFN\nppis approximately 1 or 0.3 when the\nheat transfer between magnons in YIG and electrons in\nPt is taken into account or not. We thus estimate \u0012c\n= 0.2, 0.4, 0.8. Fig.5(a) shows the SCPs as function of\nYIG thickness, where we \fnd SCPs shift to lower tem-\nperature with the increase of YIG thickness by tens of\nKelvins. To explain this feature, we refer to Eq.(13),\nwhich shows that at SCP, the ratio between the SSCs of\n\fand\u000bmodes equals to \r. The computational results\nfor these ratios as function of temperature with di\u000berent\nYIG thicknesses are shown in Fig.5(b), revealing their\nnegligible dependency on YIG thickness. Therefore, such\nlarge variation of SCPs are mainly caused by the change\nof\r. Note that the SCP for a given YIG thickness and\n\u0012cis read from Fig.5(b) by setting the ratio as the cor-\nresponding value of \rin the inset of Fig.5(a). According\nto a research in the heterostructure consisting of a ten-\nnm-thick garnet \flm and a normal metal layer[46], the\ninterface might introduce an additional anisotropy due\nto the lattice mismatch and Rashba e\u000bect. We estimate\nsuch an anisotropic \feld could cause a correction to the\nfrequency by only a few GHz, which is too small to a\u000bect\nour main results, dominated by the thermal magnons in\nTHz range.\nIV. CONCLUSION AND DISCUSSION\nIn summary, we study the properties of hybrid magnon\nmodes in YIG-GdIG hybrid bilayer structure, which is\nnaturally formed when YIG is grown on the substrate\nGGG. We \fnd that the localized and extended features\nof di\u000berent hybrid modes result in the distinct accumu-\nlations of magnons with opposite polarizations at sur-\nfaces. As magnons transfer spin angular momentum to\nelectrons in an adjacent normal metal on GdIG side by\nmagnon-electron scattering and thus cause nonzero spin\ncurrent in the normal metal, we calculate this spin cur-\nrent in the longitudinal spin Seebeck con\fguration and7\nrecover a sign change in spin Seebeck signal, previously\ndiscovered in GdIG. More interestingly, we \fnd the sign-\nchanging temperature can vary by tens of Kelvins with\nthe increase of YIG thickness.\nV. ACKNOWLEDGMENTS\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant No.11974047) and Fun-\ndamental Research Funds for the Central Universities\n(Grant No. 2018EYT02). K. X. thanks the National Nat-\nural Science Foundation of China (Grants No. 61774017,\nNo. 11734004) and NSAF (Grant No. U1930402).\nAppendix A: Matrix elements in the bosonic BdG\nHamiltonian\nThe matrix elements in Eq.(2) are de\fned as\nAij(k) = (2X\njrimj=raaJaa\nimSa;m\u00002X\njrimj=radJad\nimSd;m\n+ 2X\njrimj=racJac\nimSc;m)\u000eij\n\u00002X\njrijj=raaJaa\nijp\nSa;iSa;jeik\u0001rij;\nCij(k) = (2X\njrimj=rccJcc\nimSc;m\u0000X\njrimj=rcdJcd\nimSd;m\n+ 2X\njrimj=racJac\nimSa;m)\u000eij\n\u00002X\njrijj=rccJcc\nijp\nSc;iSc;jeik\u0001rij;\nDij(k) = (2X\njrimj=rddJdd\nimSd;m\u00002X\njrimj=radJad\nimSa;m\n\u00002X\njrimj=rcdJcd\nimSc;m)\u000eij\n\u00002X\njrijj=rddJdd\nijp\nSd;iSd;jeik\u0001rij;\nBss0\nij(k) =\u00002X\njrijj=rss0Jss0\nijp\nSs;iSs0;jeik\u0001rij: (A1)\nAppendix B: Derivation of SSCs, ATandBT\nIn this appendix, we derive the spin currents generated\nin the LSSE from the interfacial exchange Hamiltonian\nin Eq.(10). Assuming that the momentum conservation\nmight be broken by roughness at the interface, we replace\nq\u0000kby an independent vector q0in Eq.(10). Then we\napply Fermi-Golden rule to calculate transition rates [41,47]\n\u0000\"#=2\u0019\n~~J2\naX\nq;k;q0[X\ni0nk\ni0\u001cq0q\n\"#jTi0\n\u000b;kj2\u000e(Eq\n#\u0000Eq0\n\"\u0000~!k\ni0)\n+X\nj0(n-k\nj0+ 1)\u001cq0q\n\"#jTj0\n\f;kj2\u000e(Eq\n#\u0000Eq0\n\"+~!\u0000k\nj0)];(B1)\n\u0000#\"=2\u0019\n~~J2\naX\nq;k;q0[X\nj0n-k\nj0\u001cqq0\n#\"jTj0;y\n\f;kj2\u000e(Eq0\n\"\u0000Eq\n#\u0000~!\u0000k\nj0)\n+X\ni0(nk\ni0+ 1)\u001cqq0\n#\"jTi0;y\n\u000b;kj2\u000e(Eq0\n\"\u0000Eq\n#+~!k\ni0)]; (B2)\nwhere\u001cq0q\n\"#=fq0\n\"(1\u0000fq\n#) and\u001cqq0\n#\"=fq\n#(1\u0000fq0\n\").n\u0000k\nj0\n(nk\ni0) is magnon distribution function for wave vector \u0000k\n(k) and branch index j0(i0).fq\n\"(fq0\n#) is the distribution\nfunction of spin-up (spin-down) electrons of wave vector\nq(q0). Spin current is de\fned as the di\u000berence of these\ntwo processes\nIs=~(\u0000\"#\u0000\u0000#\"): (B3)\nThen we have the expression of spin current\nIs= 2\u0019~J2\naX\nk;q;q0hX\ni0nk\ni0jTi0\n\u000b;kj2\u000e(Eq\n#\u0000Eq0\n\"\u0000~!k\ni0)\u0001f\n+X\nj0n\u0000k\nj0jTj0\n\f;kj2\u000e(Eq0\n\"\u0000Eq\n#\u0000~!\u0000k\nj0)\u0001fi\n\u0000X(Te);(B4)\nwhere \u0001f=fq0\n\"\u0000fq\n#andX(Te) is the back\row from\nNM to magnetic insulator.\nAs the energy shifts of electrons are small compared to\nfermi energy, one has fq0\n\"\u0000fq\n#\u0019@EfjE=Eq(Eq0\n\"\u0000Eq\n#).\nAt low temperature, f(E)\u0019\u0002(Ef\u0000E), where \u0002( E)\nis the Heaviside step function. Therefore @EfjE=Eq=\n\u0000\u000e(Eq\u0000Ef). When the transverse area is large enough so\nas to make wave vector quasi-continuous, the summation\nsymbols of q;q0in Eq.(B4) can be transformed to integral\nasP\nq(q0)A(Eq(q0)) =R\n\u001a(E)A(E)dE, where\u001a(E) is the\ndensity of states at energy E. The expression for spin\ncurrent is therefore simpli\fed into\nIs= 2\u0019~\u001a(Ef)~J2\naX\nk[X\ni0!k\ni0nk\ni0jTi0\n\u000b;kj2\u001a(Ef\u0000~!k\ni0)\n\u0000X\nj0!\u0000k\nj0n\u0000k\nj0jTj0;y\n\f;kj2\u001a(Ef+~!\u0000k\nj0)]\u0000X(Te):(B5)\nThen the zero-order expression for spin current is\nIs\u0019D~X\nk[X\ni0!k\ni0nk\ni0jTi0\n\u000b;kj2\u0000X\nj0!\u0000k\nj0n\u0000k\nj0jTj0;y\n\f;kj2]\n\u0000X(Te); (B6)\nwhereD= 2\u0019\u001a2(Ef)~J2\na. When the system is in thermal\nequilibrium, no spin current is injected, which leads to\nX(Teq) =D~X\nk[X\ni0!k\ni0nk\ni0(Teq)jTi0\n\u000b;kj2\n\u0000X\nj0!\u0000k\nj0n\u0000k\nj0(Teq)jTj0;y\n\f;kj2]; (B7)8\nwhere T eqis the thermal equilibrium temperature (which\nshould also be the ambient temperature). In near equi-\nlibrium, the temperature of electrons, T e, approximately\nequals to T eq. 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Rev. 120, 745 (1960)." }, { "title": "2103.05365v1.Ultrafast_demagnetization_in_a_ferrimagnet_under_electromagnetic_field_funneling.pdf", "content": " 1 Ultrafast Demagnetization in a Ferrimagnet under \nElectromagnetic Field Funneling \nKshiti Mishraa, Agne Ciuciulkaiteb, Mario Zapata -Herrerac, Paolo Vavassoric,d, Vassilios \nKapaklisb, Theo Rasinga, Alexandre Dmitrieve,*, Alexey Kimela, and Andrei Kirilyu ka,f,* \naRadboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ \nNijmegen, The Netherlands \n bDepartment of Physics and Astronomy, Uppsala University, Box 516, SE -75120 Uppsala, \nSweden \n cCIC nanoGUNE BRTA, E -20018 Donostia -San Sebastian, Spain \ndIKERBASQUE , Basque Foundation for Science, E -48009, Bilbao, Spain \neDepartment of Physics, University of Gothenburg, SE-412 96 G öteborg, Sweden \nfFELIX Laboratory, Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands \n \nEmail: alexd@physics.gu.se , andrei.kirilyuk@ru.nl \n \nKEYWORDS : Plasmon nanoantennas, Ultrafast Magnetization Dynamics, Rare -Earth \nTransition Metal Alloys \n \n 2 ABSTRACT \nThe quest to improve density, speed and energy efficiency of magnetic memory storage has led to \nexploration of new ways of optically manipulating magnetism at the ultrafast time scale, in \nparticular in ferrimagnetic alloys. While all-optical magnetization switching is well -established on \nthe femtosecond timescale , lateral nanoscale confinement and thus potential significant reduction \nof the size of the magnetic element remain s an outstanding challenge. Here we employ resonant \nelectromagnetic energy -funnelin g plasmon nanoantennas to influence the demagnetization \ndynamics of a ferrimagnetic TbCo alloy thin film. We demonstrate how Ag nanoring -shaped \nantennas under resonant optical femtosecond pumping reduce the overall magneto -optical \nresponse due to demagneti zation in the underlying films up to three times compared to non -\nresonant illumination. We attribute such substantial reduction to the nanoscale confinement of the \ndemagnetization process. This is qualitatively supported by the electromagnetic simulations that \nstrongly evidence the optical energy -funneling to the nanoscale from the nanoantennas into the \nferrimagnetic film. This is the first and defining step for reaching deterministic ultrafast all-optical \nmagnetization switching at the nanoscale in such systems , opening a route to develop nanoscale \nultrafast magneto -optics. \nMAIN TEXT \nOne of the most demanding current technological challenges is the storage and processing of \nthe exponentially increasing amount of information. The quest to improve the den sity, speed and \nenergy efficiency of magnetic memory storage has led to the exploration of new ways of \nmanipulating magnetism at the ultrafast timescale. A particularly promising possibility was \nopened up by the discovery of All -Optical Switching (AOS) of magnetization [1] wherein the 3 magnetization of the Rare Earth - Transition Metal (RE -TM) alloy GdFeCo was reversed using \nultrashort l aser pulses in the absence of an external magnetic field. Following this first \nobservation, AOS has also been found to occur in a variety of materials [2]–[5]. While the \nperformance of AOS in terms of writing speed [6] and energy efficiency [7] compares favorably \nto other techniques, in terms of the smalle st attainable bit size AOS is limited to the micrometer \nscale owing to the diffraction limit of light. Attempts to downscal e the bit size reached down to \na few hundred nanometres by tighter focusing of the laser beam using a microscope objective [8] \nand by nanopatterning [9]–[11]. An important step towards nanoscale bit size was achieved by \nemploying gold two -wire nanoantennas on a TbFeCo alloy thin film , yielding an AOS -switched \nspot size of 50 nm by exploiting localized plasmons [12]. This suggests that the use of \nnanoplasmon optics could make AOS technologically via ble by making the attainable bit size \ncomparable to that, for example, of heat -assisted magnetic recording [13]. The so far revealed \nmechanism behind AOS implies that AOS proceeds via fast and efficient demagnetization [6], \n[14]–[16]. Therefore, t he first crucial step towards nanoplasmonic AOS is to follow the temporal \nevolution of the magnetization in response to plasmon nanoantenna -assisted resonant laser \nexcitation. \nHere we uncover the demagnetization dynamics in a ferrimagnetic TbCo alloy nano film assisted \nby nanoring -shaped Ag plasmon antennas. Plasmon nanorings are a widely studied system, \ndisplaying two so -called bonding and anti -bonding optical resonances: the anti-bonding (high \nenergy) dipolar mode concentrate s the electromagnetic near -field at the rims of the nanoring \nstructure [17], whereas the bonding (low energy) mode does so mostly in the center of the \nnanoring [17], [18] . The latter has a decisive advantage in the present study as leveraging on the \nchoice between the two modes , since it combines lower energy with smaller mode volume . We 4 compare the dynamics under resonant vs. off -resonant ultrafast laser pumping. In general, we \nfind qualitatively similar sub -picosecond demagnetization for both resonant and off -resonant \npumping. However, the degree of demagnetization is substantially small er for resonant pumping. \nWe attribute this to the strong electromagnetic near -field confinement and the scattered field \nfunneling by the nanoring antenna bonding mode. While the illuminating laser fluence is \npredominantly channeled to the extremely small a reas of the nanorings ’ near-field, we probe an \naveraged response largely comprising the signal from the much large r sample areas outside the \nnanorings , that receive much less fluence and consequently experience a smaller \ndemagnetization. Thus, while an all -optical pump -probe scheme cannot directly yield a complete \npicture of the nanoscale magnetization dynamics in the near -field, the effects of resonant \nexcitation are clearly detectable. Importantly, through electromagnetic simulations we get further \neviden ce of the funneling of laser fluence right into the centre of the nanoring antennas , \nsupporting the experimental observations . \n \nFigure 1. (a) SEM overview of the Ag nanoring antennas macroscopic assembly on a Tb26Co74 \nfilm; (b) A schematic of the nanoring s + ferrimagnetic film system ( indicating the thicknesses / \n 5 dimensions of the elements) and the experimental scheme of exciting the system using an ultrashort \nlaser pulse (antenna near -fields are shown schematically in red ). \nResults and Discussion: \nPlasmon nanoring antennas on ferrimagnetic film \nFigure 1a shows the scanning electron microscope (SEM ) image of the TbCo nanothin film with \nAg nanoring antennas directly on top of the few-nm thick sapphire (Al 2O3) capping layer , taking \nup about 20% of the surface. Nanoantennas are arranged with a short -range order , with the \naverage spacing between the nanorings ensuring the absence of near -field electromagnetic \ncoupling . That is, the spectral response of the entire surface is rep resentative of a single \nnanoantenna, with the correction of spectral inhomogeneous broadening due to nanoantennas \nsize variation s. This is typical for short -range -ordered arrays, produced with hole -mask colloidal \nlithography, employed here [19]. Laser illumination is done at near-normal incidence (5 ° off \nnormal), and the film structure underneath the nanoantennas is detailed in Figure 1b. \nThe optical transmission of the nanoantennas + nanofilm system shows two pronounced \nresonances (Figure 2a), corresponding to the antibondi ng (close to 480 nm) and the bonding \n(close to 920 nm) localized plasmon modes. These are well -studied dipolar modes of the \nnanoring antennas [17], [18] . Figure 2b shows the calculated charge distribution in the nanoring \nantennas when each mode is excited. For the bonding (symmetric) mode, the charge distribution \nhas the same sign on the inner and the outer edges of the nanoring, whereas for the antibondi ng \n(antisymmetric) mode the charge distribution is the opposite for the inner and the outer edges. \nStatic magnetic characterization (Figure 2c) reveals a square hysteresis loop indicating \nperpendicular magnetic anisotropy, similar to the hysteresis loops o btained for bare TbCo films 6 [20]. The coercive field increase s from 0 .2 T for the bare film to 0.26 T for the film with \nnanoantennas, possibly due to perturbations of the continuous film caused by the antennas’ \nnanofabrication , introducing domain wall pinning sites. \n \nFigure 2. (a) Experimentally measured normalized optical extinction spectra showing peaks \ncorresponding to the bonding and antibonding modes; (b) Simulated normalized surface charge \ndistributions corresponding to the two main dipolar plasmon modes of nanoring antennas; (c) \nMagnetic hysteresis loops for the Tb 26Co74 film with nanoring antennas (black) and pristine Tb 26Co74 \nfilm (red), showing square hystereses (the solid lines are a guide to the eye). \nExperimental Pump -Probe Dynamics \nThe Tb 26Co74 nanofilm has previously been reported to show multi -shot helicity -dependent AOS \n[21]. Linearly polarized pump pulse trains or single shots of any polarization induce \ndemagnetization in the film. That is, using a linearly polarized pump beam, we expect to only see \nthe effects of pump -induced heating. The symmetric shape of the nanoring antennas rules out \nany dependence on the direc tion of linear polarization of the pump. Thus, w e compare the time \nresolved dynamics of the system for resonant excitation of the dipolar bonding mode (using a \n 7 pump wavelength of 950 nm ), and off -resonant pumping (for a wavelength of 650 nm) with a \nlinearl y polarized pump. \n \n \nFigure 3. Pump -induced demagnetization (top panel) and change in transmission (bottom panels) \nmeasured for a range of incident fluences (colors shades code for different fluences) for an off -\nresonant pump wavelength of 650 nm (left pan els, black/grey data points) and a resonant pump \nwavelength of 950 nm (right panels, red/dark -red points). The solid lines in the case of the \nmagnetization traces are the bi -exponential fitting curves, whereas the solid lines in case of \ntransmission dynami cs are a guide to the eye. \n 8 \nThe nanoantennas + nanofilm system shows sub -picosecond demagnetization followed by a \nrecovery, similar to the demagnetization behaviour previously reported for Tb xCo100− x alloys \n[22]. The nanoantennas do not seem to qualitatively affect the demagnetization process, as \nevidenced from the similar behaviour for on - and off -resonance excitation (Figure 3). However, \nthe main striking difference between the two cases is the degree of demagneti zation. \nCounterintuitively, l arger demagnetization is observed for off -resonance excitation compared to \nresonant excitation. In Figure 3 the dynamics are plotted for the two cases on the same y -axis to \nvisualize the difference in the degree of demagnetizat ion for selected laser fluences. The same \ntrend is observed for pump -induced changes in transmission. \nThe difference between the responses for the two cases is even more apparent in Figure 4 where \nthe degree of demagnetization (top panel) and the maximum c hange in transmission (bottom \npanel) are plotted as a function of incident pump fluence. A linear relationship is observed for \nboth quantities for both pump wavelengths, as expected for heat -driven dynamics. However, the \nslope of degree of demagnetization as a function of fluence for the case of off -resonant pumping \n(0.077) is nearly thrice the slope for resonant pumping (0.028). Similarly, a roughly threefold \ndecrease at resonance is observed for pump -induced transmission changes, where the slope for \noff-resonant pumping is 0.05, whereas that for resonant pumping is 0.015. \n 9 \nFigure 4. Degree of demagnetization (top panel) and maximum change in transmission (bottom \npanel) plotted as a function of incident fluence for resonant (red squares) and off -resonant pumping \n(black squares). The dashed lines are the corresponding linear fits to the data points. \nThe difference , at first sight surprising, in a weaker response for resonant than off-resonant \nexcitation of the system can be rationalized by a funnel ing of th e incident pump fluence to the \ncentre of the nanorings upon exciting the dipolar bonding mode. Indeed, t he strongly reduced \npump -induced changes observed for resonant pumping can then be explained by considering the \nsize of the probe/illumination spot rela tive to the size of the nanoantennas. The signal for the \nµm-sized probe i s obtained from an area covering thousands of nanoantennas as well as from the \nferrimagnetic film outside of the nanorings ’ near-field. Crucially, nanoantenna -resonance -\nenhanced demagnetization is expected to arise only for the TbCo film in the nanorings ’ focus, \nwhere the incoming electromagn etic field of light is funneled and enhanced. However, in our \n 10 measurements the predomi nant contribution to the magnetic signal comes from the TbCo outside \nthe nanorings ’ near-field, receiving considerably less fluence and thus producing less signal . \nKnowing the surface coverage of the nanoring antennas on TbCo film allows estimation of the \nratio of surface area of the TbCo within the nanoring cavity to the surface area outside of the \nnanorings, which amounts to 1:44. The predominant contribution to the magnetic signal is thus \ncoming from the TbCo film outside of the nanoantennas near -field. These regions receive \nconsiderably less fluence under resonan t illumination than under off -resonance illumination due \nto the efficient in -coupling of the incident laser fluence by the nanoantennas. Note that the \nnanoantenna optical cross -section is substa ntially larger than their geometrical size, as it is \ncommon for plasmon nanostructures. Though the demagnetization dynamics of the TbCo film in \nthe interior of the nanorings (inner opening of 20 nm) cannot be resolved using an optical probe \nin our experime nts, the signal from areas outside the nanorings ’ near-field gives an indirect \nevidence of the electromagnetic field funneling by the plasmon nanoantennas at resonance. \nTo account for the difference in the system’s response at the two different pumping wav elengths, \nwe further take into account the wavelength -dependent absorption of the ferrimagnetic material. \nOur earlier study [21] provides the optical constants for various compositions of Tb xCo100− x \namorphous alloy films in the wavelength range 400 - 1600 nm. The values of the optical \nconstants as a function of wavelength are very similar f or the films with Tb content between 24 - \n29 %. As the composition of the films here is in this range, we use these values to extract the \nabsorption at 650 nm and 950 nm (not shown) and find that at a given fluence, the absorption at \nboth wavelengths is id entical within 1% uncertainty. That is, the observation of three times \nsmaller pump -induced changes at 950 nm compared to 650 nm strongly supports the hypothesis \nof plasmon -mediated funneling of the incident fluence. 11 The effects of plasmon resonance on dem agnetization have been previously investigated on a \nsystem of gold nanorods on a ferromagnetic permalloy film [23]. This study found enhanced \ndemagnetization of the permalloy at resonance compared to off -resonance excitation, which is \nopposite to our observations. However, an important difference between this system and the \nsystem studied here is the geometry of the plasmon element. Plasmon nanoring antennas are \nadvanced structures in terms of the resonance modes that result from the coupling of inner and \nouter wall s of the nanoring [18][24]. Specifically, the electromagnetic near -field profile of the \nnanoring antenna features a tightly confined (20 nm in size) and enhanced spot right in the \nmiddle of the nanoring, creating the field funneling effect mentioned above. In addition, \nlocalized near -fields at the nanoring opening makes this nanoantenna a very prominent candidate \nfor the highly sensitive plasmonic bio - / chemo -detector with an open and easily acc essible \nelectromagnetic cavity [25]. \nElectromagnetic simulations of the nanoantennas + ferrimagnetic nanofilm system 12 \nFigure 5. (Top) 3D intensit y maps showing the scattered electromagnetic field in the x -z plane of \nthe nanoring + ferrimagnetic film for off -resonance (left) and on -resonance (right) illumination. The \nwhite arrows represent electric field flux lines. (Middle) Cross -sectional plot of the near -field (E/E 0) \nwith dotted differently colored lines marking the distances from the Al 2O3 (capping layer) -TbCo \ninterface. (Bottom) Linear scans of the scattered field along the colored dashed lines of the middle \npanel. \n \n 13 To support the idea of the field funneling into the TbCo film by nanoring antennas, we \nperform ed electromagnetic simulations on nanoantennas + ferrimagnetic film system (Figure 5, \noff-resonance and on -resonance illumination panels are grouped to the left and right sides, \nrespectively). We visualize the intensity of the scattered field by 3D plots off - and on -resonance \n(Fig. 5 top), highlightin g field funneling by the field flux lines. Cross -sectional intensity maps \nalso show this effect (Fig. 5, mid -panels), along with a substantial field enhancement in the \nnanoring center for the on -resonance illumination. For the off -resonance case, a strong scattering \nat the corners is observed, arising from the conventional tip -effect, with the scattered field mostly \ndiffused away. \nThe strong contrast between off - and on -resonance cases can be appreciated in the magnitude \nof the scattered field, plotted as a function of position in the Tb xCo100− x film for an alumina -\nTbCo interface (Fig. 5, bottom panels). At resonance the nanoring antenna concentrates the \nscattered field within the nanoring center and directs it towards the underlying substrate of TbCo \njust below the central opening of the ring. A more defined deflection of the flux lines near the \nnanoring rims and a strong concentration (about twice the scattering field intensity compared to \nthe off -resonance illumination) at the center of the nanostructure is detected. The funneling effect \nsaturates between 15 nm and 20 nm into the depth of TbCo film, making the thickness of the \nlatter in the present study (20 nm) an exemplary case. \nConclusion \nWe find that plasmon nanoring antennas can efficiently funnel t he illuminating electromagnetic \nfield into the nanoscopic portions of the ferrimagnetic thin films and induce marked changes in \nthe demagnetization of this system. Upon the resonant excitation, the nanorings are able to 14 couple -in a substantial portion of t he incident pump -fluence and concentrate it into the 20 \nnanometers focal spot of the nanoantenna. This results in smaller pump -induced changes in \nmagnetization and transmission in the areas outside the nanoring antennas that constitute the \nmajor part of th e studied surfaces that are probed in the macroscale pump -probe experiments. \nElectromagnetic simulations of the scattered field show qualitative agreement with this picture. \nOverall, studying the nanoscale confinement of demagnetization processes under the influence of \nplasmon nanoantennas fundamentally require experiments with higher real -space resolving \npower. These are extremely experimentally demanding and are often requir ing large -scale \nexperimental facilities (such as the measurements of the results of the fs -pulsed illumination in \nferrimagnets with photoelectron emission microscopy, PEEM [10] or with X-Ray Holographic \nImaging combined with X -Ray Magnetic Circular Dichroi sm [12]). Here we essentially \ncircumvent the need for such demanding experiments by rationalizing the fundamental role of \nnanoantennas in focusing the pulsed -light illumination to the nanoscale. A further step in this \ndirection is to reduce the ferrimagnet ic film to nanoscopic elements position ed in focus of \nplasmon nanoantenna s while strictly maintaining their magneto -optical properties and required \nmagnetic anisotropy [26]. Such an experiment would provide, conversely, nanoantenna -\nenhanced demagnetization. \nIn the present case the proposed incident fluence funneling to the nanoscopic regions of the \nferrimagnet marks a promising path towards ultrafast magnetic bit min iaturization even for the \nnanofilm systems, broadly currently employed. We envision such a path in practice leading to \nfully functional ultrafast nanoscale magne tic memory architectures. \n 15 Methods \nThe sample investigated consists of a Tb 26Co74 amorphous alloy thin film grown on a glass \nsubstrate, covered with an alumina capping layer on top of which plasmonic silver nanorings \nhave been fabricated. \nThe 20 nm thick TbCo film was prepared by DC magnetron sputtering from elemental Tb and \nCo targets. To ensure uniform film deposition at room temperature, a rotating sample holder was \nused. The synthesis was performed under ultrahigh vacuum, with a base pressure of 10−10 Torr \nand an Ar+ sputtering gas pressure of 2 -3 mTorr. The TbCo film was covered with a 4nm Al 2O3 \ncapping layer. More details of the synthesis can be found in [20] and [21]. Silver nanorings of \ninner diameter 20 nm, outer diameter 70 nm and height 10 nm were fabricated on the capped \nTbCo film by hole -mask colloidal lithography, HCL [19]. Fig.1(b) shows the structural de tails of \nthe sample. \nThe fabricated sample was imaged using a Scanning Electron Microscope (SEM). To \ncharacterize the resonance modes for the sample, the optical transmission spectrum was \nmeasured in the wavelength range 350 -1050 nm. Static magneto -optical characterization was \ndone using a polar Faraday geometry at 800 nm. \nMagnetization dynamics were measured using a n all-optical two -colour pump -probe setup. \nThe probe was derived from a Ti:sapphire amplified laser system with a 1 kHz repetition rate, a \ncentral wavelength of 800 nm, and a pulse width of 100 fs at the sample position, focused to a \nspot size of 460 µm. The pump pulse was derived from the same laser by tuning the wavelength \nthrough optical parametric amplification. The off -resonance pump wavelen gth was chosen to be \n650 nm, and the spot size at the sample was 510 µm. The resonant pump wavelength was chosen 16 as 950 nm, and the spot size at the sample was 590 µm. The setup was built to have near normal \nincidence of the pump (~ 5◦ to the sample normal ). Both pump and probe beam were horizontally \npolarized i.e., in the plane of incidence. The probe beam was separated from the pump using the \nappropriate colour filters and was detected as a function of the pump -probe time delay using a \npair of balanced Si photodiodes. A static magnetic field higher than the sample coercive field \nwas applied normal to the sample throughout the course of each measurement to re -initialize the \nsaturated magnetic state of the sample before each subsequent pump pulse. \nThree -dime nsional electrodynamic calculations of the optical response and the surface charge \ndensity maps were performed by solving the Maxwell equations via the finite element method \n(FEM) implemented in the commercial COMSOL Multiphysics software [2 7]. In order to \nreproduce the experimental structures, we modeled a silver ring on a three -layered structure \nusing an analytical background field resulting from solving Fresnel equations for an incoming \nlight source on a multila yer system. After the interaction with ligh t, the scattered field due to the \nsilver ring was calculated. \nACKNOWLEDGMENT \nThe authors thank Dr. Oleg Lysenko for his contribution to sample s nano fabrication. K.M, \nA.V.K and A.K thank Dr. Sergey Semin and Chris Berkhout for technical support . This work \nwas supported by the project FEMTOTERABYTE funded from European Union’s Horizon 2020 \nresearch and innovation program under grant agreement no. 737093. PV acknowledge s support \nfrom the Spanish Ministry of Science and Innovation under the Maria de Maeztu Unit s of \nExcellence Programme (MDM -2016 -0618), and the project RTI2018 -094881 -B-I00 \n(MICINN/FEDER). V.K. acknowledges support from the Swedish Research Council (Project \nNo. 2019 -03581). 17 REFERENCES \n[1] C. D. Stanciu et al. , ‘All -optical magnetic recording with circularly polarized light’, Phys. \nRev. Lett. , vol. 99, no. 4, pp. 1 –4, 2007. \n[2] S. Mangin et al. , ‘Engineered materials for all -optical helicity -dependent magnetic \nswitching’, Nat. Mater. , vol. 13, no. 3, pp. 286 –292, 2014. \n[3] C. H. Lambert et al. , ‘All -optical control of ferromagnetic thin films and nanostructures’, \nScience (80 -. )., vol. 345, no. 6202, pp. 1337 –1340, 2014. \n[4] C. Banerjee et al. , ‘Single pulse all -optical toggle switching of magnetization without \ngadolinium in the ferrimagnet Mn 2RuxGa’, Nat. Commun. , vol. 11, no. 1, pp. 1 –6, 2020. \n[5] L. Avilés -Félix et al. , ‘Single -shot all -optical switching of magnetization in Tb/Co \nmultilayer -based electrodes’, Sci. Rep. , vol. 10, no. 1, pp. 1 –8, 2020. \n[6] I. Radu et al. , ‘Transient ferromagnetic -like state mediating ultrafast reversal of \nantiferromagnetically coupled spi ns’, Nature , 2011. \n[7] A. R. Khorsand et al. , ‘Role of magnetic circular dichroism in all -optical magnetic \nrecording’, Phys. Rev. Lett. , 2012. \n[8] M. Finazzi et al. , ‘Laser -induced magnetic nanostructures with tunable topological \nproperties’, Phys. Rev. Le tt., 2013. \n[9] L. Le Guyader et al. , ‘Demonstration of laser induced magnetization reversal in GdFeCo \nnanostructures’, Appl. Phys. Lett. , 2012. \n[10] L. Le Guyader et al. , ‘Nanoscale sub -100 picosecond all -optical magnetization switching 18 in GdFeCo microstructures’, Nat. Commun. , vol. 6, no. May 2014, 2015. \n[11] A. El -Ghazaly et al. , ‘Ultrafast magnetization switching in nanoscale magnetic dots’, Appl. \nPhys. Lett. , 2019. \n[12] T. M. Liu et al. , ‘Nanoscale Confinement of All -Optical Magnetic Switching in TbFeCo - \nCompetition with Nanoscale Heterogeneity’, Nano Lett. , vol. 15, no. 10, pp. 6862 –6868, \n2015. \n[13] W. A. Challener et al. , ‘Heat -assisted magnetic recording by a near -field transducer with \nefficient optical energy transfer’, Nat. Photonics , 2009. \n[14] J. Gorchon et al. , ‘Role of electron and phonon temperatures in the helicity -independent all -\noptical switching of GdFeCo’, Phys. Rev. B , 2016. \n[15] C. S. Davies et al. , ‘Pathways for Single -Shot All -Optical Switching of Magnetization in \nFerrimagnets’, Phys. Rev. Appl. , 2020. \n[16] C. S. Davies et al. , ‘ Exchange -driven all -optical magnetic switching in compensated 3 d \nferrimagnets ’, Phys. Rev. Res. , vol. 2, no. 3, p. 32 044, 2020. \n[17] J. Ye, P. Van Dorpe, L. Lagae, G. Maes, and G. Borghs, ‘Observation of plasmonic dipolar \nanti-bonding mode in silver nanoring structures’, Nanotechnology , vol. 20, no. 46, 2009. \n[18] J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, \n‘Optical Properties of Gold Nanorings’, Phys. Rev. Lett. , vol. 90, no. 5, p. 4, 2003. \n[19] H. Fredriksson et al. , ‘Hole -mask colloidal lithography’, Adv. Mater. , vol. 19, no. 23, pp. \n4297 –4302, 2007. 19 [20] A. Ciuciulkaite, ‘The interaction of light and magnetism in the Tb xCo100-x system’, Uppsala \nUniversity, 2019. \n[21] A. Ciuciulkaite et al. , ‘Magnetic and all -optical switching properties of amorphous \nTbxCo100-x alloys’, Phys. Rev. Mater. , vol. 4, no. 10, pp. 1 –11, 2020. \n[22] S. Alebrand et al. , ‘Light -induced magnetization reversal of high -anisotropy TbCo alloy \nfilms’, Appl. Phys. Lett. , vol. 101, no. 16, 2012. \n[23] H. Xu, G. Hajisalem, G. M. Steeves, R. Gordon, and B. C. Choi, ‘Nanorod surface plasmon \nenhancement of las er-induced ultrafast demagnetization’, Sci. Rep. , 2015. \n[24] T. Chung, S. Y. Lee, E. Y. Song, H. Chun, and B. Lee, ‘Plasmonic nanostructures for nano -\nscale bio -sensing’, Sensors , 2011. \n[25] A. Dmitriev, Nanoplasmonic Sensors . 2012. \n[26] R. M. Rowan -Robinson et al. , ‘Direction -sensitive Magnetophotonic Metasurfaces’, arXiv \npreprint arXiv:2005.14478v2 (2020). \n[27] COMSOL AB, Stockholm, Sweden. COMSOL Multiphysics R. Version 5.2. URL: \nhttp://www.comso l.com " }, { "title": "2009.11683v1.Magnetic_anisotropy_and_exchange_paths_for_octa__and_tetrahedrally_coordinated_Mn___2____ions_in_the_honeycomb_multiferroic_Mn__2_Mo__3_O__8_.pdf", "content": "arXiv:2009.11683v1 [cond-mat.str-el] 24 Sep 2020Magnetic anisotropy and exchange paths for octa- and tetrah edrally coordinated\nMn2+ions in the honeycomb multiferroic Mn 2Mo3O8\nD. Szaller,1K. Sz´ asz,2S. Bord´ acs,2,3J. Viirok,4T. R˜ o˜ om,4U. Nagel,4A. Shuvaev,1L. Weymann,1\nA. Pimenov,1A. A. Tsirlin,5A. Jesche,5L. Prodan,6,7V. Tsurkan,6,7and I. K´ ezsm´ arki7\n1Institute of Solid State Physics, Vienna University of Tech nology, 1040 Vienna, Austria\n2Department of Physics, Budapest University of Technology a nd Economics, 1111 Budapest, Hungary\n3Hungarian Academy of Sciences, Premium Postdoctor Program , 1051 Budapest, Hungary\n4National Institute of Chemical Physics and Biophysics, Aka deemia tee 23, 12618 Tallinn, Estonia\n5Experimental Physics VI, Center for Electronic Correlatio ns and Magnetism,\nUniversity of Augsburg, 86159 Augsburg, Germany\n6Institute of Applied Physics, MD-2028 Chi¸ sin˘ au, Republi c of Moldova\n7Experimental Physics V, Center for Electronic Correlation s and Magnetism,\nUniversity of Augsburg, 86159 Augsburg, Germany\n(Dated: September 25, 2020)\nWeinvestigatedthestaticanddynamicmagneticproperties ofthepolarferrimagnet Mn 2Mo3O8in\nthree magnetically ordered phases via magnetization, magn etic torque, and THz absorption spec-\ntroscopy measurements. The observed magnetic field depende nce of the spin-wave resonances, in-\ncluding Brillouin zone-center and zone-boundary excitati ons, magnetization, and torque, are well\ndescribed by an extended two-sublattice antiferromagneti c classical mean-field model. In this or-\nbitally quenched system, the competing weak easy-plane and easy-axis single-ion anisotropies of the\ntwo crystallographic sites are determined from the model an d assigned to the tetra- and octahedral\nsites, respectively, by ab initio calculations.\nI. INTRODUCTION\nStatic magneto-electric (ME) coupling, namely the po-\ntential to electrically manipulate magnetic states and\nmagnetically control electric polarization, has opened\na new path for data storage1–10. At finite frequen-\ncies, the same cross-coupling leads to fascinating op-\ntical phenomena11, such as one-way transparency12–20,\nreciprocal13and non-reciprocal21,22optical rotation.\nSincetheseMEphenomenaonlyemergeinsystemssimul-\ntaneously lacking the time-reversal and spatial inversion\nsymmetries, they have been realized in magnetically or-\ndered phases with broken inversion symmetry12–15,17–22,\nand in the paramagnetic phase of non-centrosymmetric\ncompounds when a magnetic field was applied23–26.\nIn most compounds27–30, the ME coupling arises from\nthe spin-orbit interaction, thus, the strength of the ME\ncoupling is strongly limited by its relativistic origin.\nHowever, in magnetically ordered non-centrosymmetric\ncrystals, the symmetric exchange-striction31provides an\nalternative mechanism to generate ME coupling, which\nexists even for spin-only ions with half-filled d-shell. De-\npending on the relative orientation of magnetic moments\non crystallographic sites connected by an exchange path,\nthe magnetic order can further be stabilized by distort-\ning the bond and by that modifying the strength of the\nexchange coupling. This distortion, driven by the mag-\nnetic order, also produces electric polarization in non-\ncentrosymmetric crystals, realizing the ME coupling. To\ncreate a ME monodomain state with magnetically in-\nduced macroscopic polarization, either an electric field\nis applied32, or pyroelectric polarization is necessary,\nwhen cooling the system below the magnetic ordering\ntemperature33. The latter condition is literally fulfilled\nFIG. 1.Crystal and magnetic structure of Mn 2Mo3O8.\nRed octahedra and purple tetrahedra show the oxygen coor-\ndination of Mn atoms, while green spheres represent the Mo\natoms. The zero-field ferrimagnetic spin configuration is in di-\ncated by orange and blue vectors, and relevant antiferromag -\nnetic exchange paths are shown by grey arrows.\nin type-I multiferroics34, where the onset of magnetic or-\nder takes place within a pre-existing polar state.\nThe members of the polar hexagonal (space group\nP63mc)M2Mo3O8crystal family, where Mstands for\ntransition metal ions, are ideal candidates for strong,\nexchange-striction based ME effects33. The honeycomb\nab-plane layers of Mmagnetic moments are separated by\nthe Mo4+layers (see Fig. 1), which are non-magnetic in\nthese compounds due to the formation of Mo 3O13trimer\nsinglets35. Half of the Mions are in octahedral and\nhalf in tetrahedral oxygen environment, as presented in\nFig.1. Due to a delicate balance of competing superex-\nchange paths the magnetic ordering of octa- and tetrahe-2\ndrally coordinated magnetic moments in different layers\ncan lead to various types of spin structures in this mate-\nrial class, such as collinear easy-axis antiferromagnetic36,\nferrimagnetic37and spin-flopped plane33ordered states.\nThese states can also be transformed into each other by\nan external magnetic field33,38,39.\nAt low temperatures, Fe 2Mo3O8presents the largest\nmagnetically switchable electric polarization among\nsingle-phase multiferroic crystals33,38. The ME suscep-\ntibility can be tuned by diluting Fe by Mn39or non-\nmagneticZn38. The spin excitationsofthese compounds,\nthat are classified as magnons and electromagnons40,\nshow one-way transparency in the paramagnetic phase23\nand non-reciprocal optical rotation22. However, the mi-\ncroscopic description of the sequence of magnetic phases\nand the spin-wave resonances in this material family is\nstill an open problem.\nMn2Mo3O8offers the perfect starting point to under-\nstand the magnetic properties of the M2Mo3O8com-\npounds. The half-filled 3 dshells of Mn2+ions with S=\n5/2spinand L= 0orbitalmomentallowmagneticsingle-\nion anisotropies only via higher-order interactions41.\nThus, the resulting magnetic single-ion anisotropies of\nthe tetra- and octahedral sites are expected to be weak.\nMn2Mo3O8has an easy-axis type ferrimagnetic ground\nstate below TN= 41 K where the spins of the octa-\nand tetrahedral sites are aligned antiparallel along the\nhexagonal axis of the crystal37, as shown in Fig. 1. Al-\nthoughthemagneticmomentsofthetwocrystallographic\nsites compensate each other when approaching the low-\nest temperature, the spontaneous magnetization is finite\nat higher temperatures indicating the different temper-\nature dependences of the ordered moments at the two\nsites37,42. At low temperatures, the magnetization re-\nmains zero in magnetic fields along the hexagonal axis\nup toµ0HC1= 4 T, above which it starts to smoothly\nincrease39,showinganevidentsignofaspin-reorientation\ntransition.\nIn order to gain a deeper insight into the micro-\nscopic mechanisms governing the magnetic behaviour of\nMn2Mo3O8, we followed the magnetic field dependence\nof spin-wave resonances through three magnetic phases.\nThe field dependence of the resonance frequencies and\nthe bulk magnetization were successfully described by a\nsimple microscopic model, which can serve as a starting\npoint tounderstandothersystems ofthe M2Mo3O8crys-\ntal family. The key results of the mean-field analysis are\nalso supported by our first principle calculations.\nII. METHODS\nMn2Mo3O8single crystals were grown by the chemi-\ncal transport reaction method using anhydrous TeC l4as\na transport agent. Plate-like crystals with the dimen-\nsion of about 2-3mm in the abplane and 0.5-1mm along\nthecaxis were obtained after one month transport at\n1000◦Cwithatemperaturedifferenceof50◦C. Themag-netization measurements were performed using a Squid\nmagnetometer (MPMS-5, Quantum Design) in fields up\ntoµ0H= 5 T and vibrating sample magnetometer in\nfields up to µ0H= 14 T using Physical Properties Mea-\nsurement System (PPMS, Quantum Design). The torque\nmagnetometry was also performed in a PPMS with mag-\nnetic fields of up to 9T.\nOptical transmission experiments between 60 and\n180GHz were carried out using quasi-optical terahertz\nspectroscopy43. This technique utilizes linearly polarized\nmonochromaticradiationprovidedby backward-waveos-\ncillators. Liquid He-cooled bolometer was used as detec-\ntor of the transmitted radiation. The sample was in a\nHe-cooled cryostat and the experiments were performed\natT= 3 K temperature. The magnetic field was paral-\nlel to the propagation direction of the light beam (Fara-\nday configuration). In order to increase the sensitivity\nof the absorption measurement in the frequency range\nwhere the sample dimensions ( ∼1 mm) are less than the\nwavelength of light ( λ≈3 mm), the experiments were\nperformed in the fixed frequency mode while sweeping\nthe magnetic field in the µ0H= 0−7 T range.\nFourier-transform spectroscopy was used to study the\nopticalabsorptionbetween120and6000GHzwith8GHz\nresolution. The magnetic field dependence of the spec-\ntra in magnetic fields up to µ0H= 17 T was investi-\ngated using the TeslaFIR setup of the National Institute\nof Chemical Physics and Biophysics in Tallinn.17,44This\nsetup consists of a Martin-Puplett interferometer, a mer-\ncury arc lamp as a light source and a Si bolometer cooled\ndown to 300mK as a light intensity detector. The trans-\nmission spectra at T= 3 K were measured in both the\nFaraday and Voigt configuration, i.e. in magnetic fields\nparallel and perpendicular to the direction of light prop-\nagation, respectively, using linearly polarized incoming\nbeam and unpolarized detection.\nMagnetic anisotropy was estimated by density-\nfunctionaltheory(DFT) band-structurecalculationsper-\nformed in the VASP package45,46for the experimental\ncrystal structure at 1.7K47using generalized gradient\napproximation for the exchange-correlation potential48.\nCorrelation effects in the Mn 3 dshell were included\non the mean-field via the DFT+ Ucorrection with the\nCoulomb repulsion parameter U= 5eV and Hund’s ex-\nchangeJ= 1eV49. Single-ion anisotropy for individual\nMn sites was calculated from total energies of orthogonal\nspin configurations as described in Ref. 50. Theg-factor\nvalues were estimated from calculated orbital moments.\nIII. RESULTS\nThe magnetic field dependence of the magnetization of\nMn2Mo3O8atT= 2 K is presented in Fig. 2. While for\nmagnetic fields perpendicular to the hexagonal caxis the\nmagnetization is linear up to 14 T, for H/bardblcthe magne-\ntization remainscloseto zeroup to the spin-reorientation\ntransition starting at µ0HC1= 4 T. In the field range3\n/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48/s49\n/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49\n/s48 /s49/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s91/s32 /s109\n/s66/s47/s102/s46/s117/s46/s32/s93\n/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s77/s111/s100/s101/s108/s72 /s32/s94/s32 /s99/s32/s97/s41\n/s98/s41\n/s99/s41\n/s72 /s32/s124/s124/s32/s99\n/s112 /s47/s52/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s91/s32 /s109\n/s66/s47/s102/s46/s117/s46/s32/s93/s72 /s32/s124/s124/s32/s99\n/s112 /s47/s50/s113\n/s77\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s91/s32/s84/s32/s93/s77/s113\n/s77\nFIG. 2. Magnetic field dependence of the magnetiza-\ntion of Mn 2Mo3O8atT= 2K.Magnetic field is per-\npendicular(a) and parallel(b,c) to the caxis. In all panels\nthe red solid line corresponds to the experiment, while the\nblue dashed line corresponds to the model calculation. The\norange and blue arrows illustrate the magnetic moments of\nthe two sublattices in different magnetic fields pointing hor -\nizontally/vertically for (a) and (b), respectively. The ex per-\nimental curve of the H⊥ccase is repeated in (b) as black\ndotted line for comparison. The inset in (b) magnifies the\nlow-field part of the experimental magnetization curve, usi ng\nthe same units as the main axes. Here black arrows indicate\nthe direction of the magnetic field sweep. Panel (c) shows the\ncalculated angle θMenclosed by the magnetization and the\nmagnetic field.\nbetween µ0HC1andµ0HC2= 6 T the magnetization\nsmoothly increases, asymptotically reaching the linear\nsusceptibilityofthe H⊥ccase. Tocomplementthemag-\nnetization curve presented for a limited magnetic field\nrange in Ref. [ 39], the results shown for a broader mag-\nnetic field range in Fig. 2(b) clearly indicate the isotropy\nof the magnetic susceptibility in fields above HC2.\nThe zero magnetization up to HC1for fields along\nthecaxis and the constant susceptibility for the per-\npendicular direction are characteristic of easy-axis anti-\nferromagnets. The relatively low values of the critical/s48 /s50 /s52 /s54 /s56/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s48 /s50 /s52 /s54 /s56/s74/s32 /s187 /s32/s48/s176/s74/s32 /s187 /s32/s45/s49/s176/s84/s111/s114/s113/s117/s101/s32/s91/s32 /s109 /s78/s109/s32/s93\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s91/s32/s84/s32/s93/s74/s32 /s187/s32 /s49/s176/s97/s41/s32/s32/s32/s32/s32/s32\n/s69/s120/s112/s46\n/s74/s32 /s61/s32/s45/s48/s46/s49/s176/s98/s41 /s32/s32/s32/s32\n/s77/s111/s100/s101/s108\n/s74/s32 /s61/s32/s49/s176/s74/s32 /s61/s32/s45/s49/s176/s84/s111/s114/s113/s117/s101/s32/s91/s97/s46/s32/s117/s46/s93\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s91/s32/s84/s32/s93\nFIG. 3. Magnetic field dependence of the magnetic\ntorque in Mn 2Mo3O8atT= 2K.(a) experiment and\n(b) model calculation. The angle included between the caxis\nand the magnetic field is denoted by ϑ, where the ±signs\nrepresent the directions towards the ±aaxes.\nfields along the caxis and the identical susceptibility\nforH/bardblcandH⊥cin fields above HC2indicate a\nnearly isotropic spin-system. The smooth increase of the\nmagnetization between HC1andHC2is a hallmark of\ncompeting magnetic anisotropies51. In this field region,\nthe magnetization is not parallel to the external field,\nas shown by the model calculations of the enclosed an-\ngleθMin Fig.2(c) and also evident from the magnetic\ntorque measurements in Fig. 3(a). Namely, torque ( τ) is\nproduced as the cross product of magnetization ( M) and\nmagnetic field, τ=µ0M×H. When the field is along\nthecaxis, the system is unstable since the direction of\nthe magnetization component perpendicular to the field\nhas no preferred orientation in the abplane. If the field\nis rotated by a small ϑangle towards the aaxis, either\nclockwise or anticlockwise, a characteristic peak in the\nmagnetic field dependence of the magnetic torque in the\nHC1< H < H C2field region appears. The small hys-\nteresis, only observed for some angles such as ϑ≈+1◦,\nand the sign change of the ϑ≈0◦curve are probably\ndue to the slight mechanical instability of the setup. The\napproximately saturated torque in high fields is due to\nthe compensation of the decreasing angle θM, Fig.2(c),\nby the increasing magnetization amplitude, Fig. 2(b), in\nincreasing field.\nTo explorethe magnetizationdynamics in Mn 2Mo3O8,\nwe studied the magnetic field dependence of the mag-\nnetic resonances using THz absorption spectroscopy at\nlow temperatures ( T= 3 K). The lowest-frequency res-\nonances ( ν <150 GHz), which are not accessible by far-\ninfrared spectroscopy, were investigated by the use of\nbackward-wave oscillators. Although the absorption line\nshapes were distorted by diffraction on the edges of the\nsample, and by interference effects, as seen in the inset\nof Fig.4(a), the resonance frequencies and selection rules\ncould be determined with reasonable accuracy.4\nTypicalabsorptionspectra, presentedinFig. 4(a), con-\ntain a strong absorption band between 1400GHz and\n2200GHz and additional weaker absorption peaks. We\nperformed a polarization-dependent study in order to\nclarify the selection rules of the modes. While in the\nlow-frequency region ( ν <500 GHz) the modes are sen-\nsitive to the magnetic component of light (details shown\nin Fig.5and discussed later), the high-frequency modes,\nν >1200 GHz, are electric dipole active and they can be\nexcitedbyanoscillatingelectricfieldperpendiculartothe\ncaxis. In the following, we focus on the field dependence\nof the weak resonance modes associatedwith magnon ex-\ncitations and leave the strong absorption band between\n1400 GHz < ν < 2200 GHz to later studies. More-\nover, temperature-dependent measurements show that\nthis broad excitation is not restricted to the magnetically\nordered state, though it shows some field- and tempera-\nture dependence.\nIn zero magnetic field, we observed three resonance\nbranches, ν1,ν2, andν3, as presented in the mode map of\nFig.5(a). While the lower-frequencymodes ν1andν2are\nmagnetic dipole active, the high-frequency ν3shows elec-\ntric dipole activity along the caxis. Starting from zero\nmagnetic field, in increasing fields parallel to the easy-\naxis (H/bardblc, Fig.5(a))ν1andν3shift towards lower fre-\nquencies, whilst ν2moves to higher frequencies. Within\nthe experimental precision, in all three cases the slope of\nthe shift corresponds to the free electron spin g-factor of\nge= 252. In finite fields, ν1andν2are excited by an\noscillating magnetic field, hν, perpendicular to the static\none,hν⊥H. Between µ0HC1= 4 T and µ0HC2= 6 T,\nthe slopes of the resonances ν2andν3change sign and\nonecanobserveanadditionalelectricdipole activemode,\nν4, with a positive slope at high frequencies. Above HC2,\ntheν2mode shifts towardshigher frequencies again, with\na slope corresponding to ge, while the frequencies ν3and\nν4remain roughly constant in increasing fields. In this\nphase a new electric dipole active mode, ν5, with almost\nfield-independent frequency appears. The fact that the\ng-factors of the modes are identical, further supports the\nnearly isotropic nature of the spin system.\nConsidering the smooth increase of the magnetiza-\ntion in the HC1< H < H C2phase transition region\n(Fig.2(b)), without hysteresis, the phase transition can\nbe categorized as of second order. This classification is\nfurther supported by the torque curves (Fig. 3(a), green\nandred)beingalsofreeofhysteresis. Moreover,byfitting\nthe linear field dependence of the ν1resonance frequency,\none can conclude that it softens to zero frequency at the\nHC1critical field, characteristic for second order transi-\ntions.\nIn magnetic fields perpendicular to c, see Fig. 4(b),\nν1is active for hν/bardblHand its frequency shows almost\nno field dependence. In contrast, ν2is excited by hν⊥\nHand shifts towards higher frequencies with increasing\nfield, with the slope corresponding to ge. We could not\nobserve the weak ν4resonance in the H⊥cgeometry,\nwhileν3andν5are active for oscillating electric fields,eν⊥candtheirfrequencyisindependentofthemagnetic\nfield.\nNext, we turn tothe modelling ofthe spin system. The\nmagnetization and the spin resonancesof Mn 2Mo3O8are\nconsistent with a relatively simple, two-sublattice ferri-\nmagnetic model, that we discuss in the following. The\ntwosublatticesinteractantiferromagneticallyandexperi-\nence different weak single-ion magnetic anisotropies that\ncan be attributed to the octahedral/tetrahedral enviro-\nments, respectively. The g-factors at the two crystallo-\ngraphic sites can also be different. The corresponding\nHamiltonian in the mean-field approach is\nH=JS1S2+∆1(Sc\n1)2+∆2(Sc\n2)2\n−µBµ0H(g1S1+g2S2), (1)\nwhereS1andS2arethe three-dimensionalvectorsof S=\n5/2 lengths representing the magnetic moments of the\ntwo sublattices. In the Hamiltonian the dominant energy\nscale is given by the antiferromagnetic coupling, J, while\nthe ∆ 1and ∆ 2single-ion c-axis anisotropies define the\nzero-field spin orientation. In the Zeeman term µBis the\nBohr magneton, g1andg2are theg-factors of the two\nsublattices, and µ0denotes the permeability of vacuum.\nThe ground state at a given external field value can\nbe determined by minimizing the energy given in Eq. 1.\nThe easy-axis collinear state is realized in zero field if\n∆1+∆2<0. To reproduce the smooth increase of mag-\nnetization in the H/bardblc,HC1< H < H C2case, ∆ 1and\n∆2have to have opposite signs.51In this field range the\nmagnetic moments of the two sublattices continuously\nrotate to become almost perpendicular to the increas-\ning magnetic field. Since ∆ 1and ∆ 2are different, the\nmagnetization is not parallel to the applied field, as clear\nfrom Fig. 2(c), which explains the peak of the magnetic\ntorque in the HC1< H < H C2field range, as presented\nin Fig.3.\nWithinthemodeldescribedbyEq. 1theisotropichigh-\nfield (H > H C2) differential susceptibility is\nχ=(g1+g2)2\n4J−2(∆1+∆2), (2)\nif the second-orderterms in the difference of the two sub-\nlattices, ∼(∆1−��2)2and∼(g1−g2)2, are neglected.\nFrom the lowest zero-field resonance frequencies, ν1and\nν2, the model parameters J, ∆1, and ∆ 2can be deter-\nmined, according to\nν1,2=/parenleftBig/radicalbig\n−2J(∆1+∆2)+(∆ 1+∆2)2±(∆1−∆2)/parenrightBig\nS.\n(3)\nThe slopes of ν1andν2inH/bardblc,H < H C1correspond\nto theg-factorsg1≈g2≈2. The zero-field remanent\nmagnetization at T= 2 K after a H/bardblcfield treatment,\nM0= 0.0025µB/f.u. as found in Ref. [ 39] and in our\nexperiments shown in the inset of Fig. 2(b), is a conse-\nquence of the different g-factors of the two sublattices,\nthusM0= (g1−g2)S, since sublattice 1 with the slightly5\n/s48 /s50/s48/s48 /s52/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48 /s49/s54/s48/s48 /s49/s56/s48/s48 /s50/s48/s48/s48 /s50/s50/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s48 /s50/s48/s48 /s52/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48 /s49/s54/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s110\n/s50\n/s110\n/s49\n/s110\n/s53\n/s110\n/s51/s110\n/s50\n/s110\n/s49/s110\n/s53\n/s110\n/s52\n/s110\n/s51\n/s110\n/s49\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s91/s32/s71/s72/s122/s32/s93/s72 /s32/s32\n/s101\n/s110\n/s104\n/s110/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s91/s32/s84/s32/s93\n/s97/s32 /s61/s32/s50/s48/s32/s99/s109/s45 /s49\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s91/s32/s71/s72/s122/s32/s93/s97/s32 /s61/s32/s56/s48/s32/s99/s109/s45 /s49/s72/s44 /s32/s104\n/s110/s32/s32/s101\n/s110/s101\n/s110/s32/s32\n/s104\n/s110\n/s72/s97/s41 /s32/s32/s98/s41\n/s110\n/s50/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s91/s32/s84/s32/s93\n/s49/s48/s48 /s49/s53/s48/s48/s49/s50/s109\n/s48 /s72 /s32/s91/s32/s84/s32/s93\n/s110 /s32/s91/s32/s71/s72/s122 /s32/s93\nFIG. 4. Magnetic field dependence of the magnetic absorption in Mn 2Mo3O8atT= 3K.Magnetic field is\nparallel(a) or perpendicular(b) to the hexagonal caxis. In both panels, absorption spectra are vertically shi fted in proportion\nto the magnetic field. The spectra show the absorption coeffici ent relative to the paramagnetic phase. Weak resonance mode s\nare marked by light-gray lines which are guides to the eye. Th e frequency region 550 GHz < ν <1150 GHz between the\nbreakpoints of the horizontal axis is featureless. Magneti c field dependent absorption at fixed low frequencies(a) is sh own by\nvertical red lines, where the baseline is shifted horizonta lly in proportion to the frequency. Scale of the absorption c oefficient\nfor spectra and magnetic field sweeps is indicated by blue/re d arrows in (a), respectively. The inset shows a magnified vie w of\nthe magnetic field dependent absorption at fixed frequencies . In (b), absorption for two perpendicular polarizations is shown in\norange (eν,hν⊥c,hν||H) and purple ( eν,hν⊥H,eν||c), where eνandhνstand for the electric and magnetic component\nof light, respectively. In both panels, schematic figures il lustrate the measurement geometry, where hexagonal plate s hows the\nabplane of the crystal.\nlargerg-factor aligns along the field, while sublattice 2\nturns opposite to it.\nFrom the considerations above, the field dependence\nof the magnetization, the torque and the magnon modes\ncan all be well reproduced by the following parameter\nset:J= 3 meV, ∆ 1=−0.015 meV, ∆ 2= 0.006 meV,\ng1= 2.001, and g2= 2. Using these parameters, we nu-\nmerically found the minimum of the energy in Eq. 1at\nvarious magnetic fields. The calculated magnetization of\nthe groundstate is presentedin Figs. 2(a,b) by blue lines,\nwhile Fig. 3(b) shows the calculated magnetic torque,\nboth in a good correspondence with the experiments.\nHowever, the lower critical field, HC1is slightly over-\nestimated by the model. Since the width of the peak in\nthe field dependence of the torque is proportional to theHC1field53, the model resultsa broaderpeak in Fig. 3(b)\nthan the experiments (Fig. 3(a)).\nThe model parametersare independently confirmed by\nab initio calculations that return J= 2.7meV,g1=g2=\n2.002, ∆ 1=−0.0015meV, and ∆ 2= 0.001meV, where\neasy-axis (∆ 1<0) and easy-plane (∆ 2>0) anisotropies\nare obtained at the octahedral and tetrahedral sites, re-\nspectively. The absolute values of ∆ 1and ∆ 2are under-\nestimated, though, which may be due to the systematic\nerror of DFT. Nevertheless, with the help of DFT one\ncan identify the sublattice 1 and 2 with the octahedrally\nand tetrahedrally coordinated Mn sites.\nThe magnetic resonances were determined by calcu-\nlating the response to small perturbations in the ground\nstate, as descibed in detail in Refs. [ 54] and [55]. The6\n/s110\n/s53/s32\n/s110\n/s52/s32\n/s110\n/s51/s32/s110\n/s50/s32\n/s110\n/s49/s32/s101\n/s110/s32/s94 /s32/s99 /s32\n/s104\n/s110/s32/s94 /s32/s72 /s32/s72 /s32/s124/s124/s32 /s99 /s32/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s91/s32/s84/s32/s93\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s91/s32/s71/s72/s122/s32/s93/s97/s41 /s32/s32/s32/s32/s32/s32/s98 /s41\n/s101\n/s110/s32/s94 /s32/s99 /s32/s110\n/s53/s32\n/s110\n/s51/s32/s110\n/s49/s32/s110\n/s50/s32/s72 /s32/s94/s32 /s99 /s32/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s91/s84/s93\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s91/s32/s71/s72/s122/s32/s93/s104\n/s110/s32/s94 /s32/s72 /s32/s104\n/s110/s32/s124 /s124 /s32 /s72\nFIG. 5. Magnetic field dependence of the magnetic resonance frequen cies of Mn 2Mo3O8atT= 3K as seen\nby far-infrared and microwave optical transmission spectr oscopy. Magnetic field is parallel(a) or perpendicular(b) to\nthe crystallographic caxis. In both panels black spheres correspond to the experim ental values, while blue lines to the model\ncalculation result. Resonance modes predicted by theory bu t not active in the optical experiments are shown in light blu e.\nObserved optical selection rules are indicated next to the c orresponding excitation branches.\nresults are shown in Figs. 5(a,b). The two ferrimag-\nnetically ordered classical spins have two Γ-point (zero\nmomentum) magnon excitations, which correspond to ν1\nandν2. ForH/bardblc, these two modes are excited by cir-\ncularly polarized light of opposite helicity propagating\nalong the direction of the magnetic field, thus in the lin-\nearly polarized optical experiments they are visible in\nthehν⊥Hgeometry. On the other hand, for suffi-\nciently high magnetic field H⊥ctheν1excitation cor-\nresponds to the quasi-antiferromagnetic while ν2to the\nquasi-ferromagnetic resonance, namely, ν1andν2corre-\nspond to the precession of L=S1−S2andM=S1+S2,\nrespectively. Accordingly, the selection rules are hν/bardblH\nforν1andhν⊥Hforν2, both as in the experiments and\nin the model.\nThe electric dipole active high-frequency (1200 GHz <\nν) modes, ν3,ν4andν5, can be interpreted as Brillouin\nzone-edge magnons of the simplified model system with\ntwo spins in the unit cell. However, these resonances\ncorrespond to Γ-point excitations if we consider that the\ncrystallographicunitcellcontainsfourMnsites, asshown\nin Fig.1. With the assumption of a four-spin magnetic\nunit cell, onecanextendthe modelin Eq. 1todistinguish\nbetween first-neighbour antiferromagnetic interactions of\noctahedrally and tetrahedrally coordinated spins within\ntheabplane,J1, and between adjacent abplanes,J2.\nTo numerically reproduce the zero-field splitting of the\nν4andν5resonances, we also consider a weak antiferro-\nmagnetic J3coupling between tetrahedrally coordinatedspins of neighbouring abplanes, as presented in Fig. 1.\nTo ensure the compatibility with the previous results of\nEq.1,J= 3J1+2J2has to hold, where the integers cor-\nrespond to the coordination numbers. In this context,\nν3,ν4andν5can be viewed as excitations with πphase\nshifts between the equivalently coordinated spins of adja-\ncentabplanes. For ν3the crystallographicallyequivalent\nspins oscillate in-phase within a single abplane, while\nforν4andν5four of their closest-neighbours have oppo-\nsite, and two of them the same phase, as illustrated in\nFig.6. Within this extended model, using J1= 0.8 meV,\nJ2= 0.31 meV and J3= 0.005 meV values, the mag-\nnetic field dependence of the ν3,ν4andν5resonances is\nreproduced both in the H/bardblcandH⊥cgeometries, as\npresented in Figs. 5(a,b). To explain their eν⊥copti-\ncal selection rule, one has to consider a model including\nspin-polarization coupling where the electric dipole mo-\nment of these resonances can be calculated, which is out\nof the scope of the present study.\nIn conclusion, the low-temperature static magnetic\nproperties and spin excitations of Mn 2Mo3O8were in-\nvestigated by various experimental techniques. The ob-\nserved magnetic field dependences of the magnetization,\ntorque, and spin-wave resonance frequencies are repro-\nduced by a mean-field model. The magnetic exchange\nandg-factor parameters of the model were determined\nby fitting the rich experimental dataset, and are also\nsupported by first principle calculations. The quanti-\ntative explanation of the various magnetic resonances of7\nFIG. 6.Magnetic resonance modes of Mn 2Mo3O8.Or-\nange and blue vectors, corresponding to the octa- and tetra-\nhedrally coordinated Mn spins, show snapshots of each exci-\ntation mode in H < H C1magnetic field along the positive c\naxis. The traces of the tips of all spins describe circles in t he\nabplane of the crystal. The relative amplitudes are indicated\nby the size of the curved arrows. The mode corresponding to\nthe faint cartoon was not observed in the experiments.Mn2Mo3O8can serve as a starting point to the under-\nstanding of the more complicated excitations of other\ncompounds in the M2Mo3O8family22,39.\nAcknowledgements\nThis project was supported by institutional research\nfunding IUT23-3 of the Estonian Ministry of Education\nand Research, by the European Regional Development\nFund project TK134, by the bilateral program of the\nEstonian and Hungarian Academies of Sciences grant\nNKM2018-47, by the Hungarian NKFIH grants ANN\n122879 and 2019-2.1.11-TT-2019-00029, by the Austrian\nAgency for International Cooperation in Education and\nResearch grant WTZ HU 08/2020, by the Austrian Sci-\nence Funds grant I 2816-N27, and by the grant ANCD\n20.80009.5007.19 (Rep. of Moldova). 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B 100, 024406 (2019) ." }, { "title": "1912.11634v1.Hybrid_nanophotonic_nanomagnonic_SiC_YiG_quantum_sensor__I__theoretical_design_and_properties.pdf", "content": "arXiv:1912.11634v1 [quant-ph] 25 Dec 2019Hybrid nanophotonic-nanomagnonic\nSiC-YiG quantum sensor:\nI/ theoretical design and properties.\nJ´ erˆ ome TRIBOLLET∗\nInstitut de Chimie de Strasbourg, Strasbourg University, U MR 7177 (CNRS-UDS),\n4 rue Blaise Pascal, CS 90032, F-67081 Strasbourg Cedex, Fra nce\nE-mail: tribollet@unistra.fr\nAbstract\nHere I present the theory of a new hybrid paramagnetic-ferri magnetic SiC-YiG\nquantum sensor. It is designed to allow sub-nanoscale singl e external spin sensitivity\noptically detected pulsed electron electron double resona nce spectroscopy, using an X\nband pulsed EPR spectrometer and an optical fiber. The sensor contains one single V2\nnegatively charged silicon vacancy color center in 4H-SiC, whose photoluminescence\nis waveguided by a 4H-SiC nanophotonic structure towards an optical fiber. This V2\nspin probe is created by ion implantation at a depth of few nan ometers below the\nsurface, determined by optically detected paramagnetic re sonance under the strong\nmagnetic field gradient of a YiG ferrimagnetic nanostripe lo cated on the back-side\nof the nanophotonic structure. This gradient also allow the study, slice by slice at\nnanoscale, of the target paramagnetic sample. The fabricat ion process of this quantum\nsensor, its magnetic and optical properties, its external s pins sensing properties in a\nstructural biology context, and its integration to a standa rd commercially available\npulsed EPR spectrometer are all presented here.\n1Introduction\nElectron paramagnetic resonance1(EPR) investigation of electron spins localized inside, at\nsurfaces, or at interfaces of ultrathin films is highly relevant to man y fields.2–6For example,\nin the fields of photovoltaic7and photochemistry8, EPR is useful to study the spins of\nphoto-created electron-hole pairs, their dissociation, and their e ventual transport or chemi-\ncal reaction occurring at some relevant interface. Also, in struct ural biology, it is relevant to\nstudy by EPR spin labeled proteins9,10introduced in model lipid bilayers membranes.11,12\nIn the context of the development of new theranostic agents for nanomedicine, it is also\nrelevant to study ligand-protein molecular recognition events occu rring on surfaces by EPR,\nusing for example, bifunctional spin labels.13As various nanotechnologies now allow to pro-\nduce biological, chemical or physical samples with nanoscale thicknes s, one needs to perform\nsensitive surface EPR. However, commercial EPR spectrometers have generally not enough\nsensitivity1415forstudying thosethinfilms when targetspins arediluted, and, f orsure, they\ncannot detect a single external spin. While home-made EPR experime ntal setups have been\ndeveloped recently allowing to reach the single spin sensitivity, by opt ically15–20(ODMR),\nelectrically21ormechanically22detected EPR,apowerfull upgradestrategyallowing toreach\nboth the sub-nanoscale resolution and the single external spin sen sitivity, while still using\na commercially available pulsed EPR X band spectrometer, is clearly lack ing and is highly\nrelevant for most EPR users worldwide.\nHere I present a new methodology for state of art optically detect ed pulsed double electron\nelectron spin resonance spectroscopy (OD-PELDOR), allowing SiC b ased sub-nanoscale sin-\ngle external spin sensing. This new methodology requires only a stan dard X band pulsed\nEPR spectrometer, as well as an optical fiber and a new SiC-YiG quan tum device, which\ncan be both conveniently introduced in a standard EPR tube. The 4H SiC part of the device\ncontains a nanophotonic structure with a single V2 negatively charg ed silicon vacancy color\ncenters (spin S=3/2), introduced below the surface by low energy ion implantation and used\n2as a quantum coherent ODMR spin probe19,20,27,28,42for sensing14–16external spins through\nmagnetic dipolar couplings. The V2 depth below the surface can be de termined by optical\nfiber based Optically Detected paraMagnetic Resonance (ODMR) un der the strong mag-\nnetic field gradient produced by a YiG ferrimagnetic nanostripe, app ropriately designed,26\nlocated nearby on the back-side of the 4H-SiC nano-photonic stru cture. This field gradient\nis also designed to allow the ODPELDOR investigation, slice by slice at sub -nanoscale, of\nthe target paramagnetic sample located on the 4H-SiC surface. Th e fabrication process and\nthe properties of this SiC-YIG quantum sensor are all presented h ere.\nPrinciples for sub-nanoscale single external spin sensing\nwith a V2 color center in a SiC-YiG quantum sensor.\nA single color center spin probe can sense an external spin located n earby at a nanoscale\ndistance through its dipolar magnetic coupling with this spin (fig.1a). S ince decades, the\npulsed EPR experiments called DEER (Double Electron Electron Reson ance) or PELDOR\n(Pulsed ELectron electron DOuble Resonance)1,9,10,17,30allow to measure this dipolar mag-\nneticcoupling between pairsofspins A-B using pump-probemagnetic resonance experiments\nwith two different microwave frequencies, experiments performed over macroscopic ensem-\nble of identical pairs of spins A-B. As this dipolar coupling scales with1\nR3\nAB,RABbeing\nthe distance between the two spins A and B, measuring the dipolar co upling gives acces\nto the nanoscale distance RAB. It was thus particularly used in structural biology for the\ndetermination of constraints on the 3D conformation of some spin la belled proteins. The\nsitutation becomes however much more complicated for some studie s like study of trans-\nmembrane proteins which are extremely difficult to obtain in large quan tities and even more\nin 2D model lipid bio-membranes. Thus, despite the fact that 50 perc ent of all known small\nmolecule drugs bind to transmembrane proteins, those proteins re mains difficult to study by\n3PELDOR spectroscopy by lack of sensitivity.\nFigure 1: Basic principles for external spins sensitive and nanoscale Quantum Sensing.a/\n4H-SiC-YIG sensor for ODPELDOR with a single V2 spin probe (red dot ) and a plane of\nexternal target spins, like spins labels (arrows); on the left of das hed-dot line, a 2D spin\nlabeled lipid membrane 5 nm thick, on the right, the model in green. Boz and B1y: external\nstatic and microwave magnetic fields. Bdip,z (x) is the spatially depend ent dipolar magnetic\nfield produced by the YIG ferromagnetic nanostripe located on the opposite 4H SiC surface.\nb/ OD-PELDOR pulse sequence for quantum sensing including synchr onized optical and\nmicrowave pulses. c/ Simplified scheme showing a top view of the SiC-YI G quantum sensor:\nSiC is in gray,the red dot is the V2 single color center, the dashed red r ectangle is the\nYIG nanostripe present on the opposite surface of the SiC membra ne, through which a 1D\nnanophotonic waveguide has been designed in a 2D photonic cristal m ade of air holes in SiC.\nThe 1D optical waveguide is coupled to the optical fiber by an approp riate coupler. Ky,ex,\nKy,det : excitation and PL light wavevectors.\nAs it was previously demonstrated with a single NV color center in diamo nd and a home\nmade setup,17it is now possible to detect one single or several external electron s pin by\noptically detected PELDOR spectroscopy (fig.1a). The idea is to tak e advantage of the spin\ndependent photoluminescence rate and/or of spin dependent non -radiative relaxation rates\nof some optically excited color centers found in wide gap semiconduct ors like diamond17\n(NV- center) or 4H-SiC27,28,42(V2 negatively charged silicon vacancy), firstly to perform\noptical pumping43of the ground spin state of the color center, and secondly to perf orm an\n4ultra-sensitive photoluminscence measurement19,20ofthe groundspin statepopulationof the\ncolor center by mapping this spin state population difference onto a g iven photoluminescence\namount. Single photon detectors in the visible or near infrared rang e combined with efficient\noptical pumping thus allow altogether to reach the single spin detect ion level. This is orders\nof magnitude better than the sensitivity of standard PELDOR spec troscopy performed using\na standard pulsed EPR spectrometer operating at X band (around 10 GHz). The ODPEL-\nDOR pulse sequence (fig.1b) is thus similar to the older PELDOR pulse se quence, the main\ndifference being the need of an initial optical pumping pulse and the ne ed of a second optical\nreadout pulse after the PELDOR microwave pulse sequence at two m icrowave frequencies,\nwhich ends by a last microwaveπ\n2pulse at probe frequency to convert the spin state coher-\nence into a spin state population difference. The time domain ODPELDO R curve (versus\ntd, see fig.1b for definition of td) recorded by the single V2 photolum inescence detection can\nbe either a simple oscillation at the dipolar magnetic coupling frequency if the V2 spin is\ncoupled to a single external spin, or a decaying curve if the single V2 s pin is coupled to a\nspins bath (a spin ensemble), as it was previously demonstrated with a single NV- center\nspin probe in diamond.17As I also suggested it in a previous work on quantum computing\nwith spins,26a strong dipolar magnetic field gradient produced by a ferromagnet ic nanos-\ntripe (see Bdip,z (x) on fig.1a) located nearby the V2 spin probe color center and its nearby\nexternal target spins, allows to encode the spatial position of eac h of those spins onto its\nparamagnetic resonance field or frequency, allowing to reach nano scale spatial resolution\nwith paramagnetic resonance.\nThe idea of the SiC-YIG quantum sensor presented here (fig.1c) is t o provide a convenient\nexperimental acces to this ultra-sensitive nanoscale resolution OD PELDOR spectroscopy to\nthe many worlwide pulsed EPR users, by interfacing a standard pulse d EPR spectrometer\nwith the SiC-YIG quantum sensor described here, using an optical fi ber (fig.1c and fig.2).\nThis 4H-SiC-YIG quantum sensor contains one single sub surface V2 negatively charged\nsilicon vacancy color center, whose infrared photoluminescence is w aveguided by a 4H-SiC\n5Figure 2: A laser (9) at 785 nm can conveniently excite the photolumin escence (PL) of\nthe single V2 spin probe through a dichroic miror (5), the optical fibe r (1) , the optical\ncoupler (2) and the 1D photonic waveguide of the SiC-YIG quantum s ensor (3). The PL\nis then collected by the same path in the reverse direction till the dich roic miror, and then\nit is filtered (7) before being send to the time gated sensitive photod etector (8), either a\ngated photomultiplier tube or a gated avalanche photodiode, having single photon counting\ncapabilities. The photodetector signal, which is proportionnal to th e photoluminescence\nsignal, has then to be sent in a voltage form to the pulsed EPR spectr ometer input channel\nof the transient recorder. Microwave pulses at two different freq uencies are fabricated by the\npulsed EPR/ELDOR spectrometer (10) and synchronized with a trig er voltage pulse used to\npulse the exciting laser, when necessary. For the much simpler ODMR experiments requiring\na single microwave frequency, a Si photodiode can be used for PL de tection and its voltage\nsignal send to a lock in amplifier. (4) and (6) are short focal lenses o r objectives. Static field\nB0 and microwave magnetic field B1 are perpendicular. The pulsed EPR resonator is inside\na standard pulsed EPR cryostat (4K-300K).\nnanophotonic structure (fig.1c) coupled to the optical fiber by an appropriate optical cou-\npler.47The SiC nanophotonic structure can be either a simple SiC nanobeam48,49or a 1D\noptical waveguide created in a 2D photonic crytsal,50,51itself made of air holes in a thin\n4H-SiC membrane.50–52A YIG nanostripe53on the backside of the photonic waveguide pro-\nvide a strong permanent magnetic field gradient.26Outside the pulsed EPR spectrometer, a\nsimple photoluminescence setup using a dichroic miror and few other o ptics (fig.2) is used\nto excite and collect the photoluminescence from the V2 spin probe ( at around λ= 915nm\nat low temperature), the light being waveguided by the optical fiber in both directions. A\n6commercially available pulsed EPR spectrometer operating at X band is by this way up-\ngraded into a state of art ODPELDOR spectrometer allowing single ex ternal spin sensing\nwith nanoscale resolution.\nFabrication process of the SiC-YiG quantum sensor\nThe SiC-YIG quantum sensor proposed here is an hybrid nanophoto nic-nanomagnonics de-\nvice containing a 4H-SiC1D nanophotonic waveguide and aferrimagne tic nanostripeof YIG.\nHereIpresent onepossible fabricationmethodofthequantum sen sor, but othersarepossible\nwith the advanced SiC technology.\nIt requires firstly the fabrication of a 2D photonic cristal in a thin 4H -SiC membrane, as\nshown on fig.3a. For this one can start with anappropriate 4H-SiC HP SI substrate with very\nlow n type residual doping and very few residual V2 silicon vacancy an d also very few other\nkinds of paramagnetic defects, eventually also containing a top epila yer of isotopically puri-\nfied 4H-SiC to eliminate all nuclear spins from the SiC membrane to fabr icate. If necessary\na high temperature pre-processing annealing can be used to remov e all residual V2 silicon\nvacancies in the starting 4H-SiC material. Then, using for example a fi rst RIE etching step\nor a SICOI approach, a thin 4H-SiC membrane can be produced (aro und 100 nm thickness),\nfollowing previous demonstrated methods.50–52If one wish to have the full SIC-YIG quan-\ntum sensor, then the SiC membrane need to be thin enough, typically between 100 nm and\n300nm in order that the subsurface V2 silicon vacancy feels the opt imal dipolar magnetic\nfield gradient (see next sections). But if one wishes to produced a m uch simpler SiC sensor,\nwithout the YIG nanostripe, then a thicker membrane, convenient ly produced by RIE can\nbe fabricated. Then, electron beam lithography is used to fabricat e an appropriate mask for\nthe subsequent second RIE etching step allowing to produce the air holes for the photonic\n7Figure 3: Six main steps in the fabrication process of the 4H-SiC-YIG quantum sensor\nproposed. a/ SiC membrane and then 2D photonic cristal fabricatio n; b/ oxydation of SiC;\nc/ YIG nanostripe fabrication on the backside of the SiC membrane; d/ annealing under\noxygen of YIG nanostripe; e/ Fabrication of a single V2 spin probe be low SiC surface by ion\nimplantation through a mask; f/ Removal of mask and final quantum device structure. See\ntext for details.\nnanostructure. This mask has thus to reproduce the 2D photonic crystal structure wished,\nincluding the 1D photonic waveguide. This mask for RIE can be create d either on the top\nside of SiC membrane if the membrane is thick, or on the back side of th e SiC membrane if\nthis one is very thin, using through membrane ebeam lithography54of a photoresist (fig.3a).\nThen, after this RIE step and after removing the mask used, a the rmal oxydation of the 4H\nSiC is performed (fig.3b), creating a SiO2 layer of around 10 nm. As sh own on fig.3c, the\nsingle YIG nanostripe has then to be fabricated on the backside of t he 4H-SiC 1D photonic\nwaveguide. The width Wnanobeamof this 4H-SiC 1D waveguide in the 2D photonic crystal\n(or the width of the alternative SiC nanobeam surrounded by air) ha s to be larger than\nthe width WYIGof the YIG nanostripe. As the YIG nanostripe is on the backside of t he\n8device, one has to use through membrane ebeam lithography54of a photoresist deposited\non the backside(fig.3c) of the SiC membrane. Then, room temperat ure sputtering of YIG53\non SiO2 through this photoresist mask can be used to deposit YIG on the backside of the\nmembrane. Removing the photoresist will let an amorphous YIG nano stripe deposited on\nthebackside of the4H-SiC1Dwaveguide. Asubsequent thermal an nealing inoxygen (fig.3d)\nhas then to be performed to obtain a poly-cristalline YIG nanostripe with improved spin\nwave resonance properties. Alternatively, direct Pulsed Laser De position (PLD) of YIG at\nhigh substrate temperature, followed by an appropriate backside etching of the YIG film\nthrough a mask can be used to produce such a YIG nanostripe, but in that case one should\ncreate the YIG nanostripe on the backside of the membrane befor e creating the air holes\nof the photonic nanostructure by RIE. The mask can again in this ca se be defined on the\nbackside by through membrane ebeam lithography.54\nThen, a single V2 color center has to be introduced just few nanome ters below the 4H SiC\nsurface on the front side of the SiC membrane, at a position located vertically above the\ncenter of the YIG nanostripe itself located on the backside of the S iC membrane (fig.3e).\nThe detailed ion implantation methods proposed to create such a sing le V2 spin probe in\nthe quantum sensor are described in the appendix 1 and are inspired from a recent work\nreporting the fabrication of a silicon vacancies array in 4H-SiC.55A common key point to\nthe various proposed statistical fabrication processes by ions imp lantation through a single\naperture mask is the need to evaluate each quantum sensor after fabrication, because over\naround 100 devices fabricated in parallel, around just 5 contains an appropriate single V2\nspin probe located few nm below the SiC surface, at a given depth with a precision of around\n+/- 1 nm. Firtsly, the second order photoluminescence auto-corr elation function has to be\nmeasured to test whether agiven sensor hasa singleV2 spinprobeo rnot.55Secondly, thanks\nto the strong dipolar magnetic field gradient produced by the YIG na nostripe, one can de-\ntermine by ODMR spectroscopy under such strong gradient, the d epth of this single V2 spin\nprobe in all sensors having a single V2. By time resolved ODMR, with an o ptically detected\n9hahn echo decay measurement for example, one can also determine the T2 spin coherence\ntime of this single V2 spin probe. Thus, the various post fabrication t ests combined with\nan appropriate YIG nanostripe design allow an efficient selection of th e quantum sensors\nfabricated having the desired properties (depth below the surfac e and T2 spin coherence\ntime).\nAs a last remark about the whole fabrication process, one should no te that in 4H-SiC, just\nlike for diamond, surface defects exist and can induce some magnet ic noise on the V2 sub-\nsurface single spin probe. To limit this possible spin decoherence effec t, a low temperature\npassivation treatment can be applied to the top 4H-SiC surface, like for example, a H+N\nplasma treatment40at 400 degree C, which is known to reduce its surface density of sta te to\n61010cm−2. Thermal cooling of the quantum sensor is an alternative method to reduce or\nsuppress the thermal fluctuations of those surface defects an d thus to increase in this case\nthe T2 of sub surface V2 spin probes.\nDesign of the YIG ferrimagnetic nanostripe and mag-\nnetic properties of the SiC-YiG quantum sensor\nThe quantum sensor proposed here contains a YIG nanostripe, as already said above, whose\ndesign should allow to perform ODMR test of the sub surface V2 spin p robe under a strong\ndipolar magnetic field gradient, as well as ODPELDOR under this stron g dipolar magnetic\nfield gradient to improve the spatial resolution of ODPELDOR down to nanoscale (see fig.1\nand fig.2).\nThe figure 4 summarizes the static and dynamic magnetic properties of the YIG nanos-\ntripe located on the backside of the 1D nanophotonic waveguide. Th e fig.4d shows that the\nmaximum magnetic field gradient in the x direction, perpendicular to 4H -SiC surfaces, is\n10Figure 4: YIG nanostripes: width W=500 nm , thickness T=100 nm, len gth L=100 µ m,\nBsat=1700 G; x,y,z as in fig.1. a/ and b/ EPR spectrum at X (9.7 Hz) and Q (34 GHz)\nband resp., showing, in blue, YIG SWR and, in red, the shifted EPR of g =2.00 electron\nspins placed at xopt=150 nm above the YIG nanostripe center (x=0 ) (1 G linewidth for all\nhere; SWR oscillator strength not calculated). c/ 1D eigenenergies of the spin waves along\nz axis; here z*=300+z; d/ z component of the dipolar magnetic field o f the YIG nanostripe\nas a function of x (black), as well as its gradient along x (red) multiplie d here by 100 for\nclarity. e/ and f/ Total effective Zeeman splitting at X band (dot line) , expressed in Gauss\n(thus divided by ( g µB), assuming g=2.00), as well as its two contributions: Bdz (to first\norder in blue), and Bdx (in red to second order), at xopt(e/) and a t xopt +/-10 nm (f/).\nof around 0.5 G/nm and is obtained at a distance xopt=150 nm from th e center of a given\nYIG nanostripe for the dimension chosen here (see legend of fig.4). That is why the SiC\nmembrane has to have a thickness of xopt - T/2 =100 nm here, such that the V2 spin probe\non the opposite face of the 4H-SiC nanophotonic waveguide feel th e maximum magnetic\nfield gradient. Increasing the width W of the YIG nanostripe will incre ase xopt (xopt=230\nnm for W=800 nm, corresponding to a 180 nm thick 4H-SiC membrane) , but it will also\nreduce the maximum gradient available at xopt, thus a compromise ha s to be found. The\nmagnetic field gradient produced by the YIG nanostripe considered here (W=500 nm) is\n11not rigorously one dimensional along x. However, as I previously exp lained in the context\nof quantum computing,26locally, around xopt = 150 nm here, and laterally at z=0 +/- 30\nnm along z, detailed calculations clearly show (fig. 4e) that in this port ion of plane above\nthe YIG nanostripe, the dipolar magnetic field can be considered as la terally homogeneous\nwith a precision better than 0.1 G, thus for external spins on the se nsor surface. Even in the\nportion of plane located at around xopt - 10 nm, where the V2 spin pr obe could be placed,\nand laterally at z=0 +/- 30 nm along z, the dipolar magnetic field can be c onsidered as lat-\nerally homogeneous with a precision better than 0.3 G (fig. 4f). In th e proposed fabrication\nprocess of the V2 color center, its position is well defined at +/- 10 n m along z, the diameter\nof the hole in the mask used for low energy ion implantation being 20 nm. I will also assume\nhere that it is possible to fabricate a quantum sensor device having a single isolated sub\nsurface V2 with a spin coherence time comprised between 10 µsand 50µs, in principle at\nroom temperature, and at least at a sufficiently low temperature (s ee appendix 2 for more\ndetails). Those spin coherence times thus correspond to a homoge neous V2 linewidth com-\nprised between 7 mG and 36 mG, smaller than the 50 mG variation of the dipolar magnetic\nfield which is expected with the variation of the lateral position of the V2 spin, as shown on\nfig.4f. This further implies, with a 0.5 G/nm gradient along x, that the V 2 coordinate along\nthe x axis, relatively to the center of the YIG nanostripe, or equiva lently its depth below the\n4H-SiC surface, could be determined in principle with a precision of aro und 1 Angstrom.\nAs ODPELDOR spectroscopy also allow the indirect detection of the p aramagnetic reso-\nnance of an external spin through the V2 photoluminescence signa l, and as this external spin\nfeels also the field gradient of the YIG nanostripe, its paramagnetic resonance frequency (or\nresonant magneticfield) necessary also encodeits coordinatealon gthe xaxis. Thus ODPEL-\nDOR spectroscopy performed under the strong magnetic field gra dient produced by the YIG\nnanostripe could allow to determine the coordinate along the x axis of the external spin,\nrelatively to the center of the YIG nanostripe, or equivalently its ve rtical distance above the\n124H-SiC surface. At X band (around 10 GHz), at T=100K and in water -glycerol, a trityl\nspin label56has a long longitudinal relaxation time T1= 1 ms, and also a rather long s pin\ncoherence time T2, limited by solvant proton spin diffusion and equal t o around 4 µs. For a\nsingle trityl spin labelled transmembrane protein that would be locate d just few nm above\nthe 4H-SiC surface of the SiC YIG quantum sensor, this T2 corresp onds to a homogeneous\nlinewidth of the single trityl of around 90 mG, which is slightly larger tha n the 0.05 G = 50\nmG variation of the dipolar magnetic field which is expected with the var iation of the lateral\nposition of the spin label located betwen xopt and xopt + 10 nm, as sh own on fig.4f. Thus in\nthis case, the spatial resolution along x would be limited by the 90 mG line width of the trityl\nspin label and thus to a precision of0.09\n0.5= 1.8 Angstrom. This opens exciting possibilities\nin the context of structural biology to study the structure of sp in labelled transmembrane\nprotein reconstituted in a model lipid bilayer or lipid nanodisk, itself anc hored on the 4H\nSiC-YIG sensor surface. This important application of this SiC-YIG O DPELDOR quantum\nsensor is further discussed in one of the next sections.\nAs a last remark, one can note that the YIG nanostripe design also a llows to avoid the\nspectral overlap between the many spin wave resonance (SWR) line s of YIG and the dipo-\nlar magnetic field shifted electron paramagnetic resonance lines of t he single V2 and of the\nsingle external spin label, as shown on fig.4a (at X band) and fig.4b (at Q band). As pre-\nviously explained,26this is necessary to avoid the microwave driving of some spins in the\nYIG ferromagnetic nanostripe because this could produce some un wanted decoherence ef-\nfects. Without spectral overlap between SWR and EPR lines, the re sidual V2 decoherence\nrelated to YIG is due to thermal fluctuations of the spins in YiG, which are expected to\nbe negligigle here compared to other decoherence processes for t he isolated sub surface V2\nspin probe, following my previous estimates made for a permalloy nano stripe with a higher\nsaturation magnetization in the context of spin based quantum com puting.26Note also that\nthe SWR mode which has the highest resonant magnetic field corresp onds to a SWR mode\nconfined on the lateral edges of the YIG nanostripe and is thus exp ected to have a much\n13smaller oscillator strength compared to the quasi uniform SWR mode (FMR mode) having\nthe largest oscillator strength and a much smaller resonant magnet ic field.\nDesign of the nanophotonic structure and optical prop-\nerties of the SiC-YiG quantum sensor\nThe quantum sensor proposed here contains a 4H-SiC 1D nanophot onic waveguide, as al-\nready said above, whose design should allow the efficient optical excit ation of the V2 spin\nprobe and the efficient collection of its photoluminescence, which has to be directed towards\nthe optical fiber integrated by a coupler47to this nanophotonic structure (see fig.1 and fig.2).\nThis is necessary to further integrate this sensor to a widely availab le pulsed EPR spectrom-\neter. For the coupling of the fiber to the nanophotonic waveguide,47one can use either\na tapered waveguide coupler, a diffraction grating based coupler, a n adiabatic coupler (or\nevanescent coupler), or a cylindrical GRIN microlens integrated or placed at the end of the\nfiber. Recently, a high optical coupling efficiency above 90 percent h as been demonstrated\nwith an optical fiber coupled to a diamond nanophotonic structure.57An optical coupling\nefficiency above 25 percent has also been recently obtained betwee n an optical fiber and a\nSilicon nanophotonic structure, with another approach allowing cry ogenic cooling down to\nvery low temperatures.58\nAs explained in the previous section, the optimisation of the in plane ho mogeneity and ver-\ntical strength of the dipolar magnetic field gradient produced by th e YIG nanostripe close to\nthetop4H-SiCsurfacewhere theV2spinandexternal spinsareloc atedprovides aconstraint\non the thickness of the SiC membrane that has to be fabricated (SiC membrane thickness =\nxopt - T/2, which is 100 nm for a W=500 nm YIG nanostripe width). The YIG nanostripe\nshould also be fabricated on the backside of the 1D SiC photonic wave guide, requiring that\nthe width of the 1D photonic waveguide be larger than the width of th e YIG nanostripe.\nTypically, the width of the 1D photonic waveguide has to be close to an integer number of\n14times half the wavelength of the low temperature zero phonon line of the V2 spin probe,\nwhen the 1D waveguide is a simple SiC nanobeam surrounded by air, in or der to optimize\nphoton confinement in the waveguide. With a wavelength of around 9 15 nm for the ZPL of\nV2 in 4H-SiC, this gives the following possible values of the width of such SiC nanobeam:\n457 nm, 915 nm, 1372 nm,... As we also wish to have a maximum optical elec tric field in\ninteraction with the sub surface V2 electric dipole, aligned along the c axis of 4H-SiC,20,27,28\nwe further chose a nano-beam width either of 457 nm or 1372 nm, bo th having an anti-node\nat half such a width. For a YIG nanostripe having a width of 500 nm, on e has thus to chose\na SiC nanobeam width of around 1372 nm.\nThis value can be considered as a good starting value for designing als o the width of another\nkind of 1D photonic waveguide fabricated in a 4H-SiC membrane and ma de of defects in a\n2D photonic cristal of SiC, itself made of an array of cylindrical air ho les in the SiC mem-\nbrane.50–52The main optical property of this 2D photonic crystal is its optical b andgap.50,51\nIt should be centered at the zero phonon line (ZPL) wavelength of t he V2 spin probe in\norder that the ensemble of defects (one or several lines of defec ts), that are here missing air\nholes in such a 2D photonic crystal, create a volume for photon confi nement and provide\nefficient 1D waveguiding properties to the V2 spin probe photolumines cence in this photonic\nnanostructure. The photonic band structure for TM modes prop agating in the plane of a\n2D photonic crystal with a triangular array of air holes in 4H-SiC can b e simulated numeri-\ncally.59Here I focus on TM modes, that means on electromagnetic modes ha ving an optical\nelectric field aligned along the c axis of 4H SiC and thus having the maxima l coupling with\nthe V2 electric dipole also aligned along the c axis.\nThe results of such a numerical simulation (see Appendix 3), show th at an isotorpic optical\nbandgap exist for TM mode if one fabricate such a nanophotnic stru cture with a center to\ncenter inter-hole distance equal to a= 622nmin the triangular array of air holes in SiC,\nthe air holes having a diameter of 2 ∗r=360nm. This is quite feasible with available elec-\ntron beam lithography and SiC RIE etching methods refsPCSiC. Othe r alternative designs\n15based on larger but partial optical bandgap for TM modes are also p ossible (see discussion\nin Appendix 3).\nSignal to noise ratio of the photoluminescence ODPEL-\nDOR signal obtained with the SiC-YiG quantum sensor\nUsing the normalized DEER signal expression,1,10,30Vdeer, whose value is comprised be-\ntween 0 and 1, and which is directly related to the ODPELDOR experime nt shown on\nfig.1b, considering the effect of the last additional −π\n2microwave pulse, which converts\nthe V2 spin quantum coherence into a V2 spin state populations differ ence, and assum-\ning a photon shot noise limited signal to noise ratio,60one obtains the following expres-\nsion for the photoluminescence signal to noise ratio R, in the case where ODPELDOR\nis obtained by off resonant optical excitation of the V2 spin probe, a t 785 nm for exam-\nple:R=Ropt(1−Vdeer(td, dx, C 2D,target))X, and in the case where ODPELDOR is\nobtained by a spin state selective resonant optical excitation of th e V2 spin probe, one\nfindsR=Ropt(1−Vdeer(td, dx, C 2D,target)), with Roptgiven by the formula: Ropt=\nexp/parenleftbig\n−2t0\nT2/parenrightbig/radicalBig\npcollpdetσ\nAP0T\nhν/angbracketleftΦ/angbracketright. I used above the following notations, /angbracketleftΦ/angbracketright=ΦH+ΦL\n2, and\n2X=ΦH−ΦL\n/angbracketleftΦ/angbracketright, taking into account the two possible different photoluminescence q uantum\nyield for the V2 spin probe, which depend on the spin state associate d to those optical tran-\nsisitons (see Appendix 4). The delay td is defined onfig.1b, C2D,targetis the 2D concentration\nof the target spin plane, and dxis the distance between the V2 spin probe and the target\nspin plane. σis the absorption cross section of the V2 spin probe, Ais the area on which the\noptical power P0is sent,hνis the photon energy, and Tis the integration time of the photo-\nluminescence by the photodetector over one single cycle. Xis given above, and has a value\nclose to 0.02 at room temperature according to previous ODMR expe riments on V2 with\noff resonant excitation of photoluminescence.19,20,27,28,42Vdeercan be numerically computed\nusing the linear approximation and shell factorization model.10This model was previously\n16introduced for calculating the standard DEER time domain signal in th e case of a three-\ndimensional distributions of spins. Here, this model has been adapt ed to take into account\nthebidimensional randomdistribution ofthetargetexternal spins intheir well-defined plane,\nparallel to the SiC sensor surface (see also next section). Assumin g realistic parameters esti-\nmated in the previous sections (see Appendix 4), one finds in the cas e of off resonant optical\nexcitation of the V2 spin probe, R= 90 for a single one shot one point ODPELDOR experi-\nment with off resonant optical excitation and a photoluminescence in tegration time T= 1µs.\nR can be off course increased in several ways. Firstly, R becomes 50 times larger when\nresonant optical excitation of V2 at low temperature is used, but t his is more complicated\nin practice (see appendix 4). Secondly, using averaging over 5000 c ycles of ODPELDOR\nexperiments, R is multiplied by 70, thus reaching R= 6300 for off reson ant excitation, which\ntakes in practice around 1 second (see appendix 4). Assuming Ncyc les=5000 per point and\na 100 points ODPELDOR spectrum as a function of fpump (1 point eac h 2 MHz, 200 MHz\nscanned), one could obtain such a 200 MHz ODPELDOR spectrum by o ff resonant opti-\ncal excitation (see next experimental section for an exemple of PE LDOR spectrum obtained\nwith a standard detection of EPR) in 100 s with a large signal to noise r atio R=6300, assum-\ning negligible hardware and software delays for changing the pumping microwave frequency\n(otherwise, the experimental time is determined by those delays). Thus the sensitive opti-\ncal, but indirect, detection of the paramagnetic resonance spect rum of Dark external spins\n(not photoluminescent paramagnetic centers or molecules) is poss ible by ODPELDOR spec-\ntroscopy. Note also that if the quantum sensor has many identical but isolated spin probes\nV2,NV2≥≥1, located at the same depth below the SiC sensor surface, then th e signal to\nnoise ratio is in principle also enhanced by a factor√NV2, but this last option for further\nimprovement seems difficult in practice (see appendix 4).\n17ODPELDOR quantum sensing with a SiC-YiG quantum\nsensor applied to structural biology\nThere are several ways to perform ODPELDOR quantum sensing, d epending whether the\nquantum sensor device presented here has a YIG nanostripe or no t. Also, in order to make\nthe discussion more explicit and as structural biology studies of tra nsmembranes proteins\nis expected to be an important application of this new quantum senso r, I will present the\nquantum sensing properties in the context of structural biology. The external spin consid-\nered here are thus spin labels, like nitroxide radicals,1gadolinium spin labels61and trityl\nradicals,56whose spin hamiltonians are well known.\nFigure 5: X band (9.369 GHz) EPR/ODMR spectrum of V2 in 4H SiC (in blac k) with\nan external magnetic field applied along the c axis of 4H-SiC, as well as a powder EPR\nspectrum of a/ nitroxide spin labels (red), b/ gadolinium spin labels (gr een), and c/ trityl\nspin labels (pink). Such powder spectrum is expected for spin labels in a frozen solution\nfor example. Blue squarres for probe microwave pulse and dashed d ark squarre for pump\nmicrowave pulse(s). Short pulses are spectrally broad by Fourier t ransform. Long pulse are\nspectrally selective. Multifrequency simultaneous microwave pumpin g is also now possible\nwith arbitrary waveform generators (AWG).\nSo let consider firstly the quantum sensor having one single V2 spin pr obe and no YIG\n18nanostripe. Asalreadyexplainedinthefirstsection, theODPELDOR pulsesequence(fig.1b)\nis similar to the older PELDOR pulse sequence implemented on a standar d pulsed EPR\nspectrometer. Before performing a two microwave frequencies O DPELDOR experiment in\ntime domain, one needs to record experimentally, or at least to nume rically simulate, the\nEPR spectrum of both the V2 spin probe (or several ones to get all the EPR lines) and of\nthe external spin labels. Such EPR spectrum in the first situation co nsidered here without\nany YIG nanostrip is numerically simulated at X band on fig.5, using Easy spin.62In this\nsimulation I considered many V2 all having the same uniaxial magnetic a nisotropy axis\noriented along the C axis of 4H-SiC, also assumed here to be the direc tion of the externally\napplied magnetic field. For the spin labels, a powder spectrum was simu lated with all\npossible orientations, providing broad spin labels EPR lines for nitroxid e radicals (fig.5a)\nand gadolinium radicals (fig.5b), which have anisotropic g tensor and/ or hyperfine coupling\ntensor. This is not the case for trityl spin labels, which have rather well defined magnetic\nparameters providing a narrow EPR line, typically of around 1 G at 70K in frozen water-\nglycerol.56It can be seen on fig.5a and 5b, that with nitroxyde radicals and gado linium\nspin labels, it is possible to excite a large part of their broad powder sp ectrum (dashed\ndark squarre), as it could be observed at low temperature, withou t exciting the V2 spin\nprobe. Inversely, it is possible to excite the V2 spin probe (blue squa rre) without exciting\nthe gadolinium spin labels. In the other important case of nitroxide ra idcals, which are the\nmost common radicals used in EPR based structural biology, one can excite the V2 spin\nprobe with a long selective resonant microwave pulse, thus minimally ex citing the nitroxide\nspin labels. The small amount of nitroxide radicals excited by this selec tive excitation\ncorrespond to a very diluted external spin bath of nitroxide radica ls (considering the very\nbroad EPR spectrum associated to all the nitroxide radicals), which is thus not expected to\nimpact signi���catively the spin coherence time of the V2 spin probe. On e can thus perform\nODPELDOR with such a SiC quantum sensor with nitroxide and gadolinium spin labels. In\nthe case of trityl radicals however, one can see on fig.5c that the n arrow EPR line of trityl\n19raidcals will spectrally overlap one of the EPR possible transition of th e V2 spin probe, the\ncentral one. Thus ODPELDOR is not possible with trityl radicals in this configuration with\na 4H-SiC quantum sensor which do not have a YIG nanostripe. For th e two favorable cases\nof nitroxide and gadolinium spin labels, the quantum sensing propertie s of a plane of spin\nlabels by a single V2 spin probe are numerically simulated on fig.6 for vario us experimental\nparameters.\n0 5 10 15 20 25 30 35\nd (n m)00.51\n0 5 10 15 20 25 30 35\nd (n m)00.51\n0 5 10 15 20 25 30 35\n 00.511 - V\n1 - V\n1 - V\nd (n m)1/ dx = 5 nm\n3/ dx = 15 nm2/ dx = 10 nm\nFigure 6: Dependence of the ODPELDOR normalized net signal to nois e ratio (see text),\n1-V, on the relative distance dx between the V2 spin probe and the p lane of external spin\nlabels (dx= 5 nm (1/) , 10 nm (2/), or 15 nm (3/), from top to bottom ), and on the target\nspin plane concentration ( C2D,target=1\nd2, with d in nm). Dark trace is for td=5µs, red trace\nis fortd=3µs, blue trace is for td=1µs. See fig.1 for the standard definition of td in the\n(OD)PELDOR sequence.\nIt is assumed on fig.6 that the spins labels are either anchored on lipids or on transmem-\nbrane proteins and located few nm above the SiC sensor surface, a ll in the same plane. If\nthe ODPELDOR normalized net signal to noise ratio, 1-V, is comprised between 0.5 and 1,\n20ODPELDOR is clearly feasible as explained in the previous section, wher eas if it is between\n0 and 0.5, it is much more difficult. A small inter-spin labels distance d valu e on fig.6 means\na large 2D concentration of spin labels in the target plane to sense. O ne can thus see on\nfig.6 that ODPELDOR is much easier to perform at large 2D concentra tion of spin labels,\nat smaller dx distance between the V2 and the plane of spin labels, and it is also easier if\nthe V2 has a long spin coherence time T2 allwoing to use a large value of t he time delay td\nin the ODPELDOR sequence (typically T2≥2td).\nIn practice, one would perform the quantum sensing ODPELDOR exp eriment described\nhere, firstly as a function of the microwave pump frequency, at fix edtdvalue, in order to\ndetect indirectly by the V2 photoluminescence, the EPR spectrum o f the external target\nspin labels. Then, one would perform a time domain ODPELDOR experime nt, varying td\nand using a microwave pump frequency appropriate to resonantly fl ip the external target\nspin labels. Then, the time domain decay curve obtained can be fitted with only dx and\nC2D ,target=1\nd2as fitting parameters. Note that this 2D concentration is in practic e often\nnot the total 2D concentration of spin labels, but the effective con centration if one excites\nonly one part of the target external spins spectrum, like in the cas e of nitroxide spin labels\nand gadolinium spin labels. Thus the factor associated to the propor tion of the spin labels\nEPR spectrum effectively pumped has to be taken into account in pra ctice.\nOne can note also that state of art commercial pulsed EPR spectro meters offer the pos-\nsibility to excite a given EPR spectrum over several parts by multifre quencies excitations,\nwhich is possible using an arbitrary waveform generator and which ca n provide a 2D con-\ncentration of spin labels effectivelly pumped close to the total 2D spin label concentration.\nPerforming such kind of time domain ODPELDOR experiments at sever al spin labels con-\ncentration C2D,target=1\nd2, it is thus possible to extract the parameter dx, directly related to\nthe important average insertion depth of the spin labelled proteins in the model 2D biomem-\nbrane. In order to really extract this average insertion depth of t he spin labelled proteins in\nsuch kind of SiC quantum sensor without a YIG nanostripe on the bac kside of the photonic\n210 500 1000 1500 2000 2500 3000 3500 4000 4500 5000\ntd (ns)00.20.40.60.81VDeer\nFigure 7: Time domain signal Vdeer(td), expected at dx=6nm (cont inuous line), and at\ndx=8nm (dashed dot line), each time for three different 2D externa l spins concentration:\nC2D,target=1\n(5nm)2,C2D,target=1\n(7nm)2,C2D,target=1\n(9nm)2. The lower C2D,targetis, the\nslower is the decay.\nwaveguide, a first solution consist in taking advantage of the residu al paramagnetic surface\ndefects at the SiC surface as a mean to determine the exact depth of the V2 spin probe below\nthe surface of SiC. Or, alternatively, one can fabricate on top of t he SiC surface a very thin\nSiO2 sacrificial layer on top of which a sacrificial YIG nanostripe can b e fabricated, whose\nmagnetic field gradient can encode the depth of the V2 on its resona nt frequency. Once\nthe depth of the V2 has been determined by ODMR of the V2 spin prob e under the strong\nmagnetic field gradient of the YIG nanostripe, this YIG nanostripe a nd the SiO2 ultrathin\nlayer can be removed by HF etching of the SiO2 layer. Thus simple 4H-S iC quantum sensors\nwithout a YIG nanostripe could be provided with a datasheet indicatin g the V2 depth below\nthe SiC surface and its T2 spin coherence time with a free SiC surface . Note that the case\nof a SiC quantum sensor having an ensemble of V2 spin probe and no YI G nanostripe may\nhave a larger signal to noise ratio but will probably have a lower spatia l resolution due to\nthe depth distribution expected by the ion implantation process (se e appendix 4).\nAsecond, very importantkindofquantumsensor proposedhere, istheSiC-YIG quantum\n22sensor containing a single V2 spin probe and a permanent YIG nanost ripe fabricated on the\nbackside ofthe1Dphotonicwaveguide, corresponding tothecase described inthefabrication\nprocess represented on fig.3. The experimental configuration he re corresponds to an external\nmagnetic field applied perpendicular to the c axis of 4H-SiC and here all spins are submitted\nto the strong gradient of dipolar magnetic field produced by the nea rby YIG nanostripe. The\nsimulated X band EPR/ODMR spectrum of the V2 and of the possible ta rget spin labels,\nwithout (left) and with(right) this strong gradient is described on fi g.8. This figure clearly\nshows that in this configuration and with the YIG nanostripe, it is now possible to perform\nODPELDOR spectroscopy with a single V2 and a plane of trityl spin labe ls because the\nresonant magnetic fields of the V2 and of the trityl can be spectra lly distinguished due to\nthe different YIG induced dipolar magnetic field experienced by those spins.\nThe detailed magnetic properties of this SiC YIG quantum sensor hav e been already\ndiscussed inaprevious section. Briefly, itwas shown, thatdue toth estrongdipolarmagnetic\nfield gradient produced by the YIG nanostripe, of 0.5 G per nm, and a ssumingT2,V2≥10µs\nfor a single isolated V2, and T2,trityl≥4µsfor a single isolated trityl spin label at a\ntemperature equal or below 100 K, then one can determine the x co ordinate of the V2 spin\nprobewithaspatial resolutionofaround1 A◦byODMRoftheV2under thisstronggradient,\nand the x coordinate of the trityl with a spatial resolution of aroun d 2A◦by ODPELDOR\nspectroscopy under this strong gradient, assuming a single trityl is present in the biolgical\nsample above the V2 spin probe. Thus ODPELDOR under the strong g radient of the YIG\nnanostripe of the quantum sensor allows to indirectly detect and loc ate with a high precision\nalong x a single trityl, through the measurement at a given pump freq uency of an oscillatory\nODPELDOR time domain signal (like the one obtained in standard DEER, but here with\njust two spins involved).\nOf course, it could be interesting to peform ODPELDOR spectrosco py with such an hybrid\nSiC-YIG quantum sensor having a doubly spin labelled single protein loca ted just above\nthe top SiC surface, inside a model 2D lipid bilayer or lipid nanodisc. Using the simple\n23Figure 8: X band (9.369 GHz) EPR/ODMR spectrum of V2 in 4H SiC (black ) with an\nexternal magnetic field applied perpendicular to the c axis of 4H-SiC, as well as a powder\nEPR spectrum of a/ nitroxide spin labels (red), b/ gadolinium spin labels (green), and c/\ntrityl spin labels (pink); left side: without the YIG nanostripe gradie nt, right: with the YIG\nnanostripe gradient. Such powder spectrum is expected for spin la bels in a frozen solution\nfor example. Blue squarres for probe microwave pulse and dashed d ark squarre for pump\nmicrowave pulse(s). Short pulses are spectrally broad by Fourier t ransform. Long pulse are\nspectrally selective. Multifrequency simultaneous microwave pumpin g is also now possible\nwith arbitrary waveform generators (AWG).\nODPELDORsequence describedonfig.1butunderthestrongdipolar magneticfieldgradient\nof the YIG nanostripe, then one can access, as explained just abo ve, to the X1cooordinate\nof trityl 1 and also here to the X2coordinate of trityl 2, through the value of the resonant\npump microwave frequencies at which one observe an oscillating ODPE LDOR signal. But\ntwo other bio-structural informations can be obtained through t he fourier transformation of\nthose two oscillating time domain ODPELDOR signal, because the oscillat ion period gives\naccess to the dipolar spin-spin couplings and thus to the two relative distances between the\nV2 spin probe and the two trityl spin labels anchored on the target p rotein under study,\ncalledRV2,Tri1andRV2,Tri2. Thus, ODPELDOR spectroscopy alone applied on such doubly\nspin labeled protein in a model lipid membrane provides already four con straints ( X1,X2,\n24RV2,Tri1,RV2,Tri2) on the 3D model of insertion or interaction of the protein with the m odel\nlipid biomembrane.\nConclusion\nIn this article, I have presented the theory of a new SiC-YiG hybrid q uantum sensor. I have\ndescribed a complete fabrication process, taking advantages of io n implantation methods\nand available SiC technologies, allowing to produce such a quantum sen sor. The sensor has\na single V2 negatively charged silicon vacancy color center introduce d inside a 4H-SiC 1D\nnanophotonic waveguide based on defects in a 2D photonic crytsal. The waveguide is ap-\npropriately designed to waveguide efficiently the V2 spin probe photo luminescence towards\nan optical fiber, interfacing the pulsed EPR spectrometer with an o utside standard photo-\nluminescence setup. I showed that adding a YIG ferrimagnetic nano stripe on the backside\nof the SiC photonic structure is important to determine precisely th e depth of the V2 spin\nprobe below the SiC surface with angstrom resolution. I also showed that under optimal\nconditions, and even under off resonant optical excitation of the V 2 photoluminescence,\nthis SiC-YIG quantum sensor should allow the sub-nanoscale investig ation of a single trityl\nspin labelled transmembrane protein by optically detected pulsed elec tron electron double\nresonance spectroscopy, with a large signal to noise ratio obtaine d in just one second of\nmeasurement per point, and using a simple standard X band pulsed EP R spectrometer up-\ngraded by this quantum sensor device and a simple optical fiber. This SiC-YiG quantum\nsensor should thus be of great interest for all the biophysicists, c hemists and physicists which\nare already worldwide pulsed EPR user and who wants to reach the sin gle spin sensitivity\nand sub nanoscale spatial resolution offered by quantum sensing me thodologies, just slightly\nmodifying their experimental EPR setup. The next challenges are th e fabrication of the\nproposed 4H-SiC-YIG hybrid spin-photonic-magnonic structure w ith state of art available\nSiC technology and ion implantation methods, as well as the experimen tal demonstration of\n25fiber based preparatory experiments combining an external optic al setup and a standard X\nband pulsed EPR spectrometer, both interfaced by an optical fibe r.\nAcknowledgments\nThe author thanks the University of Strasbourg and the french C NRS for the reccurent\nresearch fundings.\nAppendices\nAppendice 1: Ion implantation fabrication process.\nIt was recently demonstrated that low energy C+ ions ensemble impla ntation in 4H-SiC\nthrough an array of 65 nm diameter apertures patterned on a 300 nm thick PMMA layer\nusing electron beam lithography allows to produce, statistically, an a rray of isolated single\nsub surface V2 silicon vacancies (see main text), having, in statistic al average, a depth of 42\nnm and a longitudinal straggling of about 35 nm.\nThe idea here is thus to produce a mask having a single narrow apertu re and thus a single\naccess to the SiC substrate for implantation, to chose the energy of C+ ions implanted, the\nthickness of a sacrificial barrier layer for implantation, and also to c hose the dose of C+\nions implanted, such that, in statistical average, one single V2 color center can be produced\nper aperture and with a control over the statistical average dep th of the V2 color center. If\none also targets a lateral precision better than+/- 10 nm, this cor responds to a cylindrical\naperture in the mask used for ion implantation having a diameter equa l to 20 nm, which is\naccessible by state of art electron beam lithography. The implanted carbon ions profile and\nthe silicon vacancy concentration profile, as predicted by the SRIM software and assuming\n30 keV C+ ions implantation (with 7 degree of tilt) in a trilayer made of Zn O (60 nm), SiO2\n260\r 50\r 100\r 150\r 200\r 250\r0,0\r2,0x10\r4\r4,0x10\r4\r6,0x10\r4\r8,0x10\r4\r1,0x10\r5\r1,2x10\r5\rnormalized C+ concentration (cm\r-1\r)\r\ndepth (nm)\r120\r 140\r 160\r0,00\r0,01\r0,02\r0,03\r0,04\r0,05\rVSi number / (A˚ - ion)\r\ndepth (nm)\r\nFigure 9: Implanted carbon ions profile (left) as predicted by the SR IM software and\nassuming 30 keV C+ ions implantation (with 7 degree of tilt) in a trilayer m ade of ZnO (60\nnm), SiO2 (60 nm) and 4H-SiC (infinity); silicon vacancy concentratio n profile in 4H-SiC\n(right).\n(60 nm) and SiC (infinity), are presented on fig.9. Further assuming a dose of 30 keV C+\nions implanted equal to 4 .41012cm−2, as well as a decimation factor equal to 0.01 (to take\ninto account the more realistic amount of silicon vacancies created in practice compared to\nthe one calculted by SRIM, possibly due to interstitial-vacancy reco mbination processes or\ndefects complex formation), then I find that in statistical averag e, one single V2 spin probe\nis created through an aperture of 20 nm of diameter, the other C+ ions being stopped by\nthe 300 nm PMMA film. This also implies, in view of the Poisson Statistic exp ected, that if\none produces in parallel 100 similar quantum sensors on the same 4H- SiC wafer, 37 quantum\nsensors fabricated will have exactly 1 single V2 in the aperture, 37 w ill have 0 V2, 18 will\nhave 2 V2, 6 will have 3 V2,... Also, taking into account the V2 depth dist ribution profile\nof fig.9 (right), one can also calculate the probability that the V2 spin probe be contained\nin a thin layer of SiC of 2.5 nm thickness, as obtained by SRIM. Taking th e beginning of\nthe 4H-SiC material (absolute position 122.5 nm) as the origin of dept h, I find here P(0;\n272.5 nm)=0.21, P(2.5; 5 nm)=0.17,P(5; 7.5 nm)=0.14,P(7.5; 10 nm)=0.12, ... Thus one can\nconclude that with the fabrication process proposed here, which is statistical in essence, over\n100 quantum sensor fabricated in the same way, one will obtain arou nd 37∗0.14 = 5\nquantum sensors having a single V2 in their aperture located above t he YIG nanostripe, and\nwith a depth comprised between 5nm and 7.5 nm. Note also that at the end of this method\nof implantation through a sacrificial layer, this sacrificial layer has t o be removed. Here,\nHCl can be used to remove the ZnO sacrificial layer, while not etching the SiO2. Then HF\netching can be used to remove the SiO2 sacrificial layer. Note that t he SiO2 sacrificial layer\nbelow the dense ZnO layer is necessary to stop the Zn ions impacted b y the C+ ions im-\nplanted, avoiding Zn contamination of the SiC top surface. As an alte rnative to 30 keV C+\nions implantation through a single aperture in PMMA over such trilayer (ZnO/SiO2/SiC),\none could also perform a similar implantation but through a bilayer made of a thicker SiO2\nsacrificial layer on SiC, targeting a maximum carbon ions and V2 conce ntration close to the\nSiO2/SiC interface and then etching the SiO2 by HF. One could also per form direct 5 keV\nC+ ions implantation in 4H-SiC through a single PMMA aperture of diamet er 20 nm over\nSiC at a dose of 11011cm−2. SRIM simulation then predict in average 105 silicon vacancies\nper aperture, but taking into account once again the decimation fa ctor equal to 0.01, one\nexpect in practice in this case a single V2 spin probe per aperture. SR Im simulation (not\nshown here) also predict that the single silicon vacancy will be create d with a maximum\nprobability at zmax=6nm.\nAppendice 2: homogeneous linewidth and spin coherence time for the isolated V2 in\n4H-SiC.\nAn ensemble of V2 spins in 4H-SiC can already have a narrow inhomogen eous linewidth\nof few Gauss or even less in some previous reports. A thin epitaxially g rown isotopically\npurified layer of SiC at the top surface of the 4H-SiC device propose d could probably further\n28reduce this inhomogeneous linewidth. But, a single isolated V2 spin pro be in 4H-SiC is\nexpected to have an homogeneous linewidth of less than 1 G. In princ iple, the homogeneous\nlinewidth should be inversely proportional to the T2 spin coherence t ime of the isolated V2\nsingle spin probe. A room temperature spin coherence time for bulk ( isolated or diluted) V2\nof around 50 µsare often measured in bulk 4H-SiC HPSI, corresponding to a homoge neous\nlinewidth of around 7 mG, and previous studies reported or estimate d the T2 of such bulk\nisolated V2 to more than 300 µ sat room temperature. A much longer T2 is expected at\nlow temperature at few Kelvin, because it is ultimately limited by T1 relat ed longitudinal\nrelaxation processes, and the T1 value of isolated V2 has been show n to exceed few minutes\nat few Kelvins (see main text). Indeed, using dynamical decoupling p ulses sequences at low\ntemperature, it was recently experimentally demonstrated that t he effective T2 obtained us-\ning this methodology can reach 10 msat a temperature of few K. In practice however, for sub\nsurface V2 created by such a fabrication process, the T2 is expec ted to be much smaller due\nto the residual surface defects density, but this remained unexp lored to date and is largely\ndependent on the material and fabrication methodology used. A re sidual surface density of\nstate of 41010cm−2, correspond to an average lattice of surface defects, separat ed in plane\nby 50 nm. In the proposed quantum sensor, the V2 spin probe could be placed around 6 nm\nbelow the 4H-SiC surface. That means that over the many SiC-YiG de vices identically fabri-\ncated in parallel, there should be some of them having all the required properties, including\nsome with a long spin coherence time due to their large distance to the nearest fluctuating\nsurface defects. Note also that in all cases, it is in principle possible t o reduce the magnetic\nnoise due to surface defects by cooling down the sample to sufficient ly low temperature, thus\nincreasing their T1 relaxation time.\nAppendice 3: Optical bandgap of TM modes of the 2D photonic cryst al\nHere I present a numerical simulation of the optical bandgap prope rties of a 2D photonic\n29crystal, assumingthefollowingparameters: therefractiveindexu sedare,nair= 1,nSiC= 2.5\nandr= 0.29∗a, withrtheradiusofairholesandatheinter-center distancebetwe en neighbor\nair holes in the triangular lattice.\n00.10.20.30.40.50.60.70.80.91\nGammaGammaK M\nFigure 10: Numerical simulation of the TM modes dispersion relation an d optical bandgaps\nfound in a triangular lattice of cylindrical air holes in 4H-SiC. The norma lized frequency\nω=a\nλis plotted as a function of the in plane photon wave vector considere d. The horizontal\naxis indicates the in plane wavevector of the photon, along the direc tion ΓK,K M, and\nMΓ. Parameters: refractive index nair= 1,nSiC= 2.5 andr= 0.29∗a, with r the radius of\nair holes and a the inter-center distance between neighbor air holes in the triangular lattice.\nOne can see on fig.10 that a single isotropic very narrow optical band gap for TM modes\nexist for a normalized frequency ω=a\nλ= 0.680. Thus a good design for this 2D photonic\ncrystal consist in fabricating a triangular array of air holes in SiC, wit h a center to center\ninter-hole distance equal to a= 915∗0.680 = 622 nm, the air holes having a diameter of\n2∗r= 2∗0.29∗622=360 nm. This is quite feasible with available electron beam lithography\nand SiC RIE etching methods (see main text). One can also see on fig.1 0 that two larger but\npartial optical bandgap exist for TM modes propagating along the K Mdirection in such a\n2D photonic crystal, at normalized frequencies ω=a\nλ= 0.467 andω=a\nλ= 0.345. Thus if\nfor example, one define a line of defects in this 2D photonic crystal a long the direction Γ M,\nwhich is perpendicular to the K Mdirection, then one expects some photon confinement in\nthe direction perpendicular to this 1D photonic waveguide, and thus a still interesting col-\nlection efficiency for the V2 spin probe photoluminescence. More pre cisely, considering that\n30a photon is emitted by the V2 with an in plane wave vector forming an an gleθwith the Γ M\ndirection of this second kind of 1D photonic waveguide, then the pro bability of collecting\nthis photon in the direction of the waveguide is equal to ( cos(θ))2. The probability that the\nphoton is emitted in the orthogonal KM direction is ( sin(θ))2, and in this case it could be\nreabsorbed by the V2 spin probe and then reemitted in another dire ction more appropriate\nfor collection by the 2D waveguide. Thus, integrating over the θrange, one finds a collection\nefficiency of at least 0.5 for this 1D photonic waveguide build in a 2D phot onic crystal hav-\ning an anisotropic optical bandgap for TM modes. This is of course les s than the complete\ncollection (collection efficiency equal to 1) which is possible with the tru e isotropic optical\nbandgap case, but it is still interesting and offer an alternative optic al nanophotonic design.\nOne possible advantage here is the larger optical partial bandgap, maybe more appropriate\nfor room temperature operation where the V2 photoluminescence is broadened by phonon\nreplica. In the case of a partial optical bandgap case for TM modes withω=a\nλ= 0.467,\none thus has to fabricate a triangular array of air holes in SiC, with a c enter to center inter-\nhole distance equal to a= 915∗0.467 = 427 nm, the air holes havind a diameter of\n2∗r= 2∗0.29∗427=248 nm. This is also feasible with available electron beam lithography\nand SiC RIE etching methods. In the third case of a partial optical b andgap case for TM\nmodes with ω=a\nλ= 0.345, one thus has to fabricate a triangular array of air holes in SiC,\nwith a center to center inter-hole distance equal to a= 915∗0.345 = 316 nm, the air\nholes havind a diameter of 2 ∗r= 2∗0.29∗316=184 nm. This is still feasible with available\nelectron beam lithography and SiC RIE etching methods, but maybe it is mechanically less\nrobust.\nAppendice 4: Signal to noise ratio of the ODPLEDOR experiment\nAs already said above, coupling strategies between such a 1D photo nic waveguide and an\noptical fiber exist, with demonstrated coupling efficiencies comprise d between 0.25 and 0.90.\n31In the following, I will thus assume a V2 photoluminescence collection e fficiency pcollcom-\nprised between 0.125=0.25*0.5 (case of partial optical bandgap) a nd 0.90=0.90*1 (isotropic\noptical bandgap), thus pcoll= 0.51 in average, further assuming no loss of photons be-\ntween the SiC-YIG quantum sensor and the photo-detector. The photo-detector is assumed\nhere to have a quantum efficiency for near infrared photon detect ion at 915 nm of around\npdet= 0.4. Also, one has to note that the excitation efficiency of the photolu minescence of\nthe V2 spin probe is proportionnal to a wavelength dependent term (related to the absorp-\ntion cross section) and to the photoluminescence quantum yield, giv en quite generally by\nΦ =krad\nkrad+knot−rad, withkradthe rate of radiative recombination of the V2 spin probe from a\ngiven excited state and knot−radthe rate of non radiative recombination from a given excited\nstate, which includes the inter-system crossing rate. The V2 spin p robe color center has a\nspin3\n2, andthegroundandexcited electronicstatearebothsplittedby t heso calledzerofield\nsplitting in the ground and excited states. Thus, optical transition s noted L (like low value\nof ms) between GS (ms=+/-1/2) and ES (ms=+/-1/2) can be disting uished in energy from\nthe optical transitions between GS (ms=+/-3/2) and Es (ms=+/-3 /2), noted H (like high\nvalue of ms), even without application of a magnetic field. Thus, the e xcitation efficiency\nof the V2 spin probe can become spin state dependent by two effect s: either by spin state\nselective optical excitation of the V2 spin probe at the end of the DE ER sequence (after the\nthird microwave pulse at fprobe), which is possible by energy selective resonant excitation of a\ngiven narrow optical transition by a narrow linewidth laser, or by the fact that the quantum\nyield is spin state dependent for the V2 spin probe, due to the spin de pendent intersystem\ncrossing rate of the V2 spin probe. I now introduce the following not ations,/angbracketleftΦ/angbracketright=ΦH+ΦL\n2,\nand 2X=ΦH−ΦL\n/angbracketleftΦ/angbracketright, taking into account the two possible different photoluminescence q uan-\ntum yield. Note also that it was previously shown (see main text) that the optical power\nnecessary to obtain saturation values of optical V2 spins pumping is inversely proportional\nto their longitudinal spin-lattice relaxation time T1. AsT1increases up to several tens of\nsecond at 5K, then less than 1 mW at 780 nm spread over a 1mm*1mm s quare sample is\n32sufficient at 5K for obtaining optical pumping saturation, but more a t room temperature.\nWith the above remarks, one can estimate the signal to noise ratio R of an ODPELDOR ex-\nperiment, asdescribedbythesequence onfig.1b, assuming tdisfixedandthepumpfrequency\nfpumpvary. The photoluminescence signal Splin ODPELDOR is expected to vary depending\non the value of the microwave pump frequency chosen, because wh en the target spins are\nmicrowave manipulated on resonance, the V2 spin probe feels an acc elerated spin echo decay\nat a given tdparameter value. In optimal experimental conditions, the Noise is d ominated by\nthe optical shot noise, and is given by Npl=/radicalbig\nSpl(pB= 0), where pB= 0 means that the\nmicrowave pump frequency is off resonant with the resonant frequ ency of the target external\nspin(s).pB= 1 means on the contrary, that the microwave pump frequency is r esonant with\nthe resonant frequency of the target external spin(s). Thus, the net signal to noise ratio R\nin ODPELDOR spectroscopy is given by the formula R=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(Spl(pB=1)−Spl(pB=0))√\nSpl(pB=0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nUsing the normalized DEER signal expression (see main text), Vdeer, whose value is com-\nprised between 0 and 1, directly related to the ODPELDOR experimen t shown on fig.1b,\nand considering the effect of the last additional −π\n2microwave pulse, which converts the\nV2 spin quantum coherence into a V2 spin state populations differenc e, one obtains the\nfollowing expression for the signal to noise ratio Rin the case where ODPELDOR is ob-\ntained by off resonant optical excitation of the V2 spin probe, at 78 5 nm for example:\nR=Ropt(1−Vdeer(td,dx,C2D,target))X, and in the case where ODPELDOR is obtained\nby a spin state selective resonant optical excitation of the V2 spin p robe (assumed on the\nHH optical transition), one finds R=Ropt(1−Vdeer(td,dx,C2D,target)), withRoptgiven by\nthe formula: Ropt=exp/parenleftbig\n−2t0\nT2/parenrightbig/radicalBig\npcollpdetσ\nAP0T\nhν/angbracketleftΦ/angbracketright.\nσis the absorption cross section of the V2 spin probe, Ais the area on which the optical\npowerP0is sent,hνis the photon energy, and Tis the integration time of the photolumines-\ncence by the photodetector over one single cycle. Xis given above, and has a value close to\n0.02 at room temperature according to previous ODMR experiments on V2 with off resonant\nexcitation of photoluminescence. Thus one see immediatly that R can be greatly improved\n33by resonant spin state selective optical excitation of the photolum inescence of the V2 spin\nprobe, on the well defined Zero phonon line (ZPL) at low temperatur e, but off resonant exci-\ntation is much more convenient in practice, particularly at high tempe rature or even at room\ntemperature. Vdeercan be numerically computed using the linear approximation and shell\nfactorization model (see main text). This model was previously intr oduced for calculating\nthe standard DEER time domain signal in the case of a three-dimensio nal distributions of\nspins. Here, this model has been adapted to take into account the bidimensional random\ndistribution of the target external spins in their well-defined plane, parallel to the SiC sensor\nsurface.\nNow, assuming a sensor operating with t0= 6.25µ sand 2t0=T2= 12.5µ s, as-\nsumingC2D,targetis sufficiently large for a given td, such that Vdeer(td,dx,C2D,target) = 0 ie\n1−Vdeer= 1 (see main text for examples),further assuming, /angbracketleftΦ/angbracketright ≈1,pcoll= 0.5,\npdet= 0.4,σ\nA≈1, and choosing a photoluminescence integration time per ODPELDOR\nsequence T= 1µsfor example and an optical power at 780 nm of 20 µ Wspread over\nA≈3λ\n2λ\n10, according to the previous nanophotonic structure design, one fi nds approx-\nimatelyRopt= 5000, and thus, in the case of off resonant optical excitation of t he V2\nspin probe, R= 5000 (1 −0) 0.02≈90 for a single one shot one point ODPELDOR\nexperiment with off resonant optical excitation. R can be off course 50 times larger for\nresonant optical excitation at low temperature. The optical re-p umping time of V2 spins\nis short in practice with an appropriate laser, typicaly of around 150 µs, and much less at\nlow temperature. The ODPELDOR microwave pulses sequence after optical initialization\nof V2 spins last around 20 µs, such that the shot repetition time of the full ODPELDOR\nsequence can be taken equal here to Texp= 200µs, meaning one can perform 5000 cycles\nof averaging in one second, which further improve the signal to nois e ratio by a factor of\n√\n5000≈70. As a consequence, for a SiC quantum sensor having a single V2 sp in probe,\none finds, assuming averaging over 5000 cycles ie over 1s of experim ent in practice, a signal\nto noise ratio equal to R1sec,off optex = 6300 for off resonant excitation of the photolumines-\n34cence, while for resonant optical spin state selective excitation of photoluminescence at low\ntemperature, one finds R1sec,onoptex = 315000. In both case, ODPELDOR is clearly feasible\nand should have a high signal to noise ratio. Even if one consider some pessimistic values\nlikepcoll= 0.01 with some photon loss betwen the quantum sensor and the photo detector,\nand a low efficiency detector with pdet= 0.04, then one still finds for one second of averaging\n(5000 cycles) a signal to noise ratio of R1sec,off optex = 286. Assuming Nshot=5000 per\npoint and a 100 points ODPELDOR spectrum as a function of fpump (1 point each 2 MHz,\n200 MHz scanned), one could obtain such a 200 MHz ODPELDOR spect rum by off resonant\noptical excitation in 100 s with a signal to noise ratio R comprised betw een 286 and 6300,\nassuming negligible hardware and software delays for changing the p umping microwave fre-\nquency (otherwise, the experimental time is determined by those d elays). Thus the optical,\nbut indirect, detection of Darkexternal spins (not photoluminesc ent paramagnetic centers or\nmolecules) is possible by ODPELDOR spectroscopy. Those estimates also assume a photon\nshot noise limited noise. If some photoluminescence background is pr esent in the SiC-YIG\nquantum sensor device, of course the signal to noise ratio will be low er. But according to\nthe above estimates, it seems clearly possible to tolerate some amou nt of photoluminescence\nbackground for performing ODPELDOR spectroscopy as long as on e has sufficiently good\nphotodetection and photon collection efficiencies, and further usin g averaging. Note also\nthat the large signal to noise ratio estimated above also assume exp erimental parameters\nsuch that Vdeer(td,dx,C2D,target) = 0, but of course, values of Vdeermuch larger, between 0.5\nand 1, meaning a smaller decoherence effect of the spin bath on the V 2 spin probe, could\nalso be measured in the optimal situation of an optical shot noise limite d measurement.\nAs a last remark, it has to be also noted that if the quantum sensor h as many identical\nbut isolated spin probes V2, NV2≥≥1, located at the same depth below the SiC sen-\nsor surface, then the signal to noise ratio is in principle enhanced by a factor√NV2if it\nremains shot noise limited. 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Stoll et al., Journal of Magnetic Resonance 178, 42 (2006).\nCompeting financial interests\nThe author declare that he has no competing financial interests.\n39" }, { "title": "1906.09318v1.Magnetic_domains_without_domain_walls__a_unique_effect_of_He__ion_bombardment_in_ferrimagnetic_Co_Tb_multilayers.pdf", "content": "Magnetic domains without domain walls: a unique effect of He+ ion bombardment in \nferrimagnetic Co/Tb multilayers. \nŁukasz Frąckowiak1, Piotr Kuświk1, Gabriel David Chaves -O’Flynn1, Maciej Urbaniak1, \nMichał Matczak2, Andrzej Maziewski2, Meike Reginka3, Arno Ehresmann3, and \nFeliks Stobiecki1 \n \n1 Institute of Molecular Physics, Polish Academy of Sciences, Poznań, Poland \n2 Faculty of Physics, University of Białystok, Białystok, Poland \n3 Institute of Physics and Center for Interdisciplinary Nanostructure Science and \nTechnology (CINSaT), University of Kassel, Kassel, Germany \n \nAbstract \n \nWe show that it is possible to engineer magnetic multi -domain configurations without \ndomain wall s in a prototypical rare earth/ transition metal ferrimagnet using keV He+ ion \nbombardment. We additionally shown that these patterns display a particularly stable \nmagnetic configuration due to a deep minimum in the free energy of the system which is \ncaused by flux closure and the corresponding reduction of the magnetostatic par t of the \ntotal free energy. This is possible because light -ion bombardment differently affects an \nelements relative contribution to the effective properties of the ferrimagnet. The impact of \nbombardment is stronger for rare earth elements. Therefore, it is possible to influence the \nrelative contributions of the two magnetic subsystems in a controlled manner. The \nselection of material system and the use of light -ion bombardment open a route to engineer \ndomain patterns in continuous magnetic films much smalle r than what is currently \nconsidered possible. \n \nIntroduction \n \nThe ability to create lateral magnetic domain patterns is at the heart of a manifold of \napplications. Their use in magnetic mass memories [1–4] is absolutely straightforward but \nalso in other areas, such as magnonics [5,6] , or for the formation of defined domain \npatterns used for magnetophoresis in lab -on-a-chip devices [7–11] magnetic domain \nengineering forms a basic technology. For such applications, it is common to use \nferromagnetic layers. In these m aterials, magnetic domains are uniformly magnetized \nregions, in which the effective magnetization points in a defin ite direction. Naturally \noccurring domain patterns are formed by free energy minimization, usually as a \ncompromise between exchange, anisotropy and stray field energy terms [5,7,12,13] . The \ndomains are separated by domain walls (DWs) which are the transition regions where the \nmagnetic moment reorients from the direction within the first domain to the direction \nwithin the s econd. DW geometries depend on the ratio between the exchange coupling and \nthe anisotropy constants and typically consist of a narrow core and comparably wide \ntails [12]. Their widths constitute the natural size limit for individual domains. Therefore, \nthe lateral DW widths also constitute the critical dimension for magnetic domain \nengineering in continuous layers. Domain patterns can be engineered by local modificatio n \nof magnetic properties such as the coercive field ( HC) [14–17] or the exchange bias \ncoupling of systems composed of ferromagnetic and antiferromagnetic layers [18–21]. In \nthe past this has been achieved, e.g., by light-ion bombardment through masks [18–\n20,22,23] , by focused ion beams [24–27], by direct la ser writing [28], or by thermally \nassisted scanning probe lithography [29]. Walls between magnetic domains engineered by \nthese methods are usually non -symmetric [30] with respect to their center due to different \nanisotropies on t he two sides of the wall. However, even those methods will not be able to \nengineer domains of lateral dimensions below the respective (average) DW widths. Here \nwe describe a ferrimagnetic material system in combination with a method to engineer magnetic do main patterns without lateral DWs. This unique combination promises \nmagnetic domains in continuous layer systems of dimensions well below the typical \nferromagnetic DW widths. \nThe fundamental physics of magnetic domain formation in ferrimagnetic films is similar to \nthe one in ferromagnetic films [12], the occurrence of two magnetic moment subsystems , \nhowever, results in more involved domain formation effects . Layer systems consisting of \nrare earth (RE) transition metal (TM) alloyed layers, with alternating stoichiometric \ndomination of RE (RE+) or TM (TM+) will contain interfacial DWs at saturation [31–37] \n(Fig.1a) . This peculiar situation is possible because parallel effective magnetizations (black \narrows in Fig. 1 ) in the RE+ and TM+ layers correspond to antiparallel magnetic moments \nof the magnetic subsystems (red and blue arrows in Fig. 1) of the same type (RE or TM) in \nthe two homogeneous different layers [34]. Recently Li and coworkers investigated RE -\nTM alloy films with inhomogeneous concentrations of Tb [38] [Li2016] whose \nmagnetization reversal characteristics have also been explained by the existence of two \nnanoscale amorphous phases in a TbFeCo film with differing Tb concentration. \nHere, we demonstrate that 10 keV He+ ion bombardment allows to modify the magnetic \nproperties of ferrimagnetic Co/Tb multilayers that exhibit perpendicular magnetic \nanisotropy [39–43]. In particular, we show that with increasing dose of He+ ions the Tb \nmagnetization decreases much stronger than the Co one. This finding opens a way to \npattern RE+ ferrimagnetic films by light -ion bombardment through a mask or by light -ion \nbeam writing to locally reverse the domination from RE+ to T M+ and therefore engineer \nmagnetic domains without DWs in the two magnetic subsystems (Fig. 1b). Using this \npatterning technique, we fabricate a laterally periodic domain pattern consisting of a lattice \nof low HC TM+ squares embedded in a high HC RE+ grid (later referred to as matrix). \n \nResults and discussion \nThe subjects of our investigations are Tb/Co multilayers displaying, for small sublayer \nthicknesses, magnetic properties similar to amorphous Co -Tb alloy films [39–43]. In order \nto determine the influ ence of the 10 keV He+ ion bombardment on the properties of the \n(Tb/Co) 6 multilayers as a function of the thickness ratio between Co and Tb layers, i.e. as a \nfunction of the effective multilayer composition a particular layer system was deposited. In \nthis layer system, the nominal thickness es of the Co sublayers w ere fixed at tCo = 0.66 nm \nand the Tb sublayers were deposited as wedges with thicknesses 0 tTb 2 nm. The \nsample was bombarded with the two different He+-ion doses D of 1x1015 and 3x1015 \nions/cm² (see description in methods). The c haracterization of magnetic properties was \nperformed using a Magnetooptical Kerr Effe ct (MOKE) magnetometer in polar \nconfiguration with a probing -light wavelength of 640 nm. Fig. 2 shows changes of the \ncoercive field as a function of the Tb sublayer thickness ( HC(tTb)) for an unbombarded area \nand two areas bombarded with D = 11015 He+/cm2 and D = 31015 He+/cm2. The \nsingularit ies in the curve s HC(tTb;D) correspond to the Tb layer thicknesses tTb and the \nassociated effective Tb concentration cTb at which the magnetic moments of Co and Tb \ncompensate each other. It is easily seen that these values increase with increasing D. \nNote that the hysteresis loops for systems with Tb and Co domination have opposite \norientations. This occurs because for the light wavelength used in the MOKE set up the Co \nmagnetic subsystem determines the sign of the magnetooptical signal; the Co magnetic \nmoments are parallel to the net magnetization in Co dominated films, and antipara llel in Tb \ndominated films [39,44 –46]. After ion bombardment with D = 11015 He+/cm2 (Fig. 2c), \nthe hysteresis loop still has an orientation indicatin g the dominance of the Tb magnetic \nsubsystem; however, HC has a higher value than in the as -deposited state. Increasing D to \n31015 He+/cm² (Fig. 2d) results in a modification of the layer system such that the Co \nmagnetic subsystem starts to dominate for T b layer thicknesses tTb 1.6 nm, i.e. the Tb \nmagnetic subsystem is modified more than the Co one by the He+-ion bombardment. This is an important result, paving the way for an engineering of magnetic patterns without \nDWs. \nTo prove such a possibility , we performed local He+ ion bombardment through a resist \nmask with two doses D = 11015 He+/cm2 and D = 31015 He+/cm2 (see methods and \nsupplementary material) for a selected Tb sublayer thickness of 1.1 nm and studied the \nmagnetization reversal of the magneti cally patterned (Tb -1.1nm/Co -0.66nm) 6 multilayer . \nFor each dose f our 1x1mm² areas were patterned on the same sample with periodic ally \narranged squares of side lengths a = 3, 12.5, 25, and 100 m and distances between the \ncenters of neighboring squares of 2a (see Fig. S1 in supplementary materials). The squares \nhave been modified by ion bombardment, the rest of the samp le (matrix) remained \nunchanged. \nFull and minor P -MOKE hysteresis loops for both doses and in all patterned areas are \nshown in Figs. 3a, 3b, the magnetic moment configurations of the two magnetic \nsubsystems of the ferrimagnet corresponding to the states 1 - 4 observed in the loops are \nsketched in Figs. 3e and 3f. Additional reference MOKE -measurements were performed on \n11 mm2 square areas bom barded with D = 11015 He+/cm2 and D = 31015 He+/cm2, as \nwell as for an area of the same size, protected by the resist mask (Figs. 3c, 3d). Note that \nthe dimensions of the reference areas were much larger than the laser spot (diameter 0.3 \nmm) used for MOK E characterization. Therefore, the reference loops are not affected by \nthe border regions between bombarded and not bombarded areas. The situation is different \nfor the patterned periodic square lattices where hysteresis loops are approximately the \nsuperpos ition of the loops obtained for the reference areas. The P -MOKE signal ratio \ncorresponding to magnetization reversal of the squares and matrix is equal to the ratio of \nthe areas of these regions, which is 1/3. Only for the largest squares the observed rati o is \nnot exactly 1/3 as for these measurements the size of the individual squares is close to the \nMOKE laser spot; in consequence the signal does not fully average over several squares \nand depends on the precise position of the laser spot with respect to t he large squares. A \ncomparably small dependence of the switching fields ( HS) on the square size parameter a is \nobserved for switching between states 2 →3 and between 4 →1, whereas essentially no \ndependence is observed for the switching between states 3 →4 and 1→2 (Figs. 3a and b)). \nThis indicates a relatively weak interaction between the ion -modified regions and the \nmatrix and is caused by weak magnetostatic interactions that result from the low saturation \nmagnetization ( MS) of the studied films and their smal l thicknesses. Exchange coupling at \nthe borders between squares and matrix contributes weakly to the above interaction \nbecause of the small interaction surface (the film thickness multiplied by the total \nperimeter lengths of all the squares). \nMagnetization reversal in a RE+ matrix with embedded RE+ squares \nThe loops corresponding to this case are displayed in Fig. 3a), the magnetic moment \nconfigurations of the two magnetic subsystems and the effective magnetizations for the \nstates 1 – 4 are shown in Fig. 3 e). Switching fields indicate a dependence on the square \ndimensions only in the magnetization reversal between states 2→3 and 4→1 (Fig. 3a). \nHereinafter, the switching fields HSif, related to the transition between specific states will \nbe described using s uperscripts identifying the initial ( i) and final ( f) states, e.g., for D = \n11015 He+/cm2 (Fig. 3a) HS23 and HS41 corresponds to the magnetization reversal of areas \n(squares) subjected to ion bombardment. At this dose D, both the matrix and the squares \nshow the dominance of the magnetic moments of the Tb magnetic subsystem. Therefore, \nduring the transition between states 2→3 and 4→1 the reversals of squares correspond to \nan annihilation of domains and their corresponding DWs (cf. inset in Fig. 3e). Since t he \nDW energy released by these processes is proportional to the wall interface area, this \nreduction of HS23 (HS41) with decreasing a is understandable. Additionally, it is obvious \nthat a reduction of a produces a broadening of the transition region for the switching fields HS23 and HS41. This is related to the statistical variation of HS among the squares (see \nmovie in supplementary materials). The distribution of switching fields for squares reflects \nlocal (lateral) fluctuations of magnetic properties (mainly anisotropy and exchange \nconstants, i.e., parameters determining the energy of DWs) [12]. As the magnetization \nreversal processes 2→3 and 4→1 correspond to the annihilation of domains and DWs \nprocesses 1→2 and 3→4 are related to their creation (F ig. 3e). As the processes 1→2 and \n3→4 take place through propagation of a DW in the matrix (in this case the matrix can be \ntreated as a continuous layer [Suppl. Mat]) the magnetization reversal takes place in a very \nnarrow magnetic field range and the valu es of HS12 and HS34 are equal to the field HC of the \nmatrix (Fig. 3a, 3c). Moreover, they are practically independent of a. \nThe minor loop shift ( Hmls) (Fig. 3a), measured from the negative saturation field, show \npositive values for D = 11015 He+/cm2, revealing a ferromagnetic interaction between the \nmodified areas and the matrix [47]. This is consistent with the tendency to eliminate \nantiparallel orientations of the magnetization between the magnetic subsystems of the same \ntype (Co and Tb) on opposite sides of the border between squares and matrix [38], i.e. to \nannihilation of DWs. \nThe magnetization reversal of the magnetically textured ferrimagnetic films (Fig. 3a), in \nwhich RE+ areas (squares) are embedded in a RE+ matrix with d ifferent HS, practically \ndoes not deviate from a situation in which the ferrimagnetic film would be replaced by a \nferromagnetic one. However, the situation changes when the modified areas and the matrix \ndiffer not only in HS but also in magnetic subsystem domination. \nMagnetization reversal in a RE+ matrix with embedded TM+ squares \nThe loops corresponding to this case are displayed in Fig. 3b, the magnetic moment \nconfigurations of the two magnetic subsystems and the effective magnetizations for the \nstates 1 – 4 are shown in Fig. 3f. Fig. 3b shows measurements in a system for which th e \nion bombarded areas have lower HS and are TM+, while the matrix has a higher HS and is \nRE+. In this case, 1→2 and 3→4 magnetization reversals occur in the square areas \nmodified by ion bombardment. In states 1 and 3 (at saturation), the effective \nmagnetiz ations of squares and matrix are both oriented in the direction of the magnetic \nfield; at the same time, the magnetizations of each magnetic subsystem (Co and Tb) \nchange to the antiparallel direction across the borders between squares and matrix (Figs. \n3f, 4g). Therefore, in states 1 and 3, DWs exist at the borders of the squares. At fields HS12 \nand HS34, the squares reverse (the effective magnetizations of squares and matrix are now \nantiparallel to each other) and the DWs in the magnetic subsystems are ann ihilated. To \ncorroborate this conclusion micromagnetic simulations have been carried out to determine \nthe magnetic configuration of the Co and Tb subsystems in the transition area between \nRE+ and TM+ region . The results are shown in Fig. 4 and will be disc ussed below . \nThe comparison of the magnetic configurations of states 1 (3) and 2 (4) (Fig. 3b, 3f) \nindicates that, at remanence, states 2 and 4 are energetically more favorable than states 1 \nand 3. This is caused both by a reduction of the magnetostatic en ergy (the effective \nmagnetization in the squares is antiparallel to that of the matrix) and the annihilation of \nDWs. As a result, processes 1→2 and 3→4 involve a reduction of the free energy in the \nsystem; while the opposite processes are accompanied by an increase (2→1 and 4→3, seen \nin the minor loops). Although the individual squares reverse independently, the values HS12 \nand HS34 are close to the HC value of the modified reference area (Fig. 3b, 3d) and show a \nnarrow distribution, while HS21 and HS43the transition 4 3 is not shown in Fig. 3b) \nare greater than the HC of the reference area and have large spread (Fig. 3g). The \ninfluence of a on the above -mentioned switching fields is stronger for smaller a (or for \ngreater combined length of all DWs). In contrast to the reversal process presented in Fig. \n3a, the shift of minor loops seen in Fig. 3b is negative, indicating an antiferromagnetic \ncoupling between the TM+ squares and the RE+ matrix. However, the origin of these behaviors is the same in bot h cases and it is related to the elimination of the antiparallel \nconfiguration of magnetization in the Co and Tb magnetic subsystems (annihilation of \nDWs). The broadening of the distribution of HS21 and HS43 as a is reduced can be attributed \nto the spatial distribution of magnetic properties due to deposition and ion bombardment \nthrough resist [48–50]. \nTo support qualitatively our interpretation of the experimental data, we have performed \nmicromagnetic simulations using the publicly available OOMMF package [51] without any \nadditional extensions . Details of simulations are described in Methods. A typical full loop \nand a minor loop for the patterned strip (described in metho ds) are shown in Fig. 4a. The \nfull loop is a two -step hysteresis with intermediate states similar to those described in the \ndiscussion of Fig. 3b. Note that in Fig. 3b the dependence of the P -MOKE signal (strongly \ndominated by the magnetic Co subsystem) on the magnetic field and in Fig. 4a the one of \nthe effective magnetization is shown. State 1 corresponds to magnetic saturation in \nnegative field where the effective moments in both RE+ and TM+ regions are aligned \nparallel to the field (Fig. 4f), while for state 2 the effective moment in the bombarded area \nis opposite to that of the matrix (Fig. 4h). Close -up views of these configurations are \nshown in Figs. 4e and 4g. These transversal cross -sections show the difference between the \ntwo states: in state 1 the effective magnetization is negative everywhere but at the inter face \nthe magnetization rotates in each magnetic subsystem (i.e., the DWs are present) while; \nstate 2 does not contain a DW although the two regions have opposite effective \nmagnetizations. These two images support the key finding of our paper, namely that the \nhybrid RE+/TM+ ferrimagnetic layered system can be patterned by keV He -ion \nbombardment allowing multi -domains without DWs (stage 1). It is worth noting that, due \nto the strong antiferromagne tic interaction between the Co and the Tb magnetic \nsubsystems, the spin structure of DWs is similar to the one found in \nantiferromagnets [52,53] . \nHaving shown that in ferrimagnetic films consisting of TM+ areas embedded in an RE+ \nmatrix the antiparallel configuration of the effective magnetization can exist without DWs \nat the RE+/TM+ interfaces, now we show that at the field induced transition between states \n1 and 2 the reduction in anisotropy energy and exchange energy is accompanied by a \nreduction of the magnetostatic energy. Overall, the flux closure is achieved with the \nannihilation of the DW. Figs. 4b -d show the anisotropy, magnetostatic and exchange \nenergies as a function of field for the down s weep branch of the hysteresis loop. In state 2, \nwhich occupies the middle region of this graph ( -10 kOe < H < -1 kOe), we see that due to \nannihilation of the DWs the exchange and sum of anisotropy and magnetostatic energies \nare reduced. It is also apparent that the magnetostatic energy in state 2 is generally lower \nthan in state 1. Therefore, such magnetic configuration is very stable and is characterized \nby a deep free energy minimum, which explains the strong negative value of Hmls observed \nboth in the ex periment (Fig. 3b) and simulations (Fig. 4a). This confirms that it is possible \nto achieve flux closure in the absence of DWs which explains why the observed unique \nfeatures are particularly stable and energetically advantageous. \n \nSummary \nIt has been shown that in a prototypical rare earth (RE)/ transition metal (TM) layered \nferrimagnetic material system magnetic domains can be engineered by 10 keV He+ ion \nbombardment without DWs between the patterns. It has been shown that these patterns \ndisplay a particularly stable magnetic configuration due to a deep minimum in the free \nenergy of the system which is caused by flux closure and the corresponding reduction of \nthe magnetostatic energy part of the total free energy. As a result, a much larger magnetic \nfield is required to annihilate such a magnetic pattern than to create it. The fundamental \neffect used for engineering of such domains without DWs is the observation that the rare \nearth contribution to the effective properties of the ferrimagnetic multilayers is more sensitive to keV light-ion bombardment as compared to the contribution of the transition \nmetal. Therefore, this technique can be used in this material system to achieve a steering of \nthe relative contributions of the two magnetic subsy stems in a controlled manner . Thus, \nstarting with magnetic Co/Tb multilayers where the Tb magnetization dominates and using \nion bombardment, we created magnetic patterns where areas with Co magnetic moment \ndensity domination and small coercive fields were embedded in the matrix that retained the \nmagnetic properties of the as -deposited system. \n \nMethods \nSamples deposition. The (Tb-wedge 0 -2nm/Co -0.66nm) 6 and (Tb-1.1nm/Co -0.66nm) 6 \nlayered systems were deposited from elemental targets using magnetron sputterin g in an \nultra-high vacuum chamber (base pressure 10−9 mbar) with an argon pressure of 10−3 mbar \non 20x20 mm2 naturally oxidized Si(100) substrates coated with a Ti -4 nm/Au -30 nm \nbuffer layer [39]. The wedge -shaped sublayers were produced using a lin ear shutter. The \ngrowth of the films was carried out at RT and, in contrast to our previous \ninvestigations [39], without magnetic field. To prevent oxidation of samples an additional \n5 nm thick Au protective layer is used. \n \nIon bombardment. \nChoice of Tb thickness in Co/Tb multilayers designed for magnetic patterning. \nThe multilayer Si/Ti -4nm/Au -30nm/(Tb -wedge 0-2nm/Co -0.66nm) 6/Au-5nm sample was \nsubjected to two different doses ( D = 11015 and D = 5x1015 He+/cm2) of He+ ions. For \nboth D value s, a strip of 2mm width was bombarded across the entire sample and parallel \nto the Tb thickness gradient. \nMagnetic patterning \nA layered Si/Ti -4nm/Au -30nm/(Tb -1.1nm/Co -0.66nm) 6/Au-5nm film was magnetically \npatterned by bombardment with He+ 10keV ions with t wo different doses: D = 11015 \nHe+/cm2 and D = 31015He+/cm2 [54]. The patterning was carried out by covering the layer \nsystem with 400 nm thick photoresist (this thickness is enough to protect the film from ion \nmodification). A mask was used to photolithographically pattern four distinct areas. The \npatterns in these areas consisted of periodic arrays of squares of side a = 3, 12.5, 25, 100 \nμm, with the centers of neighboring squares separated by a distance of 2 a Fig. S1 in \nsupplementary materials). The total area of each of these arrays was 1 1mm2, i.e. large \nenough for hysteresis loops measurements using our P -MOKE magnetometer (which has a \nlaser spot of 0.3 mm in diameter). Independently, a 1x1mm2 area not covered with the \nphotoresist was also manufactured for reference. The above described pattern was \nreplicated for experiments with different value s of D. \n \nMagnetic measurements. Magnetooptical hysteresis loops in polar configuration (P -\nMOKE) were measured in the same way as described in our previous paper [39] using a \nlaser with 640 nm wavelength. Images of magnetic structure and movies illu strating \nmagnetization reversal process were recorded using a P -MOKE microscope. \n \nMicromagnetic simulations \nTo simulate the ferrimagnetic alloy film, we used cubic discretization cells with very small \nsize (1nm). For each cell a uniform random number has been assigned which determines \nthe material of which it is made. With this procedure, the alloy is modelled as a cubic \ngranular structure with random occupancy by the RE element. We believe that the granular \nstructure captures two important physical featur es: first, because the individual sublayers \nof Co/Tb mult ilayer are very thin the system does not form continuous films but tends to \nbehave as an alloy; second, the difference in atomic sizes between the RE and the TM \ncauses the structure to be amorphous r ather than crystalline. In this way, the granular \nstructure used in our simulations resembles the formation of islands during the deposition procedure. Using this approach, two features of ferrimagnets at compensation can be \ndemonstrated qualitatively: the vanishing of magnetization saturation and the unbounded \ngrowth of the coercive field. \n \nWe emphasize that the cited parameter values used in the micromagnetic modelling are \ngiven solely to facilitate replication of our micromagnetic simulations. We do not claim to \nhave obtained a quantitative agreement between simulations and experiment. We show \ninstead that the qualitative features can be reproduced in the simulation. A quantitative \nmatching would require performing numerical analys is of the errors introduced when a \ncontinuous alloy is represented by discrete grains. This task is beyond the scope of this \npaper. For this reason, in this micromagnetic simulation section we would refer to the \ndifferent elements in the structure using ge neric names (RE and TM). \nThe effect of ion bombardment in the ferrimagnet is modelled by separating the system \ninto two distinct regions. Inside the bombarded area we reduce the RE occupancy \nprobability and decrease the strength of the crystalline anisotr opy of the TM. We simulate \na strip long enough in one direction to cover a full period of the structure. The simulation \nbox is 4 m20nm5nm with periodic boundary conditions in two dimensions. \nLongitudinally, the bombarded area is placed in the central reg ion with margins of 1 m \nfrom each end of the strip; in the transversal and vertical directions it spans the whole \nsimulation box. \nTo describe RE+ and TM+ regions (corresponding, in the experiment, to protected and \nbombarded areas, respectively) four materi al regions are used to specify: first, whether the \ncell is occupied with RE or with TM elements; and second, if the cell is in the pristine (out) \nor the bombarded area (in). Any cell in the simulation belongs to one of the following \nregions: RE in, RE out, TMin, TM out. The parameters were chosen to capture the following \nknown properties of ferrimagnets [55,56] : weak ferromagnetic interaction between \nneighboring RE -RE cells, a stronger ferromagnetic interaction between neighboring Co -Co \natoms, and an even stronger antiferromagnetic interaction betwee n adjacent TM -RE cell \npairs. The magnetic moment of the RE cells was chosen to produce a compensation point \nat 22% [57]. The easy -axis of effective anisotropy is oriented perpendicular to the surface. \nThe crystalline anisotropy constant for the TM is weaker for cells located in the bombarded \narea but is everywhere larger than that of RE cells wh ich all have the same value assigned. \nThe material parameters are summarized in Table 1 \nRegion \n(x) \n\n3mMJKu \n\nmMAMS \n\nmpJAREx\n \n\nmpJATMx \nRE in 1.067 5.08 7 -24 RE out 1.067 \nTM in 1.342 1.42 -24 14 TM out 1.742 \nTable 1 Material Parameters. The exchange constants\nyxA should be read as the coupling between elements in row x \nand column y. These ad -hoc values have been chosen to reproduce the qualitative features of our experiments and should \nnot be considered as the estimations of actual material parameters. \nAcknowledgements \nThe authors would like to thank M. Schmidt and J. Aleksiejew for technical support. The \nwork was financed by the National Science Centre Poland under SONATA BIS funding \nUMO -2015/18/E/ST3/00557. M.M. and A.M. acknowledge financial support from the \nNational Science Centre Poland through the SONATINA project UMO -\n2018/28/C/ST5/00308 . \n \n Refer ences \n[1] A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y. Sonobe, Y. Ikeda, S. Sun, \nand E. E. Fullerton, J. Phys. Appl. Phys. 35, R157 (2002). [2] J. I. Martin, J. Nogues, K. Liu, J. L. Vincent, and I. K. Schuller, J. Magn. Magn. \nMater. 256, 449 (2003). \n[3] B. D. Terris and T. Thomson, J. Phys. Appl. 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Schubert, Magnetic Order and Coupling Phenomena (Springer International \nPublishing, Cham, 2014). \n[56] N. H. Duc, T. D. Hien, D. Givord, J. J. M. Franse, and F. R. de Boer, J. Magn. \nMagn. Mater. 124, 305 (1993). \n[57] M. H . Tang, Z. Zhang, S. Y. Tian, J. Wang, B. Ma, and Q. Y. Jin, Sci. Rep. 5, \n10863 (2015). \n \n \nFig. 1 a) Sketch of a layer system consisting of a stack of an RE+ and a TM+ \nferromagnetic film. Red and blue arrows indicate the magnetizations of the RE and TM \nmagnetic subsystems, respectively, black arrows indicate the effective magnetization of the \nlayers. The magnetization configuration is depicted at magnetic saturation, displaying an \ninterfacial DW (green area) between the R E+ and the TM+ layer b) Sketch of a \nferrimagnetic layer, displaying alternating RE+ and TM+ regions after local modification \nby keV light -ion bombardment. The magnetic configuration is also depicted at magnetic \nsaturation, displaying DWs (green areas) between the RE+ and TM+ regions. \n \n \nFig. 2 a) Coercive field HC as a function of Tb sublayer thicknesses of the Si/SiOx/Ti -\n4nm/Au -30nm/(Tb -wedge/Co -0.66nm) 6/Au system in the as -deposited state and after He+ \n(10 keV) ion bombardment with D=11015 He+/cm2 and D=31015 He+/cm2. The upper \nhorizo ntal axis shows the corresponding effective concentration of Tb in the whole layer \nsystem, cTb, for a given Tb thickness. The dashed line corresponds to tTb = 1.1 nm which \nwas chosen for experiments presented in Fig. 3. The hysteresis loops corresponding t o \nlarge points ( tTb=1.1 nm) in panel a) are presented in panels b), c) and d) for D=0, \nD=11015He+/cm2, and D=31015He+/cm2, respectively. \n \n \n \n \n \nFig. 3 Full and minor (full and open symbols, respectively) P -MOKE hysteresis loops \nmeasured for a Si/Ti -4nm/Au -30nm/(Tb -1.1nm/Co -0.66nm/) 6/Au-5nm system \nmagnetically patterned using ion bombardment (He+ 10 keV) with doses D = 11015 \nHe+/cm2 (a,c) and D = 31015 He+/cm2 (b,d). The different colors in panels (a, b) \ncorrespond to different sizes of patter ned squares. The hysteresis loops presented in the \nlower panels correspond to reference areas (c, d). The magn etic field corresponding to the \nminor loop shift Hmls is indicated only for a=12.5 m. The panels (e, f) show the \nmagnetization orientation in the matrix (M) and the squares (S). The black, blue and red \narrows correspond to effective magnetization, magnetization of the Co and of the Tb \nmagnetic subsystems, respectively. DWs are indicated with green. The magnetic structure \ninside the DW is shown in Fig.4e. Panel s (g,h) show differential images (difference \nbetween images recorded at a given magnetic field and at saturation in negative field) of \nmagnetic structure recorded using magnetooptic al Kerr microscope in polar configuration. \nThe photographs are arranged in rows corresponding to magnetic field ranges related to the \nminor loop reversal of the 12.5 12.5 μm squares from 1 to 2 g) and from 2 to 1 h) . \n \n \nFig. 4 a) Hysteresis loops obtained from OOMMF simulations for a patterned strip for \nrandomized distributions of Tb cells. The magnetizatio n perpendicular to the plane, mz, is \nnormalized accounting for the total number of Co and Tb cells. b) Free energy \n(magnetostatic+an isotropy+exchange), c) Sum of anisotropy and magnetostatic energies; \nand d) exchange energy as functions of applied field for the sweep of the hysteresis loop \nfrom 25 kOe to -25 kOe. To facilitate comparison, the energy terms in the saturated state \nare set to zero. (e, g) Cr oss section of the Co (red arrows) and Tb (blue arrows) \nmagnetization configuration in the region between RE+ and TM+ areas at magnetic field \nH = 25 kOe and H = -3 kOe corresponding to state 1 and 2. (f, h) Normalized mz \ncomponent at distance x away from t he boundary between the RE+ and the TM+ regions . \nIn state 1 (at saturation) a DW is present - the Co and Tb spins rotate along the x -direction \nwith continuous changes in normalized magnetization keeping its sign (f). In contrast, state \n2 shows no DW betwee n magnetic domains with antiparallel magnetization (h). The error \nbars in f) and h) correspond to the standard deviation of 11 simulations with different \nrandom distributions. \n" }, { "title": "2010.06615v1.Effects_of_spin_orbit_torque_on_the_ferromagnetic_and_exchange_spin_wave_modes_in_ferrimagnetic_CoGd_alloy.pdf", "content": "1 \n Effect s of sp in-orbit torque on the ferromagnetic and exchange spin wave \nmodes in ferrimagnetic CoGd alloy \nBoris Divinskiy ,1,* Guanxiong Chen ,2 Sergei Urazhdin ,2 Sergej O. Demokritov ,1 and Vladislav \nE. Demidov1 \n1Institute for Applied Physics and Center for Nonlinear Science, University of Muenster, 48149 \nMuenster, Germany \n2Department of Physics, Emory University, Atlanta, GA 30322, USA \n \nWe use micro -focus Brillouin light scattering spectroscopy to study the effect s of spin -orbit \ntorque on thermal spin waves in almost angular -momentum compensated ferrimagnetic CoGd \nalloy films. The s pin-orbit torque is produced by the electric current flowing in the Pt layer \nadjacent to CoGd . Both the ferromagnetic and the exchange modes are detected in our \nmeasurement s. The intensity and the linewidth of the ferromagnetic mode are modified by the \nspin-orbit torque . In contrast, the properties of the exchange mode are unaffected by the spin-\norbit torque . We also find that the frequencies and the linewidths of both modes are significantly \nmodified by Joule heating, due to the strong temperature dependence of the magnetic properties \nof CoGd in the vicinity of a ngular momentum compensation point . Our results provide insight \ninto the mechanisms that can enable the implementation of sub -THz magnetic nano -oscillators \nbased on ferrimagnetic materials , as well as related effects in antiferromagnets . \n \n \n*Corresponding author, e -mail: b_divi01@uni -muenster.de \n 2 \n I. INTRODUCTI ON \nRecent advances in the studies of spin -orbit torque s (SOT s) have opened novel \nopportunities for the fields of spintronic s and magnonic s [1-3]. In particular, SOT s have enabled \nthe development of microwave nano -oscillators based on magnetic materials [4,5], where \ncoherent oscillation emerges from thermally excited spin -wave modes. Following the initial \ndemonstration [6,7], a variety of SOT -driven nano -oscillators have been proposed and \nexperimentally realized in recent years , in effort s to improve their efficiency and coherence [8-\n14]. It was also theoretically shown [15, 16] and experimentally confirmed [5,6,8,17 ] that the \nmechanisms underlying the emergence of coherent dynamics can be elucidated by analyzing the \nevolution of the intensity and the linewi dth of thermally excited modes in the sub-critical regime . \nNano -oscillators based on ferromagnetic materials operate at frequenc ies in the range of \nabout 0.1 – 30 GHz [ 18], with the upper limit determined by the practical ly accessible \nmagnitudes of static magnetic field s. On the other hand, one of the most significant challenges in \nmodern microwave technology is the lack of compact and reliable microwave sources capable of \ngenerati ng signals in the frequency range 0.1 – 10 THz, which is commonly referred to as the \n“THz gap” [ 19-21]. It was recently proposed that the operation al frequency of SOT oscillators \nbased on antiferromagnetic (AFM) [22-25] and ferrimagnetic (FiM) [26] materials can be \nsignificantly higher than in ferromagnet -based oscillators, due to the large internal effective \nexchange fields. The latter can reach magnitudes of dozens of T esla, enabling THz -frequency \ndynamics even in the absence of external magnetic fields, and paving the way for the \nimplementat ion of SOT oscillator s capable of filling the “THz gap” . 3 \n From the point of view of technical applications, AFM materials suffer from a significant \ndisadvantage: because of the zero net magnetic moment, excitation an d detection of spin \ndynamics in these mat erials is very challengin g. In contrast, in FiM materials , the \nantiferromagnetically coupled sublattices are not equivalent . This results in a non -zero net \nmagnetic moment , enabling direct inductive excitation and probing of magnetization dynamics . \nAdditionally , because of the difference in the electronic and the optical properties of the \nelements that constitute different sublattices , spin dynamics of these sublattices can be \nselectively accessed . \nAmong the most attractive FiM system s enabling acces s to the magnetization dynamics \nof individual sublattices are transition metal -rare earth ( TM-RE) alloys [27,28]. A key benefit of \nthese materials is that their magnetic properties can be tuned in a wide range by varying their \ncomposition and temperature [27]. In particular, because of the strong temperature dependence \nof the magnetization of the RE sublattice, the magnetizations of the TM and RE sublattices \ncancel each other at a certain composition -dependent magnetization compensation temperature \nTM. Furthermore, t he angular momenta of the two sublattices cancel at the angular momentum \ncompensation temperature TA, which is different from TM because the g -factors characterizing \nthe TM and the RE sublattices are generally different . TM-RE also exhibit attrac tive electronic \nproperties. Since the magnetism of the RE atoms is mediated by the localized f electronic states \nwith energies significantly below the Fermi level , the spin -dependent electronic transport \nproperties of TM -RE alloys are dominated by the d-electrons of TM atoms . As a consequence , \nthe spin-orbit torques act predominantly on the TM sublattice , enabl ing efficient SOT -driven \ncontrol of the TM-RE alloys’ magnetization [29]. 4 \n TM-RE alloys exhibit two types of dynamic al magnetic modes , as expected for FiM \nsystems with two sublattices [30,31 ]. In the first mode , the magnetizations of the two sublattices \nremain antiparallel to each other during precession . The frequency of this mode is determined \nmainly by the external static magnetic field and the effective gyromagnetic ratio , and typically \nlies in the GHz range . These characteristics are similar to those of the dynamical modes in \nferromagnets . In the second mode , called the exchange mode , the magnetizations of the two \nsublattices do not remain antiparallel to each other , resulting in a large contributi on of exchange \ninteraction to the dynamical mode energy . Consequently , the frequency of this mode is \ndetermined by the exchange constant , and typically falls in the THz region . \nThe frequencies of the f erromagnetic and the exchange modes experience strong \nvariations at temperatures T in the vicinity of the angular -momentum compensation point TA. For \nan ideal FiM, the frequency of the ferromagnetic mode is expected to diverge at T= TA because \nof the divergence of the effective gyromagnetic ratio, while the frequency of the exchange mode \nis expected to vanish. These features are promising for the implementation of ultra-high-\nfrequency SOT oscillators. On the one hand, very high frequenc y of the ferromagnetic mode can \nbe achieved without the need for large external field s. On the other hand, the freque ncy of the \nexchange mode can be tuned down to the few-THz or sub-THz range , depending on the \napplication requirements . \nBoth the ferromagnetic and the exchange modes have been experimentally observed in \nTM-RE alloys using ferromagnetic resonance and ultrafast optical pump -probe technique s \n[28,32-37]. However, the effects of SOT on these dynamic al mode s remain unexplored . \nHere, we report an experimental study of the effect s of SOT on the magnetization \ndynamics in the CoGd /Pt bilayer in the vicinity of the a ngular -momentum compensation point . 5 \n We utilize micro -focus Brillouin light scattering (BLS) spectroscopy to detect the ferromagnetic \nand the exchange modes , and study the dependences of their characteristics on the SOT \ngenerated by electric current in the P t layer. By analyzing the intensity and the linewidth of the se \nmodes, we demonstrate that the effects of S OT are significant only for the ferromagnetic mode, \nbut there is no sizable effect o f SOT on the exchange mode. The se observ ations are consistent \nwith the general expectation that the efficiency of SOT -driven excitation is determined by the \nrelaxation rate of the dynamical modes, which is expected to be significantly higher for the high-\nfrequency exchange mode . We also show that the frequencies of both modes can be \nelectronically tuned by Joule heating . The frequenc y of the ferromagnetic mode can reach values \nof up to 50 GHz at the field of 0.4 T , while the frequency of the exchange mode can be varied in \nthe range 70 -120 GHz by varying the current . Our findings are important for the practical \nimplementation of ultra -high-frequency SOT oscillators based on FiMs , and are also likely \nrelevant to the AFM -based SOT devices . \n \nII. EXPERIMENT \nFigure 1(a) shows th e layout of our experiment . The studied system is based on a Pt(5)/ \nCo78.1Gd21.9(10) magnetic multilayer capped by Ta(3) to protect CoGd from oxidization . Here, \nthicknesses are in nanometers. The room -temperature saturation magnetization of the CoGd film , \nas determined from the vibrating -sample magnetometry measurements , is 180 kA/m. Based on \nthis value and the experimentally determined fr equencies of dynamic modes, we estimate the \nanisotropy constant of CoGd film to be equal to 0.08 MJ/m3. \nThe multilayer is patterned into a square with the side of 5 µm and electrically contacted \nby using 120 nm thick Au electrodes. Due to the large difference in the resistivities of the CoGd , 6 \n Pt, and Ta layers (1490 , 275, and 1500 nΩ*m , respectively), the electric current I flowing in the \nplane of the multilayer is predominantly transmitted through the Pt film. The electrical current is \nconverted by the spin-Hall effect ( SHE ) in Pt [38,39 ] into an out -of-plane spin current Is. The \nspin current is injected into CoGd , exerting SOT on its magnetization . According to the \nsymmetry of SHE, the effects of SOT are maximize d when the static magnetic field H0 is applied \nin plane, in the direction perpendicular to the current flow. \nSince only about 16% of the total electrical current flows through the CoGd film, we \nassume that the contribution of SOT produced by the bulk spin -orbit interaction in the \nferrimagnetic layer [40 ] is significantly smaller than that induced by SHE in Pt. The SHE in Ta \nlayer plays a negligible role in the studied system because of the partial oxidation and the large \nresistivity of the capping Ta film. We also note that the current -induced Oersted field does not \nexceed 1 .5 mT for the maximum current used in the experiment. Since this value is two orders of \nmagnitude smaller than the strength of the static magnetic field (0.1 – 0.4 T ), we assume \nnegligibl e effects of the Oersted field. \nWe characterize the effect s of the driving current on the dynamic al modes by using \nmicro -focus BLS spectros copy [41]. The probing laser light with the wavelength of 532 nm is \nfocused into a diffraction -limited spot on the surface of the CoGd film, and the spectrum of light \ninelastically scattered from the dynamical magneti zation is analyzed . The incident beam intensity \nof about 0.1 mW is sufficiently low to ensure that the perturbation of the magnetic system by the \nprobing light is negligible. The high sensitivity of BLS enables detection of thermally excited \nspin-wave modes (magnetic fluctuat ions), which are always present at nonzero temperatures \neven in the absence of the driving electric current , allowing the characterization of the magnetic \nsystem in the subcritical regime . 7 \n \nIII. RESULTS AND DISCUSSION \nFigure 1(b) shows a representative BLS spectrum of magnetic fluctuations recorded at the \nmagnetic field µ0H0 = 0.1 T, at room temperature T0 = 295 K. The spectrum exhibits a well-defined \npeak with a pronounced shoulder on its high -frequency tail , indicat ing the existence of t wo \ndynamic modes in the studied system. The spectrum is well -approximated by a sum of two \nLorentzian f unctions , enabling accurate determination of the central frequencies of the two modes . \nWe will refer to them as the low-frequency (L F) and the high -frequency (HF) mode . As the \nexternal field µ0H0 is increased from 0.1 T to 0.4 T, the central frequency of the LF mode \nmonotonically increases by a factor of 1.5 from 28 to 42 GHz , while the frequency of the HF mode \nremains nearly constant at 65 GHz , as shown in Fig. 1(c) . Note that at µ0H0 > 0.3 T, the peak \ncorresponding to the LF mode strongly overlap s with that of the HF mode , so the latter becomes \ndifficult to distinguish in the measured spectra . The obtained dependences agree well with the \ntheory of magnetization dynamics in FiMs [30] and previous experimental observations [3 4], \nallow ing us to identify the LF and the HF mode as the ferromagnetic and the exchange mode , \nrespectively. Indeed, the frequency of the ferromagnetic mode is expected to increase with the \nincrease of H0, while the frequency of the exchange mode is expected to be nearly independent of \nH0, since it is determined mainly by the effective exchange field. \nNext, we study the effects of the electric current on the characteristics of the observed \nmodes. These effects generally include variation s of the intensity of fluctuations and of the \neffecti ve damping [17], resulting in the variations of the intensity and the linewidth of the \nspectral peaks, respectively . For the direction of H0 shown in Fig. 1 (a), SOT induced by the 8 \n positive current I, as defined in this Figure, is expected to enhance magnetic fluctuations and \ndecrease the effective mode dampi ng. The opposite effects are expected for negative current. \nFigure 2(a) shows the current dependenc e of the integral intensity E of the measured BLS \nspectra. This dependence is clearly a symmetric with respect to the current direction , even though \nit is dominated by the symmetric quadratic contribution (dashed curve in Fig. 2(a)) that can be \nattributed to Joule heating. The asymmetric deviation s from the quadratic dependence become \nparticu larly pronounced at large current s |I|>10 mA. Figures 2(b) and 2(c) show the current \ndependences of the integral intensities Ef, Eex of the peaks associated with the ferromagnetic and \nthe exchange mode , respectively, obtained from the Lorentzian fits of th e measured spectra \nsimilar to that shown in Fig. 1(b). To highlight the effects of SOT, which depend on the direction \nof the current , the data for I>0 and for I<0 are shown on the same plot as a function of the \ncurrent magnitude . \nFigure 2(b) clearly demonstrates that for the ferromagnetic mode, the integral intensity is \ngenerally large r at I>0 than at the same magnitude of I<0. This result is consistent with the \neffect s of SOT, which are expected to enhance magnetic fluctuations at I>0, and suppress t hem \nat I<0 [17]. We n ote that the asymmetry between the opposite current directions is strongly \nnonlinear . In particular, the intensity at I=12 mA is only 7% larger than at I=-12 mA , while at the \nmaximum applied current Imax=17 mA, the asymmetry defined as 2(Ef(+Imax) – Ef(–Imax))/ \n(Ef(+Imax)+Ef(–Imax)) reaches about 30%. According to t he general theory of spin torque , the \ncurrent dependence of intensity associated with the SOT can be described by E=E0(1-I/IC)-1 [15]. \nHere, IC is the critical current, at which the natural damping is expected to become completely \ncompensate d by SOT . This dependence is valid for ferromagnetic material s only in the limit of \nnegligible Joule heating and current -independent mode frequency [ 15], whic h cannot 9 \n quantit atively account for our results . Nevertheless, using this dependence as an approximation, \nwe can estimate the value IC50-100 mA of the critical current for the studied system . This value \ncorresponds to the average current density of 1012 A/m2 in the bilayer, which is comparable to \nthe value s typical for the previously demonstrated SOT oscillators based on ferromagnetic metals \n[4,5]. \nIn contrast to the ferromagnetic mode, the results for the exchange mode do not indicate \nany sizable effects of SOT . The measured integral intensity Eex of the peak increases with \ncurrent, but th is increase is independent of the c urrent direction , within the experimental error \n(Fig. 2(c)) . This result is not surprising , since IC is proportional to the rel axation rate [ 15], which \nin the Gilbert damping approximation is proportional to the mode frequency. Since the frequency \nof the exchange mode is significantly higher than that of the ferromagnetic mode , the \ncorresponding critical current is expected to be much large r. One can estimate that, at Imax=17 \nmA, the intensity of the exchange mode is expected to be enhanced by no more than a few \npercent , consistent with the data of Fig. 2(c). \nSOT also modifies the mode relaxation rate, which is expected to be manifested by the \nasymmetric current dependence of the peak linewidth. T he current dependences of the spectral \nwidth s of the peaks are shown in Figs . 3(a) and 3(b) for the ferromagnetic and the exchange \nmode , respectively . Similarly to the intensities, the effect of SOT on the linewidth is sizeable \nonly for the ferromagnetic mode. At I>0 (point -up triangles in Fig. 3(a)), the linewidth is \ngenerally smaller than at I<0 (point -down triangles). In contrast , for the exchange mode (Fig. \n3(b)), the differences be tween the linewidths for the opposite current directions are within the \nexperimental uncertainty. 10 \n The data shown in Figs. 2 and 3 clearly demonstrate that, at I>0, SOT acts as the anti -\ndamping torque, increas ing the intensity of thermal fluctuations and decreas ing the spectral \nlinewidth. Furthermore , the relation between the eff ects of SOT and the frequency of the \nspecific mode is consistent with the dependences previously established for ferromagnets. Thus, \nwhile FiMs can provide high oscillation frequencies at moderate static magnetic fields, complete \ncompensation of the natural damping , needed for the excitation of magnetization auto-\noscillations at these frequencies , is expected to require very large driving currents . Thus, \npractical implementation of near-THz SOT oscillators utilizing exchange mode is a challenging \ntask that may require technological and/or scientific breakthroughs in the efficiency and \nselectivity of SOT -driven mode excitation . The latter may be accomplished by taking advantage \nof the energy and momentum selection rules involved in the excitation of magnetization \ndynamics by spin injection [ 42]. \nWe note that the effects of Joule heating clearly play a significant role in the observed \nbehaviors of the dynamical modes . These effects are expected to be particular ly large in the \nvicinity of the angular -momentum compensation point of FiM , where the frequencies of both the \nferromagnetic and the exchange mode are expected to rapidly vary with temperature . Figure 4(a) \nshows the dependence of the mode frequencies on the experimental temperature in the absence \nof current, confirming this expectation . As the temperature is increased above 295 K, the \nfrequency of th e ferromagnetic mode first increases, reaches a maximum of about 50 GHz at T = \n304 K, and then monoton ically decreases at higher T. The frequency of the exchange mode can \nbe reliably determined at T > 315 K, where it monoton ically increases from 76 to 114 GHz with \nincreasing T. 11 \n These temperature dependences agree with the previous theoretical predictions and \nreported observations for the ferromagnetic and exchange modes in FiMs [32,33,43 ]. According \nto the established models, the frequency of the exchange m ode reaches a minimum, while that of \nthe ferromagnetic mode reaches a maximum at the angular momentum compensation \ntemperature TA. Thus, the data of Fig. 4(a) indicate that, for the studied system, the angular \nmomentum compensation temperature is TA = 304 K, 9 K above the room temperature T0 = 295 \nK. We note that, in CoGd alloys, the magnetization compensation temperature TM is typically \nabout 30 – 60 K smaller than TA [33,44]. We conclude that , for our samples, the magnetization \ncompensation temperature TM is below the room temperature. Therefore, the net magnetic \nmoment of the studied films is determined by the Co sublattice within the entire temperature \nrange used in the experiments. \nThe effects of Joule heating on the frequencies of the two modes are illustrated in Fig. \n4(b). The frequency of the ferromagnetic mode increases with increasing current, reaches a \nmaximum of about 50 GHz at | I| = 8 mA, and monotonically decreases at | I| > 8 mA. The \nexchange mode becomes distinguishable in the spectrum at | I|> 12 mA , where its frequency \nmonotonically increases with increasing current magnitude . Note that the se variations are nearly \nidentical for the two opposite directions of current, confirming that they are dominated by Joule \nheating. \nBy compari ng Figs. 4(a) and 4(b), we conclude that at T0=295 K the current -dependent \ntemperature of the sample becomes equal to the angular -momentum compensation temperature \nTA= 304 K at |I| = 8 mA . At T>TA, the frequency of the ferromagnetic mode rapidly decreases, \nwhile tha t of the exchange mode decreases with increasing temperature (Fig.4(a)). This explains \nthe rapid variations of the mode linewidths at large current magnitudes, observed for both current 12 \n directions (Fig. 3). Indeed, in the Gilbert approximation, the increas e (decrease) of the mode \nfrequency is expected to result in the increase (decrease) of its linewidth. Accordingly, the \nlinewidth of the ferromagnetic mode rapidly decreases with increasing current magnitude, while \nthat of the exchange mode increases. \n \nIV. CONCLUSIONS \nIn conclusion, we have experimentally studied the dynamic magnetic modes in a n almost \ncompensated ferrimagnetic CoGd film , and their controllability by SOT induced by electrical \ncurrent in the adjacent Pt layer. Our results indicate that CoGd may be suitable for the \nimplementation of SOT oscillators with frequencies of up to several tens of GHz, achievable at \nmoderate static magnetic fields. Our data also indicate that SOT oscillators based on the \nferrimagnetic exchange mode may allow one to a chieve frequencies approaching the THz range , \nbut their operation would likely require extreme driving current densities that may be \nchallenging to achieve in real devices. We expect similar constraints to be also relevant to \nantiferromagnet -based devices . 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Atxitia, S. Wienholdt, D. Hinzke, O. Chubykalo -Fesenko, and U. Nowak, \nTemperature dependence of the frequencies and effective damping parameters of \nferrimagnetic resonance, Phys. Rev. B 86, 214416 (2012) . \n44. C. Kim, S. Lee, H. -G. Kim, J. -H. Park, K. -W. Moon, J. Y. Park, J. M. Yuk, K. -J. Lee, B. -G. \nPark, S. K. Kim, K. -J. Kim, and C. Hwang, Distinct handedness of spi n wave across the \ncompensation temperatures of ferrimagnets, Nature Materials 19, 980 -985 (2020). \n 17 \n \nFigure 1 (a) Schematic of the experiment. (b) Representative BLS spectrum measured at µ0H0 = \n0.1 T. Symbols are experimental data. Curves are the Lorentzian fits for the low-freque ncy (LF) \nand the high-frequency (HF) modes, and their sum . (c) Field dependences of the c entral \nfrequencies of the LF and HF modes . Symbols are the experimental data, curves are guides for \nthe eye. All the data were obtained at room temperature T0 = 295 K. \n \n \n18 \n \nFigure 2 (a) Current dependenc e of the total integral intensit y of the measured BLS spectra. (b), \n(c) integral intensity for the ferr omagnetic (b) and the exchange (c) modes , obtained from the \nLorentzian fits of the corresponding spectral peaks . Symbols are the experimental data . Dashed \ncurve in (a) shows the result of a quadratic fit. Curves in (b) are guides for the eye. Error bars \nshow the uncertainty of the data . The data were recorded at µ0H0 = 0.4 T. \n19 \n \nFigure 3 Current dependences of the spectral linewidth of the peaks corresponding to the \nferromagnetic (a) and the exchange (b) mode. Symbols are the experimental data, curves are \nguides for the eye. Error bars show the fitting uncertainty. For clarity, the error bars are shown \nonly if the error exceeds the size of the symbols. The data were recorded at µ0H0 = 0.4 T. \n \n \n \n \n \n \n \n20 \n \nFigure 4 Tempera ture (a) and current (b) dependences of the c entral frequencies of the \nferromagnetic and the exchange modes , as labeled . TA marks the angular -momentum \ncompensation temperature . Point -up and point -down tr iangles in (b) show the data for the \npositive and the negative currents, respectively. Symbols are the experimental data , curves are \nguides for the eye. The data were recorded at µ0H0 = 0.4 T . \n" }, { "title": "2307.06888v2.Magnon_magnon_coupling_in_synthetic_ferrimagnets.pdf", "content": "Magnon-magnon coupling in synthetic ferrimagnets\nA. Sud,1,∗K. Yamamoto,2K. Z. Suzuki,1, 2S. Mizukami,1, 3and H. Kurebayashi1, 4, 5\n1WPI Advanced Institute for Materials Research,\nTohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan\n2Advanced Science Research Center, Japan Atomic Energy Agency, 2-4 Shirakata, Tokai 319-1195, Japan\n3Center for Science and Innovation in Spintronics, Tohoku University, Sendai, 980-8577, Japan\n4London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom\n5Department of Electronic and Electrical Engineering,\nUniversity College London, London, WC1E 7JE, United Kingdom\n(Dated: August 29, 2023)\nMagnetic multilayers with interlayer exchange coupling have been widely studied for both static\nand dynamic regimes. Their dynamical responses depend on the exchange coupling strength and\nmagnetic properties of individual layers. Magnetic resonance spectra in such systems are conve-\nniently discussed in terms of coupling of acoustic and optical modes. At a certain value of applied\nmagnetic field, the two modes come close to being degenerate and the spectral gap indicates the\nstrength of mode hybridisation. In this work, we theoretically and experimentally study the mode\nhybridisation of interlayer-exchange-coupled moments with dissimilar magnetisation and thickness of\ntwo ferromagnetic layers. In agreement with symmetry analysis for eigenmodes, our low-symmetry\nmultilayers exhibit sizable spectral gaps for all experimental conditions. The spectra agree well with\nthe predictions from the Landau-Lifshitz-Gilbert equation at the macrospin limit whose parameters\nare independently fixed by static measurements.\nI. INTRODUCTION\nIn two magnetic layers separated by a thin nonmag-\nnetic spacer, conduction electrons in the spacer magneti-\ncally couple two spatially separated moments, via the so-\ncalled Ruderman-Kittel-Kasuya-Yosida (RKKY) interac-\ntion [1–3]. This interlayer exchange coupling arises from\ncoherent propagation of electron spin across the spacer\nlayer [4–6]. Due to the Friedel-like oscillation of the\nelectron phase, the exchange coupling changes its sign\nas a function of the interlayer distance, switching be-\ntween ferromagnetic and antiferromagnetic ordering of\nthe two magnetic layers [7–10]. The antiferromagnet-\nically ordered states of two identical magnetic layers,\noften called synthetic antiferromagnets (SyAFs), have\nserved as a testbed for studying antiferomagnetism where\nSyAFs’ relatively weak RKKY exchange coupling, com-\nparable to the strength of magnetic fields achievable in\nlaboratories, helps realise experiments otherwise difficult\nin atomically-ordered, crystalline antiferromagnets [11–\n13]. One such property is the magnetic resonances in\nSyAFs whose typical frequency resides within a range\nof GHz that is readily accessible by modern microwave\ntechniques [14–27].\nIn a canted static state in SyAFs under an applied\nmagnetic field, the low-lying, spatially-uniform resonance\nmodes are usually called acoustic and optical modes\nwhere the precessions of two exchange-coupled moments\nare primarily in- and out-of-phase, respectively [28]. The\nresonance frequencies of these modes exhibit different\nmagnetic field dependence, allowing them to come al-\nmost degenerate in a certain field range. Unless some\n∗aakanksha.sud.c1@tohoku.ac.jpsymmetry conditions are satisfied, there is no exact de-\ngeneracy and the spectrum is gapped as a function of\nmagnetic field [23, 29]. In the regime where the two\nmodes have well-defined in- and out-of-phase characters\naway from the degenerate range, the gap, defined as the\nminimum split with respect to the field between the two\nresonance frequencies, represents the coupling strength\nof the in- and out-of-phase oscillations and when it is\ngreater than the linewidths, it means that the energy\ntransfer between them takes place more frequently before\nthe excited state is lost, termed as the strong coupling\nregime [30]. A large coupling strength has been favoured\nfor potential magnonic applications in which such energy\ntransfer might play a crucial role [30–33].\nCoupled-moment systems offer unique research direc-\ntions and potential spintronic applications [34–38]. For\nexample, their tunable material parameters enable the\ncontrol of characteristic frequencies in nano-oscillator de-\nvices [39, 40], up to the THz regime [41]. The oscillation\nof magnetisations in SyAFs could be excited by spin-orbit\ntorques (SOTs) in planar geometories [42, 43]. SOTs\nin turn drive fast domain-wall propagation/dynamics in\ncompensated magnets [44]. Due to the compensated na-\nture, skyrmions in such a system [45] benefit from the\ncancellation of the skyrmion Hall effect, moving straight\nwithin a propagation channel [46, 47]. We also envisage\nthat some of unique properties in coupled-moment sys-\ntems can be used in exploring a variety of neuromorphic\ncomputation schemes [48–50].\nIn the most commonly studied setup of magnetisa-\ntion dynamics in SyAFs, however, the in- and out-of-\nphase oscillations are seen to remain eigenmodes for\nany magnetic field value and become fully degenerate,\nwhere symmetry of the system plays a crucial role for\nthe decoupling [25, 51–53]. MacNeill et al. presentarXiv:2307.06888v2 [cond-mat.mtrl-sci] 26 Aug 20232\n�B\n�A\nFIG. 1. (a) In the laboratory frame, we define the zdirection normal to the plane, and xdirection such that the static external\nmagnetic field lies in the x-zplane. In the canted regime when applying the field ( H) in-plane, two sub-lattice moments ( MA\nandMB) reside within the plane, canted towards H. For general static states, we introduce new coordinate axes X,Y,Z\nadapted to the two-fold rotation C2that brings the unit vector along MAto that along H. See Eq. (9) for the concrete\ndefinition. For Hin-plane and identical magnetic layers, C2combined with interchanging A and B layers is a symmetry of the\nsystem. (b) and (c) When we apply Hwith the polar angle θ̸= 90◦or two magnetic moments are not identical, C2followed\nby the magnetic layer interchange ceases to be a symmetry. This impacts on the mode coupling as discussed in this study.\nthat the coupled Landau-Lifshitz-Gilbert (LLG) equa-\ntions due to the interlayer exchange interaction are sym-\nmetric under twofold rotation around the applied field\ndirection combined with a layer swap, as long as the field\nis within the film plane (Fig. 1(a)) [51]. The acoustic\nand optical modes are odd and even upon the symme-\ntry operation respectively and therefore unable to hy-\nbridise with each other, leading to the mode degener-\nacy at a resonance point [14, 28]. This specific symme-\ntry can be broken in several ways, for example, tilting\nthe external magnetic field towards out-of-plane direc-\ntion (Fig. 1(b)) [23, 51, 52]. For general expressions\nof spin-wave mode frequencies in interlayer exchange-\ncoupled systems, Layadi presented analytical solutions\nwith a particular focus on the effect of the biquadratic\nexchange coupling and in-plane anisotropy on the spectra\nfor in-plane magnetised cases [54]. While spectroscopic\nmeasurements of interlayer exchange-coupled tri-layers\nwith different magnetic-layer thicknesses were reported\nby some groups in the past [18, 22, 55–57], there seem\nno study fully dedicated to quantitative discussions of\nthe mode hybridisation in such asymmetric interlayer-\nexchange-coupled systems.\nIn this paper, we present our detailed experimental\nand theoretical study of the magnon-magnon coupling\nphenomena in synthetic ferrimagnets where two magnet-\nically coupled layers are not identical (Fig. 1(c)). We\nsystematically compare spin-wave spectra measured by\nbroadband ferromagnetic resonance (FMR) experiments\nand calculated using magnetic parameters deduced from\nstatic magnetometry. In all cases examined, we find ex-\ncellent agreement between experiment and theory, sug-\ngesting that the coupled LLG equations at the macrospin\nlimit are indeed a reliable tool for designing and analysing\nthe spectral properties of the magnetic multilayers. Our\ncalculations further reveal dissimilar roles of quadratic\nand biquadratic exchange interactions for the size of the\ngap. Our results help design and control magnetic res-onance spectra in exchange-coupled magnetic moments\nthat can be synthetic antiferro(ferri)magnets, van der\nWaals antiferromagnets [51, 58–60] and ferromagnetic bi-\nlayers [61–63].\nII. A MACROSPIN MODEL OF SYNTHETIC\nFERRIMAGNET\nFor our purposes, a theoretical model that extends the\nresult of Ref. [54] for arbitrary direction of the static mag-\nnetic field is required, which we present in this section\nwith an emphasis on breaking of the two-fold rotation\nsymmetry. Let MA,MBbe the magnetisations of the\ntwo ferromagnetic layers. We are interested in the situa-\ntions where the two magnetic materials are not identical\n|MA| ≡MA̸=|MB| ≡MB, and the two layers have\ndifferent thicknesses dA̸=dB. The film normal is chosen\nzaxis and the film is regarded infinitely extended in the\nx, ydirections, as shown in Fig. 1(a).\nThe static state of the magentisations corresponds to\nthe minimum of free energy per unit area W. We include\nthe external magnetic field H, demagnetising field, and\nbiquadratic as well as the usual quadratic interlayer ex-\nchange interactions:\nW=dA\u001a\n−µ0MAH·nA+µ0M2\nA\n2(nz\nA)2\u001b\n+dB\u001a\n−µ0MBH·nB+µ0M2\nB\n2(nz\nB)2\u001b\n+J1nA·nB+J2(nA·nB)2. (1)\nHere we have normalised the magnetisations nA(B)=\nMA(B)/MA(B), and introduced the phenomenological ex-\nchange energies per unit area J1and J2. Without\nloss of generality, with the weak crystalline anisotropy\nbeing ignored, the magnetic field can be taken H=3\nH(ˆxsinθ+ˆzcosθ). We determine the static state\nn0\nA(B)by numerical minimisation of W, which is param-\neterised by\nn0\nA(B)=\nsinθA(B)cosϕA(B)\nsinθA(B)sinϕA(B)\ncosθA(B)\n. (2)\nIf the magnetic field is in-plane θ= 90◦and 0 <2J2<\nJ1, the static state undergoes two phase transitions at\nHsfandHffas|H|is increased from zero, where\nHsf=\f\f\f\f1\ndBMB−1\ndAMA\f\f\f\fJ1−2J2\nµ0, (3)\nHff=\f\f\f\f1\ndBMB+1\ndAMA\f\f\f\fJ1+ 2J2\nµ0. (4)\nBelow Hsf, the static state is antiferromagentic n0\nB=\n−n0\nAwithn0\nA·H≷0 according to dAMA≷dBMB.\nAbove Hff, the system is in a forced ferromagnetic staten0\nA=n0\nB=H/|H|. In between lies the spin-flop, or\ncanted, state where HcosϕA,B>0,sinϕAsinϕB<0.\nTo calculate the magnetic resonance frequencies, let us\nintroduce the linear perturbation nA(B)≈n0\nA(B)+n1\nA(B)\nwhere n0\nA(B)·n1\nA(B)= 0. The Landau-Lifshitz equations\nfollow from the free energy Wthrough the usual pro-\ncedure [64]. Although one can press on using n1\nA(B)as\nthe dynamical variables [52], we normalise them so as\nto make them canonical in the sense of Hamiltonian me-\nchanics [64], which ensures that the resulting eigenvalue\nproblem retains the correct Bogoliubov form [65]:\nδA=s\nSdAMA\nℏ|γA|n1\nA,δB=s\nSdBMB\nℏ|γB|n1\nB,(5)\nwhere Sdenotes the area of the film, and γA(B)<0\nare the gyromagnetic ratios. The linearised equations of\nmotion read\nn0\nA×dδA\ndt=γAµ0hn\nH·n0\nA−MA\u0000ˆz·n0\nA\u00012o\nδA+MA(ˆz·δA)\bˆz−\u0000ˆz·n0\nA\u0001\nn0\nA\ti\n−γA\ndAMA\b\nJ1+ 2\u0000\nn0\nA·n0\nB\u0001\nJ2\t\"\n\u0000\nn0\nA·n0\nB\u0001\nδA−s\nγBdAM A\nγAdBMB\b\nδB−\u0000\nn0\nA·δB\u0001\nn0\nA\t#\n+2γA\ndAMAJ2\b\nn0\nB−\u0000\nn0\nA·n0\nB\u0001\nn0\nA\t \nn0\nB·δA+s\nγBdAMA\nγAdBMBn0\nA·δB!\n, (6)\nn0\nB×dδB\ndt=γBµ0hn\nH·n0\nB−MB\u0000ˆz·n0\nB\u00012o\nδB+MB(ˆz·δB)\bˆz−\u0000ˆz·n0\nB\u0001\nn0\nB\ti\n−γB\ndBMB\b\nJ1+ 2\u0000\nn0\nA·n0\nB\u0001\nJ2\t\"\n\u0000\nn0\nA·n0\nB\u0001\nδB−s\nγAdBMB\nγBdAMA\b\nδA−\u0000\nn0\nB·δA\u0001\nn0\nB\t#\n+2γB\ndBMBJ2\b\nn0\nA−\u0000\nn0\nA·n0\nB\u0001\nn0\nB\t \nn0\nA·δB+s\nγAdBMB\nγBdAMAn0\nB·δA!\n. (7)\nEquations (6) and (7) describe two coupled harmonic\noscillators, i.e. there are four independent real functions\nof time to be determined. We are interested in the reso-\nnance properties, which can be analyzed in terms of any\nconsistent choice of the four independent variables. Had\nit not been for the shape anisotropy and the asymme-\ntry between dA, MA, γAanddB, MB, γB, two-fold rota-\ntion around Hwould have mapped n0\nAton0\nBand the\nsymmetry-adapted variables would have been convenient.\nFollowing MacNeil et al. [51], let C2denote the two-fold\nrotation that brings n0\nAton0\nBwhose axis coincides with\nXdirection in Fig. 1. Algebraically the action of C2on\nan arbitrary vector vis given by\nC2v=\u0000\nn0\nA+n0\nB\u0001\n·v\n1 +n0\nA·n0\nB\u0000\nn0\nA+n0\nB\u0001\n−v. (8)Although C2is not in general a symmetry of the prob-\nlem, it helps make sense of the results in terms of the\nfamiliar notions used in previous studies [23, 51]. The\ndefinition of C2becomes ambiguous for |H|< H sfand\nwhat follows does not work for |H|> H ffeither, but\nthese collinear cases are simple and separately handled\nin the Appendix. Focusing on the spin-flop phase, we\nintroduce ��±= (δA± C2δB)/√\n2 that are even and odd\neigenvectors of C2× {A↔B}. To pick out two indepen-\ndent components each for δ±, we define a new coordinate\nframe XY Z (Fig. 1) given by\nˆX=n0\nA+n0\nBp\n2 + 2n0\nA·n0\nB,ˆY=n0\nA−n0\nBp\n2−2n0\nA·n0\nB,(9)\nand ˆZ=ˆX׈Y. By construction, n0\nA·δ±= 0 so4\nthat one may write δ±=δ⊥Z\n±ˆZ×n0\nA+δ∥Z\n±ˆZ. As is\nusual in cavity magnonics, we work with the complex\nvariables α=δ⊥Z\n−−iδ∥Z\n−, β=δ⊥Z\n+−iδ∥Z\n+that wouldrepresent annihilation operators in the quantum regime.\nThis change of variables brings Eqs. (6) and (7) into\nid\ndt\nα\n−α\nβ\n−β\n=\nf1−h1 f2−h2−if3 g1 g2−ig3\nf2−h2+if3 f1−h1 g2+ig3 g1\ng1 g2−ig3 f1+h1 f2+h2−if3\ng2+ig3 g1 f2+h2+if3 f1+h1\n\nα\nα\nβ\nβ\n, (10)\nwhere overbars denote complex conjugation. Note that the equation is in the Bogoliubov form with the matrix on\nthe right-hand-side being Hermitian. For succinct expressions of the matrix coefficients, let us introduce two distinct\northogonal decompositions of the film normal ˆz=zAn0\nA+z⊥AˆZ×n0\nA+zZˆZ=zBn0\nB+z⊥BˆZ×n0\nB+zZˆZ, where\nzA=n0\nA·ˆz, z⊥A=\u0010\nˆZ×n0\nA\u0011\n·ˆz, zZ=ˆZ·ˆzand similarly for the Blayer. The coefficients are then given by\nf1=µ0H·|γA|n0\nA+|γB|n0\nB\n2−1\n2\u0012|γA|\ndAMA+|γB|\ndBMB\u0013h\nJ1n0\nA·n0\nB+J2n\n3\u0000\nn0\nA·n0\nB\u00012−1oi\n+|γA|µ0MAz2\n⊥A+z2\nZ−2z2\nA\n4+|γB|µ0MBz2\n⊥B+z2\nZ−2z2\nB\n4(11)\nf2=|γA|µ0MAz2\n⊥A−z2\nZ\n4+|γB|µ0MBz2\n⊥B−z2\nZ\n4+1\n2\u0012|γA|\ndAMA+|γB|\ndBMB\u0013\nJ2n\n1−\u0000\nn0\nA·n0\nB\u00012o\n, (12)\nf3=|γA|µ0MAz⊥A+|γB|µ0MBz⊥B\n2zZ, (13)\ng1=µ0H·|γA|n0\nA− |γB|n0\nB\n2−1\n2\u0012|γA|\ndAMA−|γB|\ndBMB\u0013h\nJ1n0\nA·n0\nB+J2n\n3\u0000\nn0\nA·n0\nB\u00012−1oi\n+|γA|µ0MAz2\n⊥A+z2\nZ−2z2\nA\n4− |γB|µ0MBz2\n⊥B+z2\nZ−2z2\nB\n4(14)\ng2=|γA|µ0MAz2\n⊥A−z2\nZ\n4− |γB|µ0MBz2\n⊥B−z2\nZ\n4+1\n2\u0012|γA|\ndAMA−|γB|\ndBMB\u0013\nJ2n\n1−\u0000\nn0\nA·n0\nB\u00012o\n, (15)\ng3=µ0|γA|MAz⊥A− |γB|MBz⊥B\n2zZ, (16)\nh1=−rγAγB\ndAMAdBMB\u00141 +n0\nA·n0\nB\n2J1+n\n2\u0000\nn0\nA·n0\nB\u00012+n0\nA·n0\nB−1o\nJ2\u0015\n, (17)\nh2=rγAγB\ndAMAdBMB1−n0\nA·n0\nB\n2\b\nJ1+ 2\u0000\n1 + 2n0\nA·n0\nB\u0001\nJ2\t\n. (18)\nThe eigenfrequencies of Eq. (10) can be calculated as\nω2=f2\n1−f2\n2−f2\n3+g2\n1−g2\n2−g2\n3+h2\n1−h2\n2 (19)\n±2q\n(f1g1−f2g2−f3g3)2+ (f1h1−f2h2)2−(g1h2−g2h1)2−g2\n3(h2\n1−h2\n2).\nOne can observe that the “couplings” g1,2,3between\nαandβall vanish if the two layers are identical and\nHis in the plane. For identical layers with θ̸= 90◦,\ng1=g2= 0, g3̸= 0 due to zA⊥=−zB⊥and the\nproblem reduces to that of Refs. [23, 51]. The variables\nα, βrepresent oscillations that are odd and even under\nC2× {A↔B}, and can be considered generalisations of\nthe acoustic and optical modes in SyAFs, respectively.\nWhen g1,2,3become comparable with f1,2,3, h1,2, how-\never, αandβevenly contribute to the eigenmodes forall values of H. This makes it meaningless to talk about\nhybridisation between odd and even modes, which would\nrequire the modes be weakly coupled away from a res-\nonance region and come almost degenerate upon tuning\nsome parameters. Indeed, there is no simple relation be-\ntween g1,2,3and the spectral gap in general.5\n-50\n0\n50\n100\n150\n200\nEqm. Angle(�)\n-0.4\n-0.2\n0.0\n0.2\n0.4\n-1.0\n-0.5\n0.0\n0.5\n1.0\nExp.\nFit\nM/MS\n(a) (b)\n(f) (g)\nExp.\nFit(d) (c) (e)\nHH\nHMA MB\nAnti-ferromagnetic Canted Forced-ferromagnetic\n(j) (i)\n(h)\nExp.\nFit\nExp.\nFit\n-0.05\n0.0\n0.05\n-0.4\n-0.2\n0.0\n0.2\n0.4\nExp.\nFit\n-0.2\n-0.1\n0.0\n0.1\n0.2\n-0.2\n-0.1\n0.0\n0.1\n0.2\n0H (T)�\nA\nB\n0H (T)�\n0H (T)�\n0H (T)�\n0H (T)�\n(k) (l) (o) (n) (m)(�)\nFIG. 2. (a-e) Normalised M-Hloops for different set of samples (a) NiFe(5)/Ru(0.4)/NiFe(3) (b) NiFe(3)/Ru(0.4)/NiFe(5) (c)\nCoFeB(3)/Ru(0.45)/NiFe(3) (d) CoFeB(3)/Ru(0.5)/NiFe(3) and (e) CoFeB(3)/Ru(0.55)/NiFe(3). The field is applied along\nthe in-plane easy axis. The solid lines are fit obtained by the theoretical static state calculations based on Eq. (1). (f-j) Static\nstate angles of magnetisation for two FM layers calculated for the best fit parameters corresponding to (a-e) respectively. (k-o)\nAngle between the two magnetisations.\nTABLE I. Summary of the VSM magnetometry parameters used to obtain the theoretical magnetisation curves shown in Fig. 2\naccording to Eqs. (1) and (20). The left column represents the sample geometry where ” ..” indicates the thermally oxidized\nSi substrate and the FM near to the substrate is the second FM layer referred to as Blayer. µ0Hex, µ0H2exare the quadratic\nand biquadratic exchange fields respectively and MA(B), dA(B)are the magnetisation and thickness for the two ferromagnetic\nlayers (NiFe/CoFeB).\nSample µ0MAµ0MBµ0Hexµ0H2exdAdB\ngeometry (T) (T) (T) ((T) (nm) (nm)\nTa/NiFe/Ru(0.4)/NiFe/Ta/.. 0.95 0.9 0.145 0.03 5 3\nTa/NiFe/Ru(0.4)/NiFe/Ta/.. 0.95 0.9 0.1 0.02 3 5\nTa/CoFeB/Ru(0.45)/NiFe/Ta/.. 1.5 1.0 0.048 0.005 3 3\nTa/CoFeB/Ru(0.5)/NiFe/Ta/.. 1.5 1.0 0.02 0.003 3 3\nTa/CoFeB/Ru(0.55)/NiFe/Ta/.. 1.5 1.0 0.03 0.002 3 3\nIII. SAMPLE GROWTH AND\nMAGNETOMETRY CHARACTERISATION\nSamples used in this study were grown by using\nmagnetron sputtering techniques inside a chamber ata base pressure better than 5 ×10-6Pa. As sum-6\nmarised in Table I, we studied five different multi-layers\nTa(5)/FM 1(dA)/Ru/FM 2(dB)/Ta(5)/thermally oxidized\nSi substrate (numbers in the brackets represent layer\nthickness in nm) after optimising growth conditions [23,\n43, 66]. Figure 2 shows normalised hysteresis loops for\nthe samples measured for static external field in the plane\nby vibrating sample magnetometer (VSM) techniques.\nThree regions distinguished by the alignment of mag-\nnetisations of the two layers are indicated in different\ncolours. As explained in the previous section, due to\nthe competition between the exchange and Zeeman ener-\ngies, our samples undergo two phase transitions. In the\nsmall magnetic field limit H < H sf(shadowed in pink),\nthe exchange energy dominates and the two moments are\naligned antiferromagnetically. As the field is increased,\nthe spin-flop transition takes place, after which the two\nmoments tilt away from the field in a canted state. Fi-\nnally, at higher field values H > H ff, the Zeeman energy\nprevails and the magnetic moments point along the field\ndirection entering the forced ferromagnetic regime as in-\ndicated in green for each plot.\nEquation (1) was used for fitting to determine the\nstatic states of each moment. For obtaining the ground\nstate, we find the values of cos ϕA,Bthat minimise Eq. (1)\nforθA=θB= 90◦in an iterative manner for each mag-\nnetic field. The orange curves in the first row of Fig. 2\nare the normalised magnetisation calculated for each field\nvalue as:\nM(H)\nMs=dAMAcosϕA+dBMBcosϕB\ndAMA+dBMB. (20)\nwhere Msis the total saturation magnetisation of the\nsample. Optimisation with respect to the experimental\ncurves yielded the best-fit values of MA, MBas well as the\nquadratic ( µ0Hex=J1/√dAdBMAMB) and biquadratic\nexchange fields ( µ0H2ex=J2/√dAdBMAMB), which\nare summarised in the Table I. While the microscopic\norigin of J1is well-explained by the RKKY interaction\nvia electrons in the spacer layer [67], the identification\nof physical origins for J2is challenging among the sev-\neral proposals [68], such as intrinsic mechanism [69, 70],\nextrinsic fluctuation [71] and magnetic-dipole origin [72].\nWe however mention that our theoretical model and spin\ndynamics measurements treat the J2term phenomeno-\nlogically and are not influenced by its microscopic origin.\nIV. SPIN DYNAMICS\nHigh frequency responses of the coupled moment sys-\ntems were characterised by broadband on-chip microwave\nabsorption techniques. As illustrated in Fig. 3(a),\neach sample chip was placed face-down on a coplanar\nwaveguide [73]. For each measurement, we fixed the\nfrequency fand swept a dc external magnetic field\nµ0Hwhile applying an ac magnetic field at 12 Hz\nfor lock-in detection techniques. Here we show our\nmeasurements on the samples NiFe(5)/Ru(0.4)/NiFe(3)and CoFeB(3)/Ru/NiFe(3), both showing avoided cross-\ning [23, 29, 51] due to the asymmetry of thickness and\nmagnetic moment size, respectively.\nA. NiFe(5 nm)/Ru(0.4 nm)/NiFe(3 nm)\nFigures 3(b) and (c) represent individual measurement\ncurves targeted at two resonance modes in the sample\nNiFe(5 nm)/Ru(0.4 nm)/NiFe(3 nm) for θ= 90◦and\ndifferent frequencies. These individual scans are used\nto produce a f-µ0Htwo-dimensional plot as shown in\nFig. 3(d) to capture the absorption spectrum. At µ0H≈\n0.25 T, instead of mode degeneracy, we observe the\navoided crossing, suggesting that the in- and out-of-phase\noscillations are strongly hybridised [23, 29]. Figure 3(e)\nplots peak positions extracted by individual curve fittings\nusing derivative Lorentzian functions [74–76]. Equa-\ntion (20) with material parameters independently ex-\ntracted in the static VSM measurements (Table I) gener-\nates curves that are in reasonable agreement with exper-\niment. This displays the applicability of the macro-spin\nmodel with the minimal set of phenomenological param-\neters for this type of experiments. To highlight the role\nof thickness asymmetry for gap opening, we also show\ntwo additional sets of model calculations for ( dA,dB)=(5\nnm, 4 nm) and (5 nm, 5 nm). The model shows that\nthe spectral gap widens as the thickness asymmetry is\nincreased and the gap disappears in a symmetric system.\nFigures 3(f-g) confirm this prediction for the symmetric\nsample NiFe(5 nm)/Ru(0.4 nm)/NiFe(5 nm) with simi-\nlar thickness of two ferromagnets. The two modes cross\neach other at µ0H= 0.15 T with an absence of gap in\nthis case. The presence of mode symmetry prevents them\nfrom hybridisation and the two modes are degenerate at\nthe crossing point. Due to the asymmetry dA̸=dB, some\nof the coupling parameters in the off-diagonal blocks in\nEq. (10), i.e. g1andg2, become non-zero, for instance\nthrough the prefactor |γA|/µ0MA−|γB|/µ0MB). There-\nfore, even for the case of θ= 90◦, the thickness asym-\nmetry generates the hybridisation of in- and out-of-phase\noscillations.\nWe can further increase the gap size by tilting the mo-\nments towards the out-of-plane, as we previously demon-\nstrated in the symmetric cases [23]. Figures 4(a-d) sum-\nmarise the experimentally-measured θdependence of the\nmagnetic resonances. We performed peak position anal-\nysis for these θ-dependent results as shown in Fig. 4(e-h),\ntogether with ∆- θrelationship plotted in Fig. 4(i). Here\n∆ is defined as the minimum of the difference between\nthe upper and lower resonance frequencies as shown in\nthe Fig. 4(i) inset. Our theoretical curves successfully\nreproduce the experimental results without any tunable\nparameters. As the out-of-plane field increases, the gap\nis enhanced in comparison with that due to the thickness\nasymmetry alone and might be attributed to an increase\nofg3(Eq. (16)) with reducing θ. The observed trend is\nfurther supported by repeated experiments with a sample7\n0.280.320.36-16-12-8-4Upper - Mode\n17 GHz 18.4GHz\n17.6GHz 19 GHz\n18 GHz 19.4GHz\nH(T)\u00010(a)\nFIG. 3. (a) Schematic of the sample structure. (b)-(c) Absorption spectra for the sample NiFe(5)/Ru(0.4)/NiFe(3) at\n(b) low and (c) high field for θ=90◦. (d) Microwave transmission as a function of frequency and field for the sample\nNiFe(5)/Ru(0.4)/NiFe(3). The field is applied within the plane, θ=90◦. A clear avoided-crossing gap is visible at field\nµ0H= 0.25 T. (e) Fitting results for data as in (d). The solid lines are fitted curves obtained from macrospin model.\nThe increasing transparencies of the lines correspond to the model calculations for the case ( dA,dB)=(5 nm, 3 nm), (5 nm, 4\nnm) and (5 nm, 5 nm) respectively. It is seen from the calculations that the spectral gap widens as the thickness asymmetry\nis increased. (f-g) Similar plots as in (d-e) for sample NiFe(5)/Ru(0.4)/NiFe(5) at θ=90◦. A clear crossing is seen at at field\nµ0H= 0.15 T. This crossing indicates that the two modes are degenerate due to the inter-layer symmetry.\nwith the inverted growth order, i.e. NiFe(3 nm)/Ru(0.4\nnm)/NiFe(5 nm), demonstrating approximately the same\nquantitative behaviour as shown in Figs. 4(i). This\nproves that the angle dependence of the gap is a robust\nfeature independent of the assignment of top and bot-\ntom layers and small fluctuations in material parameters\nacross different fabrication conditions.\nB. CoFeB(3 nm)/Ru/NiFe(3 nm)\nIn order to experimentally demonstrate the effect of\nsymmetry breaking due to the asymmetry in magnetic\nmoments ( MA̸=MB) [52], we grew multilayers of\nCoFeB/Ru/NiFe where the thickness of the two FM ma-\nterials was kept fixed at 3 nm. Figure 5 summarises the\nspectral measurements/analysis for the sample CoFeB(3\nnm)/Ru(0.45 nm)/NiFe(3 nm) for different values of θ. A\nclear avoided-crossing gap is visible in the spectra shown\nin Fig. 5(a) for θ=π/2 and the model calculations (solid\ncurves) reproduce the dispersion curves with the degree\nof moment asymmetry fixed by the static VSM measure-\nments in this stack as shown in Fig. 5(e). This is be-\ncause g1andg2become non-zero when MA̸=MB(see\nEqs. (14)-(15)). g3further adds to the coupling when\nthe two moments have out-of-plane components and thistendency is experimentally demonstrated as shown in\nFigs. 5(a)-(h).\nFigure 5(i) displays the gap size ∆ as a function of θfor\nthe samples CoFeB(3 nm)/Ru( t)/NiFe(3 nm) with three\ndifferent Ru thicknesses, i.e. t= 0.45,0.50 and 0.55 nm;\nthe magnetic parameters of these samples extracted from\nVSM measurements are listed in Table I. The Ru thick-\nness does not directly enter the free energy equation or\nLLG equation, instead mostly influencing the interlayer\nexchange coupling strength µ0Hex. Hence, comparing\nthese three samples can be a good experimental demon-\nstration of the effect of the exchange coupling strength\non GHz spectra for the coupled moments. There is in-\ndeed direct correlation between ∆ and µ0Hexas shown\nin the inset of Fig. 5(i) for θ= 90◦. We also perform fur-\nther simulations using the same parameters in the sam-\nple CoFeB(3 nm)/Ru(0.45 nm)/NiFe(3 nm), except for\nµ0Hexbeing 0.1 T. ∆ of this simulation as a function of\nθis plotted in Fig. 5(j), supporting our claim.\nWe have so far shown the reliability of our macrospin\nmodel in reproducing the experimental results of mag-\nnetic resonance spectra in coupled moments via the inter-\nlayer exchange interaction. Here we present our theoreti-\ncal predictions to discuss the magnetic-parameter depen-\ndence of ∆. The asymmetry of coupled moments, i.e. MA\nandMB, can be further enhanced in simulation and we8\n8\n12\n16\n20\n�= 80�\nf(GHz)\n(a)\n(b)\n�= 60�\n(c)\n�= 30�\n(d)\n�= 20�\n0H (T)�\n0H (T)�\n0H (T)�\n0H (T)�\n0H (T)�o\nfu - fl (GHz)\nFIG. 4. (a)-(d) Microwave transmission as a function of frequency and applied field for the sample NiFe(5)/Ru(0.4)/NiFe(3)\nfor different θ. The angle θis defined as in Fig. 1. (e)-(h) Resonance frequency as a function of field obtained by derivative\nLorentzian fitting of the experimental data. The solid lines in the figure are theoretical results obtained from the macrospin\nmodel. (i) Spectral gap as a function of θobtained from theoretical model calculations. It can be seen that a maximum gap\nof≈4.5 GHz is achieved. The spectral gap is defined as the minimum of the difference between the upper ( fu) and lower ( fl)\nresonance frequencies as a function of µ0Has shown by the dotted line in inset for the sample NiFe(5)/Ru(0.4)/NiFe(3).\nfind that ∆ is monotonically increased by enlarging the\ndifference between MAandMBfor a fixed value of µ0Hex\nas shown in Fig. 6(a), reaching up to approximately 7.5\nGHz with µ0MA= 1.5 T and µ0MB= 0.4 T. This might\nbe achieved by selecting low-moment magnets as a coun-\nterpart of CoFeB to form a stack of synthetic ferrimagnet.\nOur model simulations also suggest that in such synthetic\nferrimagnets with large moment asymmetry, µ0Hexthat\ncan be tuned by the thickness of the intermediate layer\ncan act as a knob to further enhance ∆ as presented in\nFig. 6(b). See Appendix for individual spectra for ex-\ntracting ∆. Finally, the θdependence of ∆ for different\nvalues of µ0H2exis plotted in Fig. 6(c). For these simu-\nlations, an increase of µ0H2exdecreases ∆, which is qual-\nitatively different from the role of µ0Hexon ∆, e.g. in\nFig. 5(j). This is partially because of the general competi-\ntion between J1andJ2which prefer different static state\nconfigurations and therefore combine to soften the or-\nder and decrease the scale of resonance frequency. While\nµ0H2exis not a material parameter that can be easily\ntuned by growth conditions, it is interesting to notice\nthat the biquadratic nature enters the spectral responses\nvery differently from the quadratic counterpart. In gen-\neral, when the off-diagonal block elements g1, g2, g3be-\ncome comparable with the diagonal block ones as in the\npresent case, the notion of coupling between acoustic and\noptical modes becomes inappropriate, leading to the com-\nplex dependence of ∆ on not only the asymmetry related\nparameters but also the symmetry-respecting ones suchasµ0Hexandµ0H2ex. We would also like to add that in\nour model, we did not include the mutual spin pumping\nterm between the two magnetic layers [77]. However, the\nfact that we have good agreement between experiment\nand theory without the term indicates that the contribu-\ntion of the spin-pumping term seems to be insignificant.\nV. CONCLUSION\nWe studied the dynamics of synthetic ferrimagnets and\ntheoretically and experimentally showed their magnon-\nmagnon coupling with dissimilar material and thick-\nness of two ferromagnetic layers. We presented ana-\nlytical expressions of the coupled mode resonance fre-\nquencies and used them to quantitatively discuss exper-\nimental results. Using the rich and controllable spin-\nwave spectra in interlayer-coupled magnetic moments,\nthese materials might find their important use for future\nmagnonic/spintronic applications [30–33, 78, 79].\nACKNOWLEDGEMENTS\nA. S. thanks JSPS Postdoctoral fellowship for re-\nsearch in Japan (P21777) and EPSRC for their sup-\nports through NPIF EPSRC Doctoral studentship\n(EP/R512400/1) during her PhD at UCL. K. Y. is\nsupported by JST PRESTO Grant No. JPMJPR20LB,9\n3\n6\n9\n12\n(d)\n(c)\n(b)\n�= 90�\n(a)\n�= 60�\n�= 30�\n�= 12�\n�= 30�\n0H (T)�\n0H (T)�\n0H (T)�\nf(GHz)\n�\n�\n0H (T)�\nFIG. 5. (a)-(d) Microwave transmission as a function of frequency and applied field for the sample CoFeB(3)/Ru(0.45)/NiFe(3)\nfor different θ. The spectral gap increases as θis decreased. (e)-(h) Resonance frequency as a function of field obtained by\nderivative Lorentzian fitting of the experimental data. The solid lines in (e)-(h) are theoretical results obtained from the\nmacrospin model. (i) The spectral gap as a function of θfor different Ru thicknesses, which shows a gradual increase as θ\nis decreased. Inset shows the variation of ∆ as a function of µ0Hex. (j) Spectral gap as a function of θfor sample with Ru\nthickness 0.45 nm at µ0Hex= 0.1 T and 0.05 T. The gap shows an increase as µ0Hexis increased. The spectra used for\nextracting the spectral gap is given in Appendix C.\nJapan and JSPS KAKENHI (No. 21K13886). SM thanks\nto CSRN in CSIS at Tohoku Univ. and to JSPS KAK-\nENHI (No. 21H04648, 22F21777, 22KF0030)Appendix A: Collinear ground states\nThe coordinate axes we used in the main text, Eq. (9)\nare not well-defined for n0\nA·n0\nB=±1, namely when the\ntwo magnetisations are collinear in the static state. This\nhappens for H≤HsfandH≥Hffif the magnetic field10\n0.40.81.21.60369120\n.060.120 3 06 09 02.42.62.83.0(a)/s109 0MA= 1.5 T/s109\n0Hex= 0.048 T/s113\n = 90o/s68 (GHz)/s109\n0MB (T)(b)/s109\n0MA= 1.5 T/s109\n0MB= 0.5 T/s113\n = 90o/s109\n0Hex (T)(c)/s109\n0H2ex= 0.005 T/s109\n0H2ex= 0.002 T/s109\n0H2ex= 0.001 T/s113\n (o)\nFIG. 6. Spectral gap obtained from simulations by varying (a) MBand (b) µ0Hex. The other fixed parameters used for the\nsimulations are indicated on the plot. As the asymmetry is increased, a very large spectral gap ≈12 GHz is obtained for µ0Hex\n= 0.15 T as shown in (b). (c) Spectral gap as a function of θfor different biquadratic exchange field values µ0H2exfor the\nsample with Ru thickness of 0.45 nm. The other parameters used for the simulation are the same as given in Table I. For low\nµ0H2exvalues, the increase in gap size is not prominent as θis varied.\nis in-plane θ= 90◦, and more generally at high fields if\nthe two layers are identical.\nLet us first discuss the antiferromagnetic state n0\nA·\nn0\nB=−1, for which we can assume H=Hˆx. In place\nofC2given in Eq. (8), the static state satisfies C′\n2n0\nA=n0\nB\nwhere\nC′\n2v= (ˆy·v)ˆy−v. (A1)\nOne may still then define δ±= (δA± C′\n2δB)/√\n2 and\ndecompose them as δ±=δ⊥z\n±ˆz×n0\nA+δ∥z\n±ˆz. The rest\ndoes not have to be changed with zA=z⊥A=zB=\nz⊥B= 0, zZ= 1 and n0\nA=−n0\nB=±ˆxaccording to\ndAMA≷dBMB.\nFor the ferromagnetic state n0\nA·n0\nB= 1, ˆX=n0\nA\nis well-defined and one may redefine ˆY=ˆy. With this\nprovision, C2is simply a two-fold rotation about ˆXand\nδ±=δA∓δB. Again nothing needs to be modified in\nEq. (10) and beyond with z⊥A=z⊥B= 0.\nAppendix B: Additional magnetisation-dynamics\nresults for other samples measured in this study\nThis section provides supplementary results for sam-\nples measured in this study, which further supports the\nobservations and claims as described in the main text.\nTop panels (a-b) in Figs. 7 and 8 show measurements for\nsome remaining angles not shown in the main text for\nthe samples with stacking pattern CoFeB/Ru(0.45)/NiFe\nand NiFe(5)/Ru(0.4)/NiFe(3) respectively. The fittings\nproduced by our macrospin model are shown in bottom\npanel, which agree well with the experimental data.\nThe measurements were repeated for other sets of sam-\nples following the procedure outlined in the main text\nand we saw similar behaviours of spectral gap variation\nwith change in applied field angle towards out-of-plane\n36912f (GHz)(a)/s113 = 50°-\n14-21022P\n (a.u.)(b)/s113 = 20°0\n.080.1636912f (GHz)(c)/s109\n0H (T)0.150.30/s109\n0H (T)(d)FIG. 7. (a)-(b) Extra plots of microwave transmis-\nsion as a function of frequency and applied field for\nCoFeB(3)/Ru(0.45)/NiFe(3) for different θvalues. The spec-\ntral gap increases as θis decreased. Figures (c-d) shows reso-\nnance frequency obtained using derivative Lorentzian fitting\nof the experimental data and the solid lines are the theoretical\ncurves obtained from macrospin model.\nas shown in Fig. 9 for sample CoFeB/Ru(0.5)/NiFe and\nFig. 11 for sample CoFeB/Ru(0.55)/NiFe. The resonance\nfrequency obtained by fitting of experimental data using\nderivative of Lorentzian function along with the theoreti-\ncal predictions are plotted in Figs. 10 and 12 correspond-\ning to Figs. 9 and 11 respectively. For samples with Ru\nthickness 0.5 and 0.55 nm the gap opening is smaller than\nthe sample with Ru thickness 0.45 nm as shown in Fig. 9\nand Fig. 11. This is due to the lower value of µ0Hex\nin these samples. These results further support our ob-\nservation of spectral gap controlled by the out-of-plane11\n8121620/s113 = 70°f (GHz)(a)( b)/s113 = 40°-\n113P\n ( a.u.)0\n.10.20.30.48121620/s109\n0H (T)(c)f (GHz)0\n.20.40.6/s109\n0H (T)(d)\nFIG. 8. (a)-(b) Extra plots of microwave transmis-\nsion as a function of frequency and applied field for\nNiFe(5)/Ru(0.4)/NiFe(3) for different θvalues. Figures\n(c-d) shows resonance frequency obtained using derivative\nLorentzian fitting of the experimental data and the solid lines\nare the theoretical curves obtained from macrospin model for\nthe experimental data as in (a-b).\n0.000.020.040.062468(a)f (GHz)/s113 = 90°0\n.020.040.06/s113 = 60°( b)0\n.040.080.122468/s109\n0H (T)f (GHz)/s113 = 30°( c)0\n.080.160.24/s109\n0H (T)-1.4-0.9-0.40.10.61.11.62.12.6P\n (a.u.)/s113\n =12°( d)\nFIG. 9. (a)-(d) Microwave transmission as a function of fre-\nquency and applied field for CoFeB(3)/Ru(0.5)/NiFe(3) for\ndifferent θ. The gap opening is smaller as compared to sam-\nple with Ru thickness 0.45 nm due to smaller µ0Hexof this\nsample.\nangle θand exchange field strength µ0Hexas mentioned\nin the main text.\n0.000.020.040.062468f (GHz)/s113 = 90°( a)0\n.020.040.06(b)/s113 = 60°0\n.040.080.122468/s109\n0H (T)(c)f (GHz)/s113 = 30°0\n.080.160.24/s109\n0H (T)(d)/s113 = 12°FIG. 10. (a)-(d) Resonance frequency extracted from deriva-\ntive Lorentzian fitting of experimental data as a func-\ntion of applied field along with theoretical prediction for\nCoFeB(3)/Ru(0.5)/NiFe(3). These correspond to the data\nshown in Fig. 9.\n0.030.06246810(a)/s113 = 90°f (GHz)(\ng)0\n.030.060.09(b)/s113 = 60°0\n.040.080.12246810/s109\n0H (T)f (GHz)(c)/s113 = 30°0\n.080.160.24/s109\n0H (T)-5-3-113579P\n (a.u.)(\nd)/s113 = 12°\nFIG. 11. (a)-(d) Microwave transmission as a function of\nfrequency and applied field for CoFeB(3)/Ru(0.55)/NiFe(3)\nfor different θvalues. A small variation in spectral gap is\nseen as the θvaried.\nAppendix C: Numerical simulations to study the\nimpact of varying parameters on coupling gap\nUsing numerical simulations, we explored different pa-\nrameter regimes beyond the experimental conditions. 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Phys. 11, 453 (2015)." }, { "title": "1705.10027v2.Electronic_Structure_and_Magnetic_Properties_of_Half_metallic_Ferrimagnet_Mn___2__VAl_Probed_by_Soft_X_ray_Spectroscopies.pdf", "content": "arXiv:1705.10027v2 [cond-mat.str-el] 18 Jun 2017preprint\nElectronic Structure and Magnetic Properties of the Half-m etallic Ferrimagnet\nMn2VAl Probed by Soft X-ray Spectroscopies\nK. Nagai,1H. Fujiwara,1,∗H. Aratani,1S. Fujioka,1H. Yomosa,1Y. Nakatani,1\nT. Kiss,1A. Sekiyama,1F. Kuroda,2,3,4H. Fujii,3,4T. Oguchi,3,4A. Tanaka,2J.\nMiyawaki,5,6Y. Harada,5,6Y. Takeda,7Y. Saitoh,7S. Suga,4and R. Y. Umetsu8\n1Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan\n2Department of Quantum Matter, ADSM, Hiroshima University, Higashi-hiroshima, Hiroshima 739-8530, Japan\n3CMI2-MaDIS, National Institute for Materials Science, 1-2-1 Se ngen, Tsukuba, Ibaraki 305-0047, Japan\n4Institution of Scientific and Industrial Research,\nOsaka University, Ibaraki, Osaka 567-0047, Japan\n5Institute for Solid State Physics (ISSP), University of Tok yo, Kashiwanoha, Chiba 277-8581, Japan\n6Synchrotron Radiation Research Organization, University of Tokyo, Sayo-cho, Sayo, Hyogo 679-5198, Japan\n7Materials Sciences Research Center, Japan Atomic Energy Ag ency (JAEA), Sayo, Hyogo 679-5148, Japan\n8Institute for Materials Research, Tohoku University, 2-1- 1 Katahira, Sendai 980-8577, Japan\n(Dated: June 17, 2021)\nWe have studied the electronic structure of ferrimagnetic M n2VAl single crystals by means of soft\nX-ray absorption spectroscopy (XAS), X-ray absorption mag netic circular dichroism (XMCD) and\nresonant soft X-ray inelastic scattering (RIXS). We have su ccessfully observed the XMCD signals\nfor all the constitute elements. The Mn L2,3XAS and XMCD spectra are reproduced by spectral\nsimulations basedondensity-functionaltheory, indicati ngtheitinerantcharacter oftheMn3 dstates.\nOn the other hand, the V 3 delectrons are rather localized since the ionic model can qua litatively\nexplain the V L2,3XAS and XMCD spectra. This picture is consistent with local ddexcitations\nrevealed by the V L3RIXS.\nPACS numbers: 71.20.Be,75.50.Gg,78.70.Dm\nINTRODUCTION\nHalf-metals are characterized by a peculiar electronic\nstructure as one of the spin sub-bands is metallic and\nthe other is semiconducting with a gap at the Fermi level\n(EF). The expected 100%spin-polarizationat EFis suit-\nable for functional spintronic applications. In particular,\nfull-Heusler alloys with the chemical formula X2YZ(X\nandY: transition metals, Z: main group element) have\nbeen extensively studied since the Currie temperature\n(TC) is , forexample, ashigh as 985K forCo 2MnSi [1, 2].\nL21ordered Mn 2VAl is a ferrimagnet with TC∼760\nK[3, 4], and is predicted to be half-metallic [5, 6]. The\nsaturation magnetic moment ( Ms) is reported as 1.94 µB\nper formula unit (f.u.) for poly crystalline samples [4, 7],\nwhich is much smaller than that of 4.78 µB/f.u. for\nCo2MnSi [8]. The small Msdue to the antiparallel spin\ncoupling of V and Mn offers an advantage for saving\nthe electric power of the current-induced magnetization\nswitching, since the switching current is proportional to\nM2\ns[9, 10].\nTorevealthedetailedmagneticpropertiesandtheelec-\ntronic structure of the Heusler alloys, intensive studies\nhave been performed by means of X-ray absorption spec-\ntroscopy (XAS) and X-ray absorption magnetic circular\ndichroism (XMCD) [11–19]. The electronic structure of\nCo2MnSi has been known as the Co 3 delectrons are itin-\nerant, while the Mn 3 delectrons contribute to the local\nmagnetic moment [13, 14, 17]. Nevertheless, only fewworks have been performed on Mn 2VAl [20–22] because\nof the difficulty in synthesizing high quality crystals. It\nhas been reported as the epitaxially grown Mn 2VAl films\nare prone to various disorder [20]. Since the disorder\neffects change the magnetic properties and the local elec-\ntronic structure [21], the line shapes of XAS and XMCD\nspectra varied depending on the individual samples [20–\n22]. Therefore, the electronic structure of Mn 2VAl is still\ncontroversial.\nIn this work, we present the detailed electronic struc-\nture and magnetic properties of the high quality sin-\ngle crystals of L2 1-ordered Mn 2VAl, revealed by means\nof XAS and XMCD for all constitute elements. The\nmagnetic moments of the Mn sites evaluated by using\nthe magneto-optical sum-rule [23, 24] were qualitatively\nconsistent with the results of density functional theory\n(DFT) within the experimental accuracy. We found that\nthe line shapes of Mn L2,3XAS and XMCD spectra\nof Mn 2VAl were deviated from those of Mn spectra in\nCo2MnSi, and are well explicable by the spectral simu-\nlation based on DFT, indicating the itinerant character\nof the Mn 3 delectrons. On the other hand, the V L2,3\nXAS and XMCD spectra were qualitatively described in\nthe ionic model. Moreover, the ddexcitation in the V\n3dstates was probed using resonant soft-X-ray inelas-\ntic scattering (RIXS) [25, 26], suggesting that the V 3 d\nelectrons are localized.2\nEXPERIMENTAL\nCharacterization of single crystals\nPolycrystallinemotheringotsofMn 2VAlandCo 2MnSi\nwerepreparedbyinductionmeltinginanargongasatmo-\nsphere. Since the melting temperature of V is very high\nas∼2180 K and the vapor pressure of V is high enough\nduring the induction melting process, the excess Mn ions\nare contained in the mother ingot of Mn 2VAl. Single\ncrystalsofboth Mn 2VAl andCo 2MnSi weregrownbythe\nBridgeman method with a size of 12 mm in diameter and\nabout30mminlength. Obtainedingotswereannealedat\n1473 (1373) K in the case of Mn 2VAl (Co 2MnSi) to gain\nthe size of single crystal grains. For Mn 2VAl, a two-step\nannealingprocessat1123Kand873Kwasemployedand\nthe crystal was finally slow-cooled to room temperature\nto control the degree of order. The specimen of Co 2MnSi\nwas obtained by slow cooling from 1373 K to room tem-\nperature. Crystal orientation was checked with a Laue\ncamera and the specimen was cut in a strip form in the\ndirection parallel to <100>for Mn 2VAl and <110>for\nCo2MnSi. Composition of the specimens was confirmed\nwith an electron probe microanalyzer(EPMA) to be Mn:\n50.5, V: 26.9, Al: 22.6 for Mn 2VAl and Co: 50.0, Mn:\n25.9, Si: 24.1 for Co 2MnSi. Moreover, the long-range or-\nder parameter for the L2 1structure (S L21) [27, 28] was\nevaluated as 0.84 for Mn 2VAl and 0.90 for Co 2MnSi by\nusing the X-ray diffraction measurements. Note that the\nSL21value for Mn 2VAl wasmuch higher than that for the\nepitaxial films of S L21∼0.5[20, 21], indicating the well\nordered single crystal.\nThe bulk magnetization was measured with a super-\nconducting quantum interface device (SQUID) magne-\ntometer. We obtained a total magnetic moment of\nmSQUID=1.82µB/f.u. forMn 2VAlsinglecrystals,which\nis slightly smaller than the value of 1.94 µB/f.u for poly-\ncrystalline samples [4, 7] and the expected value of 2\nµB/f.u. from the Slater-Pauling rule [1]. This devia-\ntion would be due to off-stoichiometric effects; the small\namount of excess Mn and V ions substituted into Al\nsites can reduce the total magnetic moment, (see AP-\nPENDIX).\nSoft X-ray spectroscopies\nSingle crystalline samples were fractured in situin\nultrahigh vacuum to obtain the clean surface. XAS\nand XMCD measurements were performed at BL23SU\nin SPring-8 [29]. The spectra were recorded in total-\nelectron-yieldmodewithanenergyresolutionbetterthan\n0.1 eV using a superconducting magnet in fields up to 2\nT along the incident beam direction. To eliminate any\nexperimental artifacts arising from system errors, each\nXMCD spectrum was measured for opposite orientationsof the applied magnetic field and the resulting spectra\nwere averaged. The measurement temperature was set\nto 20 K. The RIXS measurements were performed at the\nSPring-8BL07LSU ‘HORNET’ end-station [30–32]. The\ntotal energy resolution was set to ∼200 meV at the mea-\nsurement temperature of 300 K.\nDENSITY FUNCTIONAL THEORY\nThe electronic structure calculation based on DFT\nhas been performed using the HiLAPW code, which\nis based on the all-electron full-potential augmented\nplane-wave (FLAPW) method[33]. The generalized gra-\ndient approximation (GGA) using the Perdew-Burke-\nErnzerhofschemehasbeen used forthe exchangecorrela-\ntion potential[34, 35]. We have used the second-variation\nprocedure to include the spin-orbit coupling in addition\ntoascalar-relativisticscheme, givinganaccuratedescrip-\ntionequivalenttosolvingtheDiracequationforrelatively\nlight elements as 3 dtransition metals. Plane-wave ex-\npansion cutoffs were set to 20 Ry for wave functions and\n80 Ry for charge density and potential functions. The\nmuffin-tin sphere radius was chosen as 1.1 ˚A for all ele-\nments. For the Brillouin-zone integration, a 16 ×16×16\nmesh was used with the tetrahedron integration tech-\nnique. The atomswereplaced onthe generalform X2YZ\nof the L2 1structure with Xat (1\n4,1\n4,1\n4) and (3\n4,3\n4,3\n4),Yat\n(0,0,0), and Zat (1\n2,1\n2,1\n2). The lattice constant was set\nto 5.875 ˚A [6].\nFigure 1(a) shows the calculated spin-polarized den-\nsity of states for Mn 2VAl with an half-metallic gap for\nthe majority-spin states, which is qualitatively consis-\ntent with previous reports [5, 6]. The partial density\nof states (PDOS) in Figs. 1(b) and 1(c) show that the\nMn 3dminority-spin sub-bands contribute mainly to the\nstates in the vicinity of EF. Meanwhile, PDOS of the V\n3dstates are only ∼10% of PDOS of the Mn 3 dstates\naroundEFas shown in the insets. Clear splitting of\nthe V 3dstates into the occupied t2gand unoccupied eg\nstates is seen with the band splitting energy of ∼3.2 eV\nas shown in Fig. 1(d). The PDOS of the Al 3 pstates are\neven smaller and ∼10% of the PDOS of the V 3 dstates\natEF.\nThe whole XMCD spectra were simulated within\nthe dipole approximation on the basis of the FLAPW\nmethod, where the transition matrix elements were cal-\nculated only within the muffin-tin sphere. The core-\nhole potential in the final states was neglected in the\npresent calculations. To take into account the experi-\nmentalcore-holelifetimebroadening,thecalculatedspec-\ntra were broadened using Lorentzian functions with γ=\n0.97(0.36) eV and 0.78(0.28) eV for the L2(L3) edges of\nMn and V, respectively, while 0.15 eV was employed for\nthe AlKedge. The detailed information about the cal-\nculation has been given in Ref. 36.3\n-2 02(c) Mn d PDOS \n eg \n t2g\n-2 02(d) V d PDOS\n eg \n t2g-4 -2 024\n-4 -2 0 2 4(b) PDOS \n Mn\n V \n Al-10 -5 0510 DOS (states / eV spin cell) -4 -2 0 2 4(a) Total \nmajority \nminority \n-0.3-0.2-0.10.00.10.20.3\n-4 -2 0 2 4\nEnergy relative to E F (eV) (e) Al PDOS\n p -0.3-0.2-0.10.0\n0.4 0.2 0.0 -0.2\n-0.03-0.02-0.010.00\n0.4 0.2 0.0 -0.2 -1.5-1.0-0.50.0\n0.4 0.2 0.0 -0.2Mn VAl 2\nFIG. 1: (Color online) (a) Total density of states of L2 1or-\ndered Mn 2VAl. (b) PDOS of Mn, V, Al sites. (c), (d) PDOS\nof Mn and V 3 dstates projected onto the egandt2gorbital.\n(e) PDOS of Al 3 pstates. The insets show the enlarged view\nof PDOS around EF.\nRESULTS AND DISCUSSIONS\nMagnetic properties probed by XMCD and\nsum-rule analysis\nFigures 2(a) and 2(b) show the experimental results\nof Mn and V L2,3XAS and XMCD spectra of Mn 2VAl.\nµ+(µ−) denotes the XAS absorption intensity for par-\nallel (antiparallel) alignment of the photon helicity and\nsample magnetization direction. The XAS spectra are\nnormalized by the L3peak intensity of the polarization-\nsummed XAS ( µ−+µ+). There are fine structures in the\nXMCD ( µ−−µ+) spectra for both Mn and V L2,3edges,\nand the sign of the XMCD signals is opposite between\nthem, reflecting ferrimagnetic coupling between the Mnand V spins (Mn along applied field). The detailed line\nshape is discussed in the next subsection (IV. B).\nMoreover, we can clearly see sizable XMCD signals\neven in the Al Kedge as shown in Fig. 2(c) despite its\nnon-magnetic atomic character [37, 38], indicating a non\nnegligible orbital polarization of the valence states with\nAl 3pcharacter. The most prominent XMCD peak is lo-\ncated just at the threshold. The XMCD signals originate\nfrom a combination of the spin-orbit and exchange split-\ntings of the Al 3 pstates. The integration of the XMCD\nsignals over the K edge, which is proportional to the or-\nbital magnetic moment [25, 39], falls to zero within the\nexperimental accuracy.\nThe theoretical simulations are consistent with the ex-\nperimental spectra, although there are some discrepan-\ncies in XMCD. The DFT calculation yields an orbital\nmoment of mL= 0.001µB/Al with an orbital-to-spin\nmagnetic moment ratio of mL/mS=−0.05. The devia-\ntion between the experiment and theory especially on the\nhigher energy region is possibly due to the energy depen-\ndent lifetime broadening effects, which are not taken into\naccount in the simulation, since the broadening width is\ngiven by fitting the edge step of the XAS spectrum.\nTo reveal more detailed magnetic properties of the\ntransition metal sites, the orbital and spin magnetic\nmoments per dhole were estimated by applying the\nmagneto-optical sum-rule [23, 24, 40, 41] using the equa-\ntions,\nmspin\nnh=−6p−4q\nrPcC, (1)\nmorb\nnh=−4q\n3rPc, (2)\nwherenhis the number of unoccupied dholes,p(q) is\nthe integral of the XMCD signal over the L3(L2,3)-edge,\nandris the integral of the polarization summed XAS\nintensity after subtracting the background. As shown in\nthe bottom of Figs. 2(a) and 2(b), the cut off energy\nto obtain pwas set to the minimum of the polarization\nsummed XAS intensity between the L2andL3-edges.\nTheqandrvalues were evaluated at the energy of 694\neV for Mn and 570 eV for V. Pcdenotes the degree of\ncircular polarization of 0.97 [29]. In addition, we must\ntake into account the so-called jjmixing effect by us-\ning the correction factor C for the spin sum-rule [42].\nAssuming C = 1.5 for Mn[43], we obtained a spin (or-\nbital) magnetic moment of mMn\nspin(mMn\norb) = 0.27 (0.005)\nµB/Mn/hole, which isconsistentwith that obtained with\nourDFT calculationas summarizedin TableI. Assuming\nnh= 5.20 estimated from the DFT, we obtained a total\nMn moment of 2.86 µB/f.u.\nUsing the orbital sum-rule for V, we obtained an or-\nbital magnetic moment of mV\norb= 0.005 µB/V/hole as\ngiven in Table I. Meanwhile, it is not straight forward to\nadopt the spin sum-rule for V since the C value is not\nestablished and ranges from 3.6 to 5.0 due to the small4\n0.7 \n0.6 \n0.5 \n0.4 \n0.3 \n0.2 \n0.1 \n0.0 \n-0.1\n-0.2Intensity (arb. unit) \n680 660 640Mn \n2 Tesla\n20 K\n \n μ−\nμ+\n bg(a) 0.7 \n0.6 \n0.5 \n0.4 \n0.3 \n0.2 \n0.1 \n0.0 \n-0.1\n-0.2\n560 540 520V\n2 Tesla \n20 K \n(μ−– μ+) 2(b)\n10 \n8\n6\n4\n2\n0r\n− + ∫ ( μ + μ − bg )6\n4\n2\n0XAS Integral r\n ∫ ( μ− + μ+\n − bg )\n-0.2-0.10.0 0.1 0.2 XMCD Integral 680 660 640\nPhotonp q ∫ ( μ− − μ+\n )\n-0.10.00.1\n560 540 520\nEnergy (eV) pq ∫ (μ− − μ+\n ) \n μ−\nμ+\n bg \n(μ−– μ+) 2\n0\n0Mn VAl 2\nIntensity (arb. unit) \n1568 1564 1560(c) Al K-edge \n 2 Tesla\n 20 K \nXAS \nExperiment \nXAS \nCalculation (DFT) \nXMCD 100 \nExperiment \nXMCD 100 \nCalculation (DFT) \nFIG. 2: (Color online) XAS and XMCD spectra of Mn 2VAl recorded at T = 20 K with H = 2 T. (a) Mn and (b) V L2,3XAS\nand XMCD spectra (top), together with the integrated curves of XAS (middle) and XMCD (bottom). (c) Al KXAS and\nXMCD spectra with the DFT-based simulation.\nTABLE I: Spin and orbital magnetic moments of Mn 2VAl in\nunit ofµBper hole for an Mn (V) site obtained by the sum-\nrule analysis and DFT.\nMn2VAl\nmspin/nhmspin/nhmorb/nhmorb/nh\n(XMCD) (DFT) (XMCD) (DFT)\nMn 0.27 0.28 0.005 0.006\nV -0.15 -0.11 0.005 0.001\nspin-orbit splitting at the V L2,3edges [42, 44]. If we\nevaluate mV\nspinusing the spin sum-rule assuming C = 1,\nwe obtained mV\nspin= -0.04µB/V/hole. In this case, we\nobtained a total V moment of -0.26 µB/f.u., assuming nh\n=7.39 estimated from the DFT. On the other hand, we\ncan evaluate the total V moment as mV=−1.04µB/f.u.\nusing the total Mn moment (2.86 µB/f.u.) and the total\nbulk moment (1.82 µB/f.u.) obtained by SQUID. There-\nfore, we found that a likely correction factor for V is C =\n3.8, beingcomparabletothatof3.6giveninRef. 42. Thespin (orbital) moment for V is deviated from the DFT\nvalue ofmV\nspin(mV\norb) = -0.11 (0.001) µB/V/hole as listed\nin Table I, implying the limitation of DFT for describing\nthe electronic structure of the V 3 dstates as discussed\nlater.\nFigure 3 (a) shows the relative intensity of the Mn L3\nXMCD signal at 640.2 eV on sweeping the field from -2\nto 2 T, giving the element specific magnetization profile\nof the Mn sites. Meanwhile, the field dependence of the\nVL3XMCD intensity at 514.4 eV shows the opposite\ntrend reflecting the antiferromagnetic spin coupling to\nthe Mn moment as shown in Fig. 3(b). The enlarged\nviews around 0 T in the insets of Figs. 3(a) and 3(b)\ngive a coercivity of 10 mT for both Mn and V sites,\nwhich is about 10% forthe epitaxial films [20]. Moreover,\nthe XMCD intensity does not changeafter saturating the\nmagnetization,andthustheparamagneticcomponentre-\nported in Ref. 20 is negligible. Therefore we stress that\nthe impurity and disorder effects are successfully sup-\npressed in our single crystalline samples.5\n-1.0 -0.5 0.00.51.0\n-2 -1 0 1 2\nMagnetic Field (Tesla)(b)\nV 3-edge \nhν = 514.4 eV \n20 K\n-1.0-0.50.00.51.0Relative intensity (arb. unit) -2 -1 0 1 2(a)\nMn 3-edge\nhν = 640.2 eV \n20 K\n-40 -20 020 40 \nMagnetic Field (mT) -40 -20 020 40 \nMagnetic Field (mT)L L\nFIG. 3: (Color online) (a), (b) Magnetic field dependence of\nthe XMCD intensity at the Mn and V L3peak. The inset\nshows the expanded view around the 0 T.\nXAS and XMCD line shapes: itinerant v.s. localized\ncharacter in the electronic states\nTo discuss the characteristic electronic properties of\nMn2VAl, we focus on the detailed line shapes of the\npolarization-summed XAS and XMCD spectra for both\nMn and V L2,3edges. Figure 4(a) displays the enlarged\nview of the Mn L2,3XAS spectrum of Mn 2VAl, showing\na metallic line shape as reported in Ref. 22. Compared\nwith the XAS spectrum of Co 2MnSi, in which the Mn\n3dstates have a localized 3 d5character [13, 14, 17], the\nL3main line of Mn 2VAl shifts to higher energies even\nthough the leading edges start at the same energy of 638\neV. Moreover, the L2peak of Mn 2VAl does not show the\ndoublet structure due to the atomic multiplets observed\nin Co2MnSi [12–14, 17, 46, 47].\nIn contrast, the XMCD spectrum of Mn 2VAl shows\nclear doublet peaks in the L2edge as well as a clear\nshoulder structure on the low energy side of the L3main\npeak as shown in Fig. 4(b). This line shape is qualita-\ntively consistentwith that reportedin Ref. 21. Note that\nthe MnL3XMCDspectrum ofCo 2MnSi showsthe single\npeak locatedat the middle ofthe L3main line in contrast\nto the low-energy shoulder present in Mn 2VAl, illustrat-\ning a clear difference in the local electronic structure on\nthe Mn sites between these two materials.\nIn Fig. 5 we show the V L2,3XAS and XMCD spectra\nof Mn 2VAl together with those of V metal reported in\nRefs. 44 and 48 to discuss the electronic structure of the\nY-site element. The V L2,3XAS spectrum of Mn 2VAl is\nqualitatively similar to that for V metal. Nevertheless,\nthe XMCD spectrum of Mn 2VAl significantly deviates\nfrom that of V metal. The single XMCD peak before the\nL3mainline observedin Vmetal issuppressed, whiletwo\ntinyshoulderstructuresshowupbeforethe L3mainpeak\nin Mn2VAl [20, 21]. In addition, the L2XMCD shape for\nMn2VAl is clearly different from that of V metal, which\nis well reproduced by DFT-based simulations [44, 48].\nThese results suggest the localized character of the V 3 d\nstates in Mn 2VAl.\nTo obtain further insight of the electronic structure for\nthe Mn and V 3 dstates, we haveanalyzedthe line shapes660 655 650 645 640 635\nPhoton Energy (eV)Mn XMCD\n \n \n Mn 2VAl\n Co 2MnSi(b)Intensity (arb. unit) 660 655 650 645 640 635 Mn XAS (a)\n0 \n Mn 2VAl\n Co 2MnSi\nFIG. 4: (Color online) Mn L2,3-edges XAS (a) and XMCD\n(b) spectra for Mn 2VAl(solid line) and Co 2MnSi(dashed line).\n530 525 520 515 510 V XAS\n \n (a) \n530 525 520 515 510V XMCD\n \n Mn 2VAl\n V metal (ref. 48)(b )\nPhoton Energy (eV)Intensity (arb. unit) \n0 Mn 2VAl\n V metal (ref. 48)\nFIG. 5: (Color online) V L2,3-edges XAS (a) and XMCD\n(b) spectra for Mn 2VAl(solid line) and V metal (dashed line)\nreported in the reference [48]. The XMCD spectra are nor-\nmalized at the L3peak.6Intensity (arb. unit) \n655 650 645 640 635\nPhoton Energy (eV) 0\n0\n0Experiment Mn L2,3 XAS\nDFT \n 10Dq = -0.5 eV\n 10Dq = -1.5 eV\n \n 10 Dq = -0.5 eV\n10 Dq = -1.5 eV\nXMCD (x6) \nExperiment \nDFT \nion model ion model Mn 2VAl \nFIG. 6: (Color online) Mn L2,3-edge XASand XMCD spectra\ncompared with the simulation based on DFT and the ionic\nmodel. Spectra are normalized by the L3-peak intensity of\nthe polarization summed XAS spectra.\nof XAS and XMCD spectra using simulations based on\nthe DFT and the ionic model, representing the itiner-\nant and localized limit of the 3 dstates. The ionic cal-\nculations based on the full multiplet theory were imple-\nmented using the XTLS 9.0program[49]. The local crys-\ntalline electric field (CEF) was taken into account for the\nMn2+ion with tetrahedral ( Td) symmetry and the V2+\nion with octahedral ( Oh) symmetry. All the intra-atomic\nparameterssuch as the 3 d-3dand 2p-3dCoulomb and ex-\nchange interactions (Slater integrals) and the 2 pand 3d\nspin-orbit couplings have been obtained using Cowan’s\nHartree-Fock code [50] and reducing the Slater parame-\nters to 80%.\nIn Fig. 6, we show the results of spectral simulations\nfor the Mn L2,3XAS and XMCD. The ionic model XAS\ncalculation with 10Dq = -0.5 eV [14] shows multiplet\nstructures with a shoulder on the higher energy side of\ntheL3main line and the L2doublet. Moreover, an ad-\nditional low-energy satellite at the L3peak appeared,\nreflecting the CEF splitting of 10Dq = -1.5 eV. However,\nsuch multiplet structures are not observed in the experi-\nmental XAS spectrum. Furthermore, the XMCD simula-\ntion based on the ionic model is obviously deviated from\nthe experimental result, whereas the DFT-based simu-\nlation better reproduces the XAS and XMCD spectra,\nparticularly for the low energy structure at the L3edge,525 520 515 510\nPhoton Energy (eV) 000V L2,3 XAS \nDFT \n 10Dq = 2.0 eV \n 10Dq = 0.5 eV \n Experiment \nIon model XMCD (x8) \nExperiment \nDFT Ion model \n 10Dq = 2.0 eV \n 10Dq = 0.5 eV Intensity (arb. unit) Mn 2VAl \nFIG. 7: (Color online) V L2,3-edge XAS and XMCD spectra\ncompared with the simulation based on DFT and the ionic\nmodel. Spectra are normalized by the L3-peak intensity of\nthe polarization summed XAS spectra.\nindicating the itinerant character of the Mn 3 dstates in\nMn2VAl.\nOn the other hand, the situation is different in the\ncase of V L2,3XAS spectra as shown in Fig. 7. The\nDFT-based simulation fails to explain the relative peak-\nintensity ratio of L3andL2XAS. Moreover, the line\nshape of XMCD spectrum is not explained by DFT es-\npecially for the low-energy shoulders of the L3XMCD.\nIn addition, one may also notice that the L2XMCD is\nalso deviated from DFT in contrast to the case of V-\nmetal [44, 48] (Fig. 5(b)), implying the limitation of the\nband picture for the V 3 dstates in the case of Mn 2VAl.\nMeanwhile, the ionic calculation reproduces the exper-\nimental results rather well with a CEF parameter of\n10Dq = 2 eV for Ohsymmetry, which is not dramati-\ncally smaller than the band-splitting energy between the\nV 3d t2gandegstates obtained by the DFT as shown in\nFig 1. These results suggest that the V 3 delectrons are\nrather localized and thus the atomic multiplets should be\ntaken into account for the spectral simulations.\nTo check the localized nature of the 3 delectrons, we\nhave performed Mn and V L3RIXS, which is a pow-\nerful tool to probe the local electron excitation in the\n3dstates [51, 52]. As shown in Fig. 8(a), the energy-\ndispersive fluorescence components are dominant in Mn\nL3RIXS, supportingthe itinerantpicture [53] for the Mn7\n644 \n642 \n640 \n638 \n12 8 4 0\n518 \n516 \n514 \n512 \n12 8 4 0high \nlow Mn L3-RIXS V L3-RIXS \nEnergy Loss (eV) (a) (b) Excitation Photon Energy (eV) \nFIG. 8: (Color online) Intensityplots of hνin-dependentRIXS\nat the Mn (a) and V (b) L-edges. Dashed circle is the guide\nfor the eye indicating the resonant excitation.\n3dstates. Meanwhile, the V L3RIXS in Fig. 8(b) clearly\nshows the clear resonant inelastic excitation ( ddexcita-\ntion) with the constant energy-loss component around 2\neV, which is consistent with the 10Dq value of ionic cal-\nculations forthe V L2,3XAS and XMCD spectra. There-\nfore, we stress that this ddexcitation indicates the local-\nized character of the V 3 delectrons in Mn 2VAl.\nCONCLUSIONS\nWe investigated the element specific magnetic proper-\nties and electronic structure of three different constitute\nelements of high quality single crystal Mn 2VAl by using\nXAS and XMCD. The successful observation of the Al\nKXMCD signals suggests the very small spin unbalance\nof the Al pstates near EFinduced by the hybridiza-\ntion with the transition metal 3 dstates. The Mn and\nVL2,3edge XMCD spectra show the ferrimagnetic spin\ncoupling between Mn and V states, and the magnetic\nmoments evaluated by the sum-rule analysis has shown\nan excellent agreement with DFT calculation for its Mn\nsites. The electronic structure analysis for the Mn L2,3\nXAS and XMCD spectra with the DFT-based simula-\ntion reveals the itinerant character of the Mn 3 dstates,\nwhereas the V L2,3XAS and XMCD spectra are qual-\nitatively explained by the ionic model. In addition, V\nL3RIXS showed the ddexcitation, indicating the local-\nized character of the V 3 dstates. Our results give the\nsignificant remark to design the magnetic properties by\ncontrolling the itinerant and localized character of the d\nstates depending on the site symmetry in Heusler alloys.\nACKNOWLEDGMENTS\nWe thank T. Kanomata, S. Imada, A. Yamasaki, K.\nYamagami, M. Kawada, A. Koide and A. Kimura for\nfruitful discussions. The measurements were performed\nunderthe approvalofBL23SUandBL07LSUatSPring-8\n(Proposal Nos. 2016A3832 (E28) and 2016B7512). FK,HF, and TO would like to thank support by Materialsre-\nsearch by Information Integration Initiative from CMI2-\nMaDIS, NIMS. This work was financially supported by a\nGrant-in Aid for Scientific Research (JP16H04014) from\nthe Ministry of Education, Culture, Sports, Science and\nTechnology, Japan. A part of this work was supported\nby Japan Science and Technology Agency, Precursory\nResearch for Embryonic Science and Technology (JST-\nPRESTO).\nAPPENDIX: OFF-STOICHIOMETRY EFFECTS\nThe off-stoichiometry effects in Mn 2VAl were numer-\nically simulated by the Korringa-Kohn-Rostoker (KKR)\nmethodincorporatedwiththecoherent-potentialapprox-\nimation (CPA) in the local density approximation(LDA)\nimplemented by Machikaneyama-2002 package [54] as\ndemonstrated in Fig. 9. The atomic positions and the\nlattice constant were set to the same values as Section\nIII. Assuming that the Al sites are substituted by excess\nions with 2% for Mn and 8% for V, which is evaluated\nby EPMA for the measured specimen, we obtained the\ntotal magnetic moment of 1.71 µB/f.u. This value was\nsmaller than that of the stoichiometric composition of\n1.96µB/f.u. The obtained spin polarization at EFde-\nfinedasP= (N↑−N↓)/(N↑+N↓), whereN↑(N↓)denotes\nDOS atEFfor the majority (minority) spin subband in\nFig. 9, is obtained as -0.93 (-0.91) for off-stoichometric\n(stoichometric) composition. Note that the total density\nof states are not significantly modified due to the off-\nstoichiometry effects within the simulation as shown in\nFig. 9.\n-10-50510 DOS (states/eV spin cell)\n-4 -2 0 2 4\nEnergy Relative to E F (eV)/s32/s33/s34/s35/s36/s37/s38/s35/s39/s40/s34/s41/s36/s37\n/s32/s35/s42/s42/s43/s33/s34/s35/s36/s37/s38/s35/s39/s40/s34/s41/s36/s37/s32\n/s32/s32/s32/s44/s45/s32/s46/s47/s32/s48/s49/s50/s32/s51/s49/s32/s52/s47/s32/s33/s53/s54/s33/s34/s36/s34/s53/s34/s36/s35/s49/s32/s36/s49/s32/s55/s56/s32/s33/s36/s34/s40/s33/s57majority\nminorityMn2VAl\nTotal DOS \n(KKR-CPA-LDA)\n \nFIG. 9: (Color online) Off-stoichiometric effects on total de n-\nsity of states of Mn 2VAl obtained by the KKR-CPA-LDA\nmethod.8\n∗Electronic address: fujiwara@mp.es.osaka-u.ac.jp\n[1] I.Galanakis, in Heusler Alloys , editedbyC. Felser andA.\nHirohata, Springer Series in Materials Science (Springer\nInternational Publishing, Switzerland, 2016), Vol. 222.\n[2] P. J. Brown, K.U. Neumann, P.J. 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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": "2009.05742v1.Ferrimagnetic_States_of_Na_K_Alloy_Clusters_in_Zeolite_Low_Silica_X.pdf", "content": "arXiv:2009.05742v1 [cond-mat.soft] 12 Sep 2020Ferrimagnetic States of Na-K Alloy Clusters in Zeolite Low- Silica X\nTakehito Nakano,1,2,∗Shingo Araki,3,†Luu Manh Kien,4,2Nguyen Hoang Nam,5\nDuong Thi Hanh,2Akihiro Owaki,2Ken Goto,2Akira Matsuo,6Koichi Kindo,6and Yasuo Nozue2,‡\n1Institute of Quantum Beam Science, Graduate School of Scien ce and Engineering, Ibaraki University,\n2-1-1 Bunkyo, Mito, Ibaraki 310-8512, Japan\n2Department of Physics, Graduate School of Science, Osaka Un iversity,\n1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan\n3Department of Physics, Okayama University, Okayama 700-85 30, Japan\n4Nano and Energy Center, Hanoi University of Science,\nVietnam National University, 334 Nguyen Trai, Thanh Xuan, H anoi, Vietnam\n5Center for Materials Science, Faculty of Physics,\nHanoi University of Science, Vietnam National University,\n334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam\n6Institute for Solid State Physics, University of Tokyo,\n5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan\n(Dated: September 15, 2020)\nIn zeolite low-silica X (LSX), β-cages with the inside diameter of ≈7˚A are arrayed in a diamond\nstructure. Among them, supercages with the inside diameter of≈13˚A are formed and arrayed in a\ndiamondstructurebythesharingofwindows withtheinsided iameterof ≈8˚A.Thechemicalformula\nof zeolite LSX used in the present study is given by Na xK12−xAl12Si12O48per supercage (or β-cage),\nwhereNa xK12−xandAl 12Si12O48aretheexchangeablecationsofzeolite LSXandthealuminos ilicate\nframework, respectively. Na-K alloy clusters are incorpor ated in these cages by the loading of guest\nK metal at nK atoms per supercage (or β-cage). A N´ eel’s N-type ferrimagnetism has been observed\natn= 7.8 forx= 4. In the present paper, optical, magnetic and electrical p roperties are studied\nin detail mainly for x= 4. Ferrimagnetic properties are observed at 6 .5< n <8.5. At the same\ntime, the Curie constant suddenly increases. An optical refl ection band of β-cage clusters at 2.8 eV\nis observed at n >6.5 in accordance with the sudden increase in the Curie constan t. An electrical\nresistivity indicates metallic values at n/greaterorapproxeql6, because a metallic state is realized in the energy\nband of supercage clusters. The ferrimagnetism is explaine d by the antiferromagnetic interaction\nbetween the magnetic sublattice of itinerant electron ferr omagnetism at supercage clusters and that\nof localized moments at β-cage clusters. The electrical resistivity in ferrimagnet ic samples at n= 8.2\nforx= 4 increases extraordinarily at very low temperatures, suc h as≈106times larger than the\nvalue at higher temperatures. Observed anomalies in the ele ctrical resistivity resembles the Kondo\ninsulator, but itinerant electrons of narrow energy band of supercage clusters are ferromagnetic\ndifferently from the Kondo insulator.\nPACS numbers: 82.75.Vx, 71.28.+d, 75.30.Mb, 75.50.Xx, 75.75.-c, 36.40. -c\nI. INTRODUCTION\nZeolite crystals have free spaces of regular cages for\nguest materials [1]. There are many different types of ze-\nolite structures [2]. Alkali metal clusters incorporated in\ncages of zeolites have a wide variety in electronic proper-\nties, such as a ferrimagnetism, a ferromagnetism, an an-\ntiferromagnetism, and an insulator-to-metal transition,\ndepending on the kind of alkali metals, their loading den-\nsity, and the structure type of zeolite frameworks [1, 3].\nIn zeolite low-silica X (LSX), supercages and β-cages\nwith the inside diameters of ≈13and≈7˚A, respectively,\narearrayedinadiamondstructure,namelythedoubledi-\namond structure. Up to now, detailed studies have made\n[1, 3–14]. A N´ eel’s N-type ferrimagnetism has been ob-\nserved in Na-K alloy clusters incorporated into zeolite\n∗takehito.nakano.phys@vc.ibaraki.ac.jp\n†araki@science.okayama-u.ac.jp\n‡nozue@phys.sci.osaka-u.ac.jpLSX, where an antiferromagnetic interaction works be-\ntween nonequivalent magnetic sublattices of supercages\nandβ-cages [1, 3, 6, 9]. In the present paper, their op-\ntical, electrical and magnetic properties are studied in\ndetail.\nBesides the N´ eel’s N-type ferrimagnetism, a ferromag-\nnetism has been observed in Na-rich Na-K alloy clusters\nin zeolite LSX [12]. In pure Na clusters in zeolite LSX,\na metallic phase has been observed with the increase in\nNa loading density [1, 5, 8, 10, 13]. In pure K clusters in\nzeolite LSX, a ferrimagnetic property at higher K load-\ning densities has been observed in a metallic phase [1, 9].\nUnder the pressure loading of K-metal into zeolite LSX,\nan itinerant electron ferromagnetism has been newly ob-\nserved at the loading pressure of ≈0.9 GPa [14].\nAfter the discovery of ferromagnetic properties in K\nclusters in zeolite A [15], detailed studies have been\nmade [1, 3, 16–38]. In zeolite A, α-cages with the in-\nside diameter of ≈11˚A are arrayed in a simple cubic\nstructure. A spin-cant model of Mott-insulator antifer-\nromagnetism of K cluster array in α-cages is proposed2\n[1, 29, 32, 38]. In Rb clusters in zeolite A, a ferrimag-\nnetism has been observed [25, 39, 40]. An antiferromag-\nnetism of Mott insulator in alkali metal clusters in so-\ndalite has been clearly observed [41], and detailed stud-\nies have been made [42–55]. In sodalite, β-cages are ar-\nrayed in a body centered cubic structure. Alkali met-\nals in quasi-low-dimensional systems, such as the quasi-\none-dimensional metallic system in channel-type zeolite\nL [56–59], has been studied.\nA. Zeolite LSX\nZeolite X is one of the most typical aluminosilicate\nzeolites, and is nonmagnetic insulator unless guest mate-\nrials are loaded. Zeolite LSX is the zeolite X with Si/Al\n= 1 in aluminosilicate framework. The framework of ze-\nolite LSX is negatively charged and illustrated in Fig. 1\ntogether with typical sites of exchangeable monovalent\ncations ( As). Al and Si atoms are alternately connected\nby the sharing of O atoms. The space group is Fd¯3 with\nthe lattice constant of 25 ˚A. The chemical formula per\nunit cell is given by A96Al96Si96O384before the loading\nof guest materials. The number of cations is the same\nas that of aluminium atoms in framework. The frame-\nworkstructure type ofzeolite LSX is called FAU (IUPAC\nnomenclature [2]). The framework of FAU is constructed\nofβ-cages arrayed in a diamond structure. Among β-\ncages, “supercages (cavities) of FAU” are formed and\nalso arrayed in a diamond structure. The distance be-\ntween adjoining β-cages (or supercages of FAU) is 10.8\n˚A. Hereafter, we call “supercage of FAU” simply by “su-\npercage”. There are eight supercages (or eight β-cages)\nin the unit cell, and the chemical formula per supercage\n(orβ-cage) is given by A12Al12Si12O48. Zeolite LSX\nused in the present study contains Na and K cations,\nand the chemical formula per supercage (or β-cage) is\ngiven by Na xK12−xAl12Si12O48. Hereafter, we call it by\nNaxK12−x-LSX.\nInordertoacquireanintuitiveunderstandingofframe-\nwork structure, a polyhedral form is illustrated in Fig. 2.\nEachβ-cageis connectedto fouradjoining β-cagesbythe\nsharing of hexagonal prisms (double 6-membered rings,\nD6Rs). Supercages share windows of twelve-membered\nrings (12Rs) with adjoining supercages. The inside di-\nameters of 12R and 6R are ≈8 and≈3˚A, respectively.\nEachβ-cage shares 6-membered rings (6Rs) with four\nadjoining supercages.\nB. Alkali metal loading into zeolite Na xK12−x-LSX\nAlkali metals are easily loaded into zeolite by the va-\npor phase for unsaturated condition or by the direct con-\ntact with alkali metal for the saturated condition. In\nthe present paper, we loaded guest K metal at natoms\nper supercage (or β-cage) into Na xK12−x-LSX, and dis-\ncribe it as K n/NaxK12−x-LSX. The average number of!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nAl\nO\nSi\nβ-cage A\nsupercage of FAU 10.8 /uni00C5.169\nFIG. 1. (Color online) Aluminosilicate framework structur e\nof zeolite LSX and typical sites of exchangeable Acations\nwithout guest materials. β-cages are arrayed in a diamond\nstructure. Among them, supercages of FAU are formed and\narrayed in a diamond structure. The distance between ad-\njoiningβ-cages (or supercages of FAU) is 10.8 ˚A. See also the\npolyhedral illustration of the structure in Fig. 2.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nβ-cage \nsupercage \nof FAU \n6R 12R D6R \nFIG. 2. (Color online) Schematic illustrations of framewor k\npolyhedra of zeolite LSX. Each β-cage is connected to four\nadjoining β-cages by the sharing of double 6-membered rings\n(D6Rs), and arrayed in a diamond structure. Supercages of\nFAU are arrayed in a diamond structure by the sharing of\ntwelve-memberedrings(12Rs)withfouradjoiningsupercag es.\ns-electrons provided by the loading of alkali metal is also\nnper supercage (or β-cage).\nAn outermost s-electron of an alkali atom has a large\nsize and a small ionization energy, so that s-electrons in\nbulk alkali metals are well described by the free-electron\nmodel.s-electrons introduced in zeolite by the loading of\nguest alkali atoms move freely over cations distributed in\ncages. The aluminosilicate framework, however, is neg-\natively charged and has high-energy conduction bands.\nTherefore, s-electrons are repulsed by the framework.\nThes-electrons successively occupy quantum states of\nclusters formed in cages. If we assume a spherical quan-\ntum well (SQW) potential for cage, quantum states, such\nas 1s, 1pand 1dstates, are formed in the increasing or-\nderofenergy, andtwo, six andten s-electronscanoccupy\nrespective quantum states successively [1]. Schematic il-\nlustrations of cluster in supercage and quantum states3\nofs-electron in the SQW potential with the diameter of\n13˚A are given in Fig. 3. A large sphere in supercage\nis a schematic image of s-electron wave function. 1 s, 1p\nand 1dquantum states have energies 0.9, 1.8 and 3.0 eV\nfrom the bottom ofthe SQWpotential, respectively. The\nnumber in each parentheses indicates the degeneracy in-\ncluding spin. The optical excitations (dipole transitions)\nare allowed between 1 s-and-1pand between 1 p-and-1d\nstates. That between 1 s-and-1dis forbidden.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \n3.0 eV \n1.8 eV \n0.9 eV \nsupercage of FAU 1s(2) 1d(10) \n1p(6) \n13 Å ≈13 Å \nFIG. 3. (Color online) Schematic illustrations of alkali me tal\ncluster in supercage of FAU and the quantum states of s-\nelectron in the SQW potential with the diameter of 13 ˚A.\nThe SQW potential, however, is primitive for the su-\npercage cluster, because of large 12R windows. The\nspheres of s-electron wave functions in adjoining su-\npercages largely overlap with each other, because the\ndistance between adjoining supercages is 10.8 ˚A which\nis shorter than the inside diameter of supercage ≈13˚A.\nNevertheless, we use 1 s, 1pand 1dquantum states of the\nSQW potential, because of a convenient model to think\nabout quantum states localized in supercage. In zeolite\nA, K clusters are well localized in α-cages with the inside\ndiameter of ≈11˚A, and the SQW model well explains ex-\nperimental results, because of rather narrow windows of\nα-cages [1, 16, 17, 38]. Electrons in regular supercages of\nzeolite LSX are expected to construct the energy band, if\nthe contributions of the electron-phonon interaction and\nthe electron correlation are not significant. Because the\nsupercage has the Tdsymmetry which has no inversion\nsymmetry at the cage center, 1 s, 1pand 1dstates hy-\nbridize with each other. The electronic states of energy\nband are constructed of these hybridized states depend-\ning on the positions in the Brillouin zone. For example,\nthe electronic states at the bottom of the lowest band are\nmainly constructed of 1 sstates.\nSchematic illustrations of cluster in β-cage and quan-\ntumstatesof s-electronintheSQWpotentialwiththedi-\nameterof7 ˚AaregiveninFig.4. Alargespherein β-cage\nis a schematic image of s-electron wave function. 1 sand\n1pquantum states have energies 3.1 and 6.3 eV from the\nbottom of the SQW potential, respectively. These ener-\ngies are much higher than respective states in supercage,\nbecause of a narrow size of β-cage. As adjoining β-cages\nare well separated by D6Rs as shown in Fig. 2, s-electron\nwave functions in adjoining β-cages scarcely overlapwith\neach other, but a finite overlap occurs through 6Rs be-tween supercages and β-cages.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \n6.3 eV \n3.1 eV \nβ-cage 1s(2) 1p(6) \n7 Å ≈7 Å \nFIG. 4. (Color online) Schematic illustrations of alkali me tal\ncluster in β-cage and the quantum states of s-electron in the\nSQW potential with the diameter of 7 ˚A.\nC. Electronic properties of Na-K alloy clusters in\nKn/NaxK12−x-LSX\nElectronic properties of Na-K alloy clusters in\nKn/NaxK12−x-LSXlargelydependon xaswellas n. The\ncontributions of Na atoms are the larger ionization en-\nergy and the smaller cation size, compared with those\nof K atoms. In K n/K12-LSX (namely x= 0), pure K\nclusters show a metallic phase at n/greaterorapproxeql6 and a ferrimag-\nnetic property at the saturation loading density n≈9 at\nambient pressure [1, 9]. Under the pressure loading of K-\nmetal into zeolite K 12-LSX, the disappearance of the fer-\nrimagnetism has occurred and an itinerant electron fer-\nromagnetism have been newly observed at n≈15 at the\nloading pressure ≈0.9 GPa [14]. In Na-K alloy clusters\nin Kn/Na4K8-LSX (namely x= 4), the N´ eel’s N-type\nferrimagnetism has been observed [1, 3, 4, 6]. Under the\npressure loading of K-metal into zeolite Na 4K8-LSX, a\nnew ferrimagnetism have been observed at the loading\npressure ≈0.5 GPa [7]. In K n/Na7.3K4.7-LSX (namely\nx= 7.3), a nearly pure ferromagnetism in an insulating\nphase has been observed at n≈9 [12]. The origin of the\nferromagnetismis assigned to the ferromagnetic superex-\nchange coupling between magnetic moments at β-cage\nclusters via sp3closed-shell clusters at supercages.\nPure Na clusters are generated by the Na metal load-\ning into zeolite Na 12-LSX (namely x= 12). Insulating\nand non-magnetic states of pure Na clusters have been\nobserved in Na n/Na12-LSX for n/lessorapproxeql11. A metallic phase\nhas been observed with the increase in n. A thermally\nactivated paramagnetic susceptibility has been observed\nsignificantly at n≈16, and is assigned to the thermal\ndistribution of metastable small polarons [1, 5, 8]. The\ntemperaturedependenceofthe paramagneticsusceptibil-\nity has been observed in the shift of23Na NMR narrow\nline [10, 13], although there are many nonequivalent Na\nsites in Na n/Na12-LSX [1, 11]. This result indicates that\nNa cations are hopping thermally over many Na sites at\nhigher temperatures during the NMR time window, and4\nnuclei of relevant Na cations feel average paramagnetic\nfield of thermally metastable small polarons.\nIn the present paper, optical, magnetic and electri-\ncal properties in K n/NaxK12−x-LSX are studied in de-\ntail mainly for x= 4. Ferrimagnetic properties are ob-\nserved at 6 .5< n <8.5 in K n/Na4K8-LSX. At the same\ntime, the Curie constant suddenly increases, and a re-\nflection band of β-cage clusters at 2.8 eV is observed at\nn >6.5. An electrical resistivity indicates metallic value\natn/greaterorapproxeql6. The electrical resistivity increases extraor-\ndinarily at very low temperatures in ferrimagnetic sam-\nples, such as ≈106times larger than the value at higher\ntemperatures. The ferrimagnetism is explained by the\nantiferromagnetic interaction between the magnetic sub-\nlattice of itinerant electron ferromagnetism at supercage\nclusters and that of localized moments at β-cageclusters.\nWe try to explain these anomalies of electrical resistivity\nby the analogy of the Kondo insulator, where itinerant\nelectron spins of supercage clusters interact with local-\nized electron spins of β-cage clusters. Itinerant electrons\nof narrow energy band of supercage clusters, however, is\nferromagnetic, differently from the Kondo insulator.\nII. EXPERIMENTAL PROCEDURES\nZeolites are crystalline powder of few microns in grain\nsize. The as-synthesized zeolite LSX was x= 9. Na\ncations were fully exchanged to K 12-LSX in KCl aqueous\nsolution. K 12-LSX was partly ion-exchanged in aque-\nous NaCl solution in order to get Na xK12−x-LSX. The\nvalue ofxwas estimated by means of inductively coupled\nplasma (ICP) spectroscopy. Zeolite Na xK12−x-LSX was\nfully dehydrated in vacuum at 500◦C for one day. Dis-\ntilled potassium metal was set into a quartz glass tube\ntogether with the dehydrated Na xK12−x-LSX in a glove-\nbox filled with a pure He gas containing less than 1 ppm\nof O2and H 2O. The potassium metal in the quartz glass\ntube was adsorbed into the Na xK12−x-LSX at 150◦C.\nThe thermal annealing was made for enough time to get\nthe homogeneous K-loading. The value of nwas esti-\nmated from the weight ratio of K-metal to Na xK12−x-\nLSX powder.\nThe optical diffuse reflectivity rwas measured at room\ntemperaturebythe useofanFTIRspectrometer(Nicolet\nMagna 550) and a double monochromator-type UV-vis-\nNIR spectrometer (Varian Cary 5G). KBr powder was\nused for the reference ofwhite powder. Since samples are\nextremely air-sensitive, optical measurements were per-\nformed on samples sealed in quartz glass tubes. The dif-\nfuse reflectivity rwas transformed to the optical absorp-\ntion spectrum by the Kubelka-Munk function (1 −r)2/2r\nwhich gives the ratio of the absorption coefficient to the\nreciprocal of powder size. The sum of the normal reflec-\ntivityRand the transmission coefficient Trwas obtained\nby the transformation R+Tr= 4r/(1+r)2[16]. The nor-\nmalreflectivityspectrumwasobtainedas R= 4r/(1+r)2\nat the spectral region for Tr≪R.A SQUID magnetometer (MPMS-XL, Quantum De-\nsign) was used for magnetic measurements in the tem-\nperature range 1.8-300 K. A diamagnetic signal from the\nquartz glass tube is included in the SQUID signal as the\ntemperature-independent background, and is subtracted\nfrom measured magnetization.\nFor an electrical resistivity measurement, powder sam-\nples were put between two gold electrodes, and an ade-\nquate compression force ≈1 MPa was applied during the\nmeasurements. Because of the extreme air-sensitivity\nof samples, they were kept in a handmade air-proof\ncell. These setting procedures were completed inside\nthe glovebox. The cell was set into Physical Property\nMeasurement System (PPMS, Quantum Design), and\nthe temperature was changed between 2 and 300 K.\nThe electrical resistivity of the cell was measured by the\nfour-terminal method with the use of Agilent E4980A\nLCR meter at the frequency range from 20 Hz to 2\nMHz and DC. The frequency dependence of the complex\nimpedance was analyzed by the Cole-Cole plot, and the\nDC or 20 Hz electrical resistivity ρwas obtained by the\nmultiplication of the dimensional factor (area/thickness)\nofcompressed powder. Due to the constrictionresistance\n[60] at connections between powder particles as well as\nthe low filling density of powder particles, the observed\nresistivity is about two orders of magnitude larger than\nthe true value. The relative values in different samples,\nhowever, can be compared with each other within an am-\nbiguity of factor, because of the constant compression\nforce. Fortunately, values in the present study change\nin the several orders of magnitude. Detailed experimen-\ntal procedures are explained elsewhere [8]. The upper\nlimit of the present resistivity measurement was ≈109\nΩcm, and obtained values for ρ/greaterorapproxeql109Ωcm are unreli-\nable. The ionic conductance of dehydrated zeolites un-\nder the low compression force is expected in the order of\n10−9Ω−1cm−1at room temperature [61], and is negligi-\nble at lower temperatures in the present study. A small\nresistivity of the short circuit in the cell ( <0.1 Ωcm) is\nincluded in the measured value, but is negligible in the\npresent study.\nThe high-field magnetization was measured by using\nan induction method with a multilayer pulse magnet at\nthe Institute for Solid State Physics, the University of\nTokyo. A non-destructive pulsed magnet for 70 T was\nused for this measurement. Sample sealed in a high-\nquality quartz glass tube with a diameter of 2 mm was\nset in the pickup coils. The observed magnetization is\nnormalized by the results obtained by the SQUID mag-\nnetometer at H <5×104Oe.\nIII. EXPERIMENTAL RESULTS\nA. Optical properties\nOpticalresonantabsorptionand reflectionspectrapro-\nvide an important information on the dipole transition of5\nelectronicstatesincludingnonmagneticones. Absorption\nspectra of dilutely K-loaded K n/NaxK12−x-LSX (n≪1)\nat room temperature (RT) are shown in Fig. 5 for x= 0,\n1.5, 4 and 7.3. Spectra in K n/K12-LSX, K n/Na1.5K10.5-\nLSX and K n/Na4K8-LSX have continuous peaks above\n≈0.6eV. These peaksare assignedto the excitation from\n1s-like states to the empty energy bands of supercage\nnetwork [3]. A new band appears at ≈2.6 eV with mark\nin Kn/Na7.3K4.7-LSX, in addition to above mentioned\ncontinuous peaks. This new band is assigned to the ex-\ncitation from 1 s-like states to 1 p-like ones of clusters in\nβ-cages [12].\nFIG. 5. (Color online) Absorption spectra of dilutely K-\nloaded K n/K12-LSX (x= 0), K n/Na1.5K10.5-LSX (x= 1.5),\nKn/Na4K8-LSX (x= 4) and K n/Na7.3K4.7-LSX (x= 7.3) at\nroom temperature, where n≪1.\nIf we assume strict SQW potentials shown in Figs. 3\nand 4, the 1 s–1pexcitation energies are expected at 0.9\nand 3.2 eV in clusters localized in supercage and β-cage,\nrespectively. Because of the lack of the inversion symme-\ntry at the center of supercage, 1 s, 1pand 1dstates hy-\nbridize partly with each other in the energy band. Con-\ntinuous DOS of the hybridized energy band of supercage\nnetwork are expected, because of the electron transfer\nthrough large 12R windows with the size ≈8˚A. In prin-\nciple, the absorptioncoefficient of the band-to-band exci-\ntation is proportional to the joint-density-of-states times\nthe transitiondipole momentsbetween groundstatesand\nexcited states. The observed gap energy of continuous\nabsorption bands, ≈0.6 eV, originates from the forma-\ntion energy of small bipolarons at supercages at low K-\nloading densities, as stated in Section IVA. Small bipo-\nlarons are optically excited to the extended states of the\nhybridized energy band of supercage network. The β-\ncagepotentialprovideswell-isolatedelectronicstates, be-cause of narrow windows. The optical excitation from 1 s\nto 1pstates is expected at 3.2 eV in Fig. 4, but the effec-\ntive potential size is expected to be slightly larger than 7\n˚A, such as 7.8 ˚A, in order to fit the observed excitation\nenergy≈2.6 eV. As discussed in Section IVB, the sur-\nrounding cations are expected to extend the confinement\npotential.\nFIG. 6. (Color online) Reflection spectra of K n/Na4K8-LSX\nat room temperature. The value of nis indicated for each\nspectrum.\nReflection spectra of K n/Na4K8-LSX (x= 4) at room\ntemperature are shown in Fig. 6. The K-loading density\nnis indicated for each spectrum. A reflection band of\nnearly metallic s-electrons of supercage clusters is seen\nbelow≈1 eV in each spectrum. The plasma edge of\nmetallic s-electrons is estimated to be ≈1 eV. With the\nincrease in n, theβ-cage cluster bands grow around ≈2.3\nand≈2.8 eV. The 2.3 eV band grows at lower values of\nn. As shown in Section IIIB, a ferrimagnetism and a\nsudden increase in the Curie constant are observed si-\nmultaneously at 6 .5< n <8.5. The 2.8 eV band of\nβ-cage clusters is assigned to the magnetic K-rich clus-\nters (small polarons) for 6 .5/lessorapproxeqln/lessorapproxeql8.5 and nonmagnetic\nK-rich clusters (small bipolarons) for 8 .5/lessorapproxeqln, as dis-\ncussed in Section IVB. The 2.3 eV band is assigned to\nnonmagnetic Na-rich clusters at β-cages.\nIn Kn/Na1.5K10.5-LSX (x= 1.5), similar reflection\nspectra are observed at room temperature, as shown in\nFig.7. Reflectionbandsof β-cageclustersareobservedat\nsimilar energies 2.2 and 2.8 eV. The 2.8 eV band appears\natn/greaterorapproxeql7.5. As shown in Section IIIB, a ferrimagnetism\nand an increase in the Curie constant are observed si-\nmultaneously at 7 .8< n/lessorapproxeql9.5. The 2.8 eV band is\nassigned to the K-rich magnetic clusters (small polarons)6\natβ-cages, as discussed in Section IVB. Reflection bands\nat 2.2, 2.3 and 2.4 eV are expected to be nonmagnetic\nNa-richβ-cage clusters with different configurations of\ncations.\nFIG. 7. (Color online) Reflection spectra of K n/Na1.5K10.5-\nLSX at room temperature. The value of nis indicated for\neach spectrum.\nB. Magnetic properties\nTemperature dependences of magnetization in\nKn/Na4K8-LSX under the low magnetic field of 10\nOe are shown in Fig. 8. The value of nis indicated\nfor each curve. The observed large magnetization\noriginates from the spontaneous magnetization, because\nof an applied magnetic field is very weak. The Curie\ntemperature increases and decreases with n. A typical\nN´ eel’s N-type ferrimagnetism with the zero minimum of\nmagnetization at the compensation temperature Tcomp\nis seen at n= 7.6, 7.8 and 7.9. A similar zero minimum\nmay be expected below 1.8 K at n= 6.7 and 7.0. A\ngradual increase in magnetization around the Curie\ntemperature is seen at n= 7.6 and 7.8 with the decrease\nin temperature, indicating that a weak inhomogeneity\nis expected to exist in the temperature of the magnetic\nphase transition. The zero minimum at Tcomp, however,\nis clearly seen.\nThe N´ eel’s N-type ferrimagnetism is explained by an\nantiferromagnetic interaction between two nonequivalent\nmagnetic sublattices A and B, one of which (A) has both\na very weak internal magnetic interaction and the satu-\nration magnetization which is larger than the magnetiza-\ntion of the other sublattice (B). The sublattice B has a\nstronger internal interaction. Below the Curie tempera-\nture, the sublattice B increases the spontaneous magne-tization. The magnetization of sublattice A follows the\nsublattice B with the opposite direction. At Tcomp, mag-\nnetizations of sublattices A and B have the same mag-\nnitude with opposite directions, and the total magneti-\nzation becomes zero. Below Tcomp, the sublattice A has\nthe magnetization larger than that of sublattices B. As\ndiscussed later in Section IVC, we introduce a model of\ntwo magnetic sublattices A and B constructed by local-\nized magnetic moments of β-cage clusters and an itin-\nerant electron ferromagnetism of supercage clusters, re-\nspectively. In Fig. 8, Tcompseems to approach the Curie\ntemperature relatively, indicating that an antiferromag-\nnetic interaction between magnetic sublattices A and B\nand/or the magnetization of sublattice A increase with n\nat the ferrimagnetic condition.\nn-dependences of the asymptotic Curie temperature\nTC, the Weiss temperature TWand the Curie constant\nin Kn/Na4K8-LSX are shown in Fig. 9. The Curie con-\nstanthasasuddenincreaseattheferrimagneticcondition\n6.5< n <8.5, as colored in blue. TWis positive and neg-\native at lower and higher values of n, respectively. The\n2.8 eV band of β-cage clusters grows at n/greaterorapproxeql6.5 in Fig. 6\nin accordance with the sudden increase in the Curie con-\nstant.\nFIG. 8. (Color online) Temperature dependences of magneti-\nzation in K n/Na4K8-LSX under the magnetic field of 10 Oe.\nThe value of nis indicated for each curve.\nThe sudden increase of the Curie constant in Fig. 9 is\nestimated to be ≈5×10−5Kemu/cm3. If we assume\nlocalized magnetic moments of β-cage clusters with spin\ns= 1/2 andg= 2, the Curie constant Cβis given by\nCβ=Nβg2µ2\nBs(s+1)\n3kB=Nβµ2\nB\nkB, (1)\nwhereNβandkBare the number density of magnetic\nclusters at β-cages and the Boltzmann constant, respec-7\nFIG. 9. (Color online) n-dependences of the asymptotic Curie\ntemperature TC, the Weiss temperature TWand the Curie\nconstant in K n/Na4K8-LSX.\ntively. The estimated value of Nβamounts to ≈15% of\nβ-cages and the saturation magnetization becomes ≈0.7\nG.\nThe background Curie constant in Fig. 9 is ndepen-\ndent, for example, ≈1.3×10−4Kemu/cm3atn≈7.5.\nThe Curie constant of an itinerant electron ferromag-\nnetism for supercage clusters, Cs, is given by\nCs=N0peff2µB2\n3kB, (2)\nwhereN0andpeffµBare the number density of su-\npercages and the effective local magnetic moment per\nsupercage, respectively. The value of peffestimated from\nthe background Curie constant is ≈1.1 which corre-\nsponds to the saturation magnetization of ≈5.3 G. In\ncase of the itinerant electron ferromagnetism, however,\nthe spontaneous magnetization at low magnetic fields is\nmuch smaller than that estimated from the Curie con-\nstant, such as ≈1/3 in the itinerant electron ferromag-\nnetism in the pressure loading of K metal into K 12-LSX\n[14]. If we assume a similar ratio, the spontaneous mag-\nnetization of supercage clusters will be ≈1.8 G at low\ntemperatures. The total magnetization will be ≈2.5 G.\nAt very high magnetic fields, the saturation of total mag-\nnetization is observed at 2.7 G as shown later in Fig. 12.\nIn order to explain the N´ eel’s N-type ferrimagnetism ob-\nserved in Fig. 8, the spontaneous magnetization of su-\npercage clusters at low temperatures will be smaller than\n≈0.7 G of the saturation magnetization at β-cage clus-\nters.\nThe temperature dependence of magnetization in\nKn/Na1.5K10.5-LSX under the magnetic field of 10 Oe\nis shown in Fig. 10. The value of nis indicated for\neach curve. The Curie temperature increases and de-\ncreases with n. The magnetization has a minimum at\nthe temperatures lower than the respective Curie tem-\nFIG. 10. (Color online) Temperature dependences of magne-\ntization in K n/Na1.5K10.5-LSX under the magnetic field of 10\nOe. The value of nis indicated for each curve.\nFIG. 11. (Color online) n-dependences of the asymptotic\nCurie temperature TC, the Weiss temperature TWand the\nCurie constant in K n/Na1.5K10.5-LSX.\nperatures, indicating that this is the N´ eel’s P-type ferri-\nmagnetism, where the magnetization of β-cage clusters\nis smaller than that of supercage clusters at any tem-\nperature. n-dependences of the asymptotic Curie tem-\nperature TC, the Weiss temperature TWand the Curie\nconstant are shown in Fig. 11. The Curie constant is\nmuch larger than that in K n/Na4K8-LSX. The Curie\nconstant has an increase at the ferrimagnetic condition\n7.8< n/lessorapproxeql9.5, as colored in blue. TWis positive and\nnegative at lower and higher values of n, respectively,\nat the ferrimagnetic condition. The 2.8 eV band of β-8\ncage clusters grows at n/greaterorapproxeql7.5 in Fig. 7. The increase\nin the Curie constant at n≈8.5 is roughly estimated to\nbe≈1×10−4Kemu/cm3which corresponds to localized\nmagnetic moments with spin 1/2 distributed at ≈30%\nofβ-cages and the saturation magnetization of ≈1.5 G.\nThebackgroundCurieconstant ≈3×10−4Kemu/cm3at\nn≈8.5correspondsto peff≈1.7. This valuecorresponds\nto the saturation magnetization of ≈8 G. As explained\nabove in K n/Na4K8-LSX, the spontaneous magnetiza-\ntion of supercage clusters will be much smaller than ≈8\nG. At very high magnetic fields, the saturation of total\nmagnetization is observed at 4.2 G, as shown later in\nFig. 12.\nFIG. 12. (Color online) The magnetization process up to\nhigh magnetic fields at 1.3 K for K n/NaxK12−x-LSX, where\nthe respective values of ( x,n) are (4, 7.7), (1.5, 8.75) and (0,\n8.9). The corresponding magnetic moment per supercage (or\nβ-cage) is indicated in the axis on the right in units of µB.\nThe magnetization process up to high magnetic fields\nat 1.3 K is shown for K n/NaxK12−x-LSX in Fig. 12,\nwhere the respective values of ( x,n) are (4, 7.7), (1.5,\n8.75) and (0, 8.9). The corresponding magnetic mo-\nment per supercage (or β-cage) is indicated in the axis\non the right in units of µB. The magnetization process in\nK7.7/Na4K8-LSX displays a weak hump around 3 .5×104\nOe, and the saturation at 2.7 G after the clear bend at\n22.7×104Oe. A hump in K 8.75/Na1.5K10.5-LSX is un-\nclear, but is expected around ≈8×104Oe. The magneti-\nzation process in K 8.9/K12-LSX displays a hump around\n16×104Oe, and the saturation at ≈6 G after the bend\nat≈32×104Oe. As discussed later in Section IVD, the\nmagnetization process of ferrimagnetism in the model of\nclassical magnetic moment has a flat magnetization up\nto the spin-flop field, and a constant slope up to the sat-\nuration field. The observed results, however, have round\nshapes at the beginning of magnetization and above thespin-flop field. This shape is explained by the increase in\nmagnetization of the itinerant electron ferromagnetism\nof the supercage clusters.\nC. Electrical properties\nAn electrical conductivity and its temperature depen-\ndence give an important information on carriersin solids,\nespecially in correlated polaron systems. The electrical\nconductivity σwith different types of carriers are given\nby\nσ=/summationdisplay\njeµjNj, (3)\nwheree,µjandNjare the elementary electric charge,\nthej-th carrier mobility, and the number density of j-th\ncarriers, respectively. There are following two limiting\nmodels in the electrical conductivity having the Arrhe-\nnius law [62]. In the band gap model with nearlytemper-\nature independent mobility, the conductivity is propor-\ntional to the number density of thermally activated free\ncarriersNj, and is expressed by the Arrhenius law. The\ngapenergyisgivenbytwotimesofthethermalactivation\nenergy. In the small polaron hopping model, the Arrhe-\nnius law can be applied to the temperature dependence\nof mobility approximately, where the thermal activation\nenergy is related to the polaron formation energy, etc. A\ndisorder and an electron correlation can have important\ncontributions to the electrical conductivity in addition to\nabove mentioned mechanisms. The electrical resistivity\nρis given by 1 /σ.\nThe temperature dependences of ρin Kn/Na4K8-LSX\nat various values of nare shown in Fig. 13. The value of\nnis indicated for each curve. The temperature of sam-\nple was decreased from 300 K. The value of ρat 300 K\ndecreases with n. With the decrease in temperature, ρ\nbasically increases, because of the decrease in the mo-\nbility of small polaron hopping. A weak anomaly is seen\naround150K. A similar anomalyand a temperature hys-\nteresis in ρhave been clearly observed around 150 K in\nKn/Na7.3K4.7-LSX [12]. In Fig. 13, ρatn= 5.1 and\n5.8 slightly increases at low temperatures, and is finite\nat the lowest temperature 2 K. This result indicates that\na finite number of free carriers (large polarons) are dis-\ntributed at low temperatures. Although ρatn= 7.1\nand 7.8 is much lower than that at n= 5.1 or 5.8 above\n≈50 K,ρatn= 7.1 and 7.8 quickly increases at very\nlow temperatures and exceeds values at n= 5.1 or 5.8.\nThe electrical conductivity σ= 1/ρin Kn/Na4K8-LSX is\nplotted for n= 7.1 and 8.2 in Fig. 14 as a function of the\nreciprocal of temperature, 1 /T. The thermal activation\nenergy depends on temperature. The activation energy\nEgis roughly estimated to be ≈1.2 and≈4 meV around\n3 and 15 K, respectively, for n= 8.2.\nThen-dependence of ρin Kn/Na4K8-LSX is plotted\nfor 2, 20 and 100 K in Fig. 15. The value of ρat 2 K\ndecreases with nup ton= 5.8, but increases extremely9\nFIG. 13. (Color online) Temperature dependences of the elec -\ntrical resistivity ρin Kn/Na4K8-LSX at various values of n,\nwhere temperatures are decreased from 300 K. The value of\nnis indicated for each curve.\nFIG. 14. (Color online) Temperature dependences of electri -\ncal conductivity 1 /ρatn= 7.1 and 8.2 in K n/Na4K8-LSX.\natn= 7.1 and 8.2 at the ferrimagnetic condition 6 .5<\nn <8.5 shown in Figs. 8 and 9. The value of ρat 2K for\nn= 8.2 is≈106times of that at 100 K. The increase is\nnot significant at n= 9.0. A similar increase in ρat low\ntemperatures has been observed in K n/K12-LSX at the\nferrimagnetic condition of n, and the value of ρat 2K for\nn= 9.0 is≈102times of that at 100 K [1, 9].\nFIG. 15. (Color online) n-dependence of electrical resistivity\nρat 2, 20 and 100 K in K n/Na4K8-LSX.\nIV. DISCUSSIONS\nA. Model of correlated polaron system\nIfs-electron wave functions of alkali metal clusters\nare well localized quantum-mechanically in zeolite cages,\nthe tight-binding approximation can be applied to them\n[30, 31]. A narrow energy band of s-electrons with a\nstrong electron correlation is expected in the supercage\nclusters in K n/K12-LSX, because of a large mutual\nCoulomb repulsion energy within supercages and the\nelectron transfer through 12R windows [14]. Further-\nmore,s-electrons have an interaction with the displace-\nment of alkali cations distributed in cages. Hence, s-\nelectrons have an electron-phonon deformation-potential\ninteraction as well as the electron correlation.\nIn order to take an overview of the electronic proper-\nties of alkali metals in zeolites, it is effective to introduce\nfollowing coarse-grainedparameters of the correlated po-\nlaron system given by the so-called Holstein-Hubbard\nHamiltonian [1, 63, 64]\nH=−/summationdisplay\ni,j,σtija†\niσajσ+U/summationdisplay\nini↑ni↓\n+/summationdisplay\ni/parenleftbiggP2\ni\n2m+1\n2mω2Q2\ni/parenrightbigg\n−λ/summationdisplay\niQi(ni↑+ni↓),(4)\nwhereaiσ(a†\niσ) is the annihilation (creation) operator\nof the electron with the spin σat thei-th site, and\nniσ=a†\niσaiσ.tijis the electron transfer energy between\nthei-th and the j-th sites. Uis the on-site Coulomb re-\npulsion energy (the Hubbard U). The localized phonons\n(Einstein phonons) with the mass mand the frequency\nωare assumed in the third term. QiandPiare the lat-\ntice distortion and the conjugated momentum at the i-th\nsite, respectively. In the last term, the on-site electron-\nphonon interaction is introduced by the assumption of10\nthe site diagonal coupling constant λ. Here, we define\nthe lattice relaxation energy Sas [63]\nS=λ2\nmω2. (5)\nIf we consider the electron transfer between the nearest\nneighbor sites for /angbracketlefti,j/angbracketrightonly, the first term of the right-\nhand side of Eq. (4) can be written as\n−t/summationdisplay\n/angbracketlefti,j/angbracketright,σa†\niσajσ, (6)\nwheret(>0) is the transfer energy of electron to the\nnearest neighbor site. The t-U-S-ncoarse-grained model\nof correlatedpolaron system is introduced to alkali-metal\nloaded zeolites, where nis the average number of elec-\ntrons provided by alkali atoms per site.\nSchematic illustration of the Holstein-Hubbard model\nis given in Fig. 16. Red arrows indicate spins of elec-\ntrons. If tis large enough, large polarons migrate as free\ncarriers. In cases of U > SandU < Sat small t, small\npolaron with the energy −S/2 and small bipolaron with\nthe energy U−2S, respectively, become stable as the\nself-trapped states. Small polarons and small bipolarons\ncontribute to the conductivity by their hopping process\nat finite temperatures.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \n−t\nU\nsmall polaron large polaron \nsmall bipolaron U − 2S− S/2 \nFIG. 16. (Color online) Schematic illustration of theHolst ein-\nHubbard model. Red arrows indicate spins of electrons. If t\nis large enough, large polarons are stabilized as free carri ers.\nIn cases of U > SandU < Sat small t, small polaron with\nthe energy −S/2 and small bipolaron with the energy U−2S,\nrespectively, are stable.\nIn zeolites, tis introduced through windows between\nadjoining cages. The energy band width 2 Bis given by\n2B= 2ht, where his the number of nearest neighbor\nsites.his 4 for supercage or β-cage in zeolite LSX. The\nenergy of the band bottom is located at −B. IfB <\nS/2, an electron relaxes into small polaron. The value\nof 2Bfor supercage network is roughly estimated to be\n≈2 eV for 1 pstates in LSX from the spectral width of\nthe supercage band in Fig. 5. tforβ-cage network is\nnegligibly small, because of the large separation by D6Rs\nas shown in Fig. 2. Therefore, clusters generated in β-\ncages relax into self-trapped states because of a finite\nS, and become small polarons with magnetic moment or\nsmall bipolarons without magnetic moment, as discussed\nin Section IVB.Twos-electrons in the same cage have a Coulomb re-\npulsion energy U. The value of Udepends on the size\nof cage, but is almost independent of the configuration\nof cations. The unscreened Ubetween two electrons in\nthe 1sstate is estimated to be ≈3 eV for supercage with\nthe inside diameter of ≈13˚A and≈6 eV forβ-cage with\nthe inside diameter of ≈7˚A [1]. A finite screening ef-\nfect reduces the value of unscreened U. A qualitative\ninterpretation has been given by the t-U-S-nmodel for\nvarious properties of alkali metals in different zeolites [1].\nAt lower loading densities, tis relatively small because\ns-electrons occupy lower quantum states of clusters, such\nas 1sstates, and the electron-phonon interaction Sdom-\ninates the system. Hence, small bipolarons are stabilized\nat lower loading densities. A gap energy ≈0.6 eV in ab-\nsorptionspectra ofdilutely K-loadedK n/NaxK12−x-LSX\ninFig.5isassignedtotheformationenergyofsmallbipo-\nlarons at 1 sstates in supercages. An effective value of t\nfor the energy band near the Fermi energy is expected to\nincrease with n, because s-electrons occupy higher quan-\ntum states of clusters, such as 1 pand 1dstates, and the\nmetallic states are realized at large ndepending on the\nkindofalkalimetals, etc. [1]. Ametallicstateisexpected\natn/greaterorapproxeql6 in K n/Na4K8-LSX as shown in Fig. 13, indi-\ncating that free carriers of large polarons are generated\nby 1pelectrons in supercage clusters. A similar metallic\ntransition has been observed in K n/K12-LSX [1, 9].\nB. Clusters at β-cages\nMagnetic moments of clusters in β-cages play a cru-\ncial role in magnetisms of K n/NaxK12−x-LSX. The value\noftbetween β-cages is negligibly small. If an electron\noccupies the 1 sempty state with the energy E1satβ-\ncage, a small polaron with the energy E1s−S/2 is gen-\nerated by the electron-phonon interaction according to\nthe Holstein-Hubbard model, as illustrated in Fig. 17. If\nthe second electron occupies the small polaron site, the\nsecond electron has the energy E1s+U−3S/2. As shown\nin Fig. 17(a), small polarons with magnetic moments are\ngenerated in β-cages at U > S, if the Fermi energy EF\nsatisfies\nE1s−S\n2< EF< E1s+U−3S\n2. (7)\nWith the increase in EFwithn, small bipolaron with the\nenergy2E1s+U−2Sin the spin-singletstate isgenerated\nby the occupation of the second electron, if EFsatisfies\nE1s+U−3S\n2< EF. (8)\nThis model means that small polarons with the mag-\nnetic moments are stabilized only at the condition given\nby Eq. (7) for EF. On the other hand, there is no choice\nforEFatU < Sin Eq. (7), and small polarons are un-\nstable at any value of EF. This is because the pairing\nof small polarons forms small bipolarons with the energy11\n2E1s+U−2Swhich is more stable than the separatepair\nof small polarons with the total energy 2 E1s−S, as illus-\ntrated in Fig. 17(b), indicating that small polarons with\nthe magnetic moments are not stabilized at any value of\nnatU < S.\n(β-cage) (β-cage)−S2U−3S2\nU−S−S2\nU−3S2U−S(U>S) (U Sand (b)\nU < S, accordingtotheHolstein-Hubbardmodel. Redarrows\nindicate spins of electrons. The value of tis negligibly small\nbetween β-cages. Small polarons and small bipolarons in β-\ncages are formed depending on the relative magnitudes of U\nandSand the Fermi energy EF. See text in detail.\nThe value of Sstrongly depends on the kind of cations\nand their arrangement such as the number and the loca-\ntions of cations. Generally, Sfor Na-rich cluster is larger\nthan that for K-rich one, because of the larger ionization\nof Na atom. The value of Sincreases with the number\nof cations which contribute to the formation of cluster.\nGenerally, cations in zeolites are located near the\naluminosilicate framework, because of the attractive\nCoulomb force between cations and negatively charged\nframework. However, cations keep the mutual distance,\nbecause of the repulsive Coulomb force among them. In\neachβ-cage of zeolite LSX, there are three cation sites,\nI, I’ and II, which are located at the center of D6R, the\njust side of D6R in β-cage and the center of 6R in su-\npercage, respectively, as illustrated in Fig. 18 [11]. There\nare 12 cation sites for β-cage (four sites of I, four sites of\nI’ and four sites of II). Because site I is shared with ad-\njoiningβ-cages, there are 10 cation sites per β-cage. By\nthe loading of guest alkali metal, the number of cation\nincrease. At the same time, the locations of cations are\nadjusted by the interaction with the s-electronsshared in\nclusters, as expressed by the electron-phonon interaction\nin the Holstein-Hubbard model.\nAccording to the structure analysis in hydrated\nNaxK12−x-LSX, Na cations occupy preferably site I [65].\nSites I and II have the full occupancy, but site I’ has a\nhalf occupancy. According to the structure simulation\nof dehydrated zeolite LSX, the simultaneous occupations\nat sites I and I’ are expected, unlikely in other zeolites\n[66, 67]. The total average number of cations is ≈10 for\noneβ-cage, and the average number becomes ≈8 perβ-\ncage because of the sharing of site I between adjoining\nβ-cages. The 8 of 12 cations are distributed around each!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nSite I Site I' \nSite I' Site II \nFIG. 18. (Color online) Schematic illustration of cation si tes\nI, I’ and II around β-cage.\nβ-cage. Other 4 cations are distributed in supercage.\nBy the loading of guest alkali metal, s-electrons are\nshared with cations, and the metallic bonding among\ncations stabilizes cation-rich clusters in β-cages. In\nNan/Na12-LSX, a full occupation of Na cations are ob-\nserved simultaneously at sites I, I’ and II for n= 9.4 and\n16.7,namely12cationsforeach β-cagecluster(10cations\nperβ-cage cluster) [11]. In the simplest model, the pos-\nsible numbers of cations for the cluster in β-cage are 10,\n11 and 12 with the increase in n, where the numbers of\ncations at site I’ are 2, 3 and 4, respectively. According\nto this model, three kinds of β-cage clusters are expected\nwith respective optical excitation energies. The optical\nexcitation energy from 1 sto 1pstates is mainly deter-\nmined by the confinement potential size of s-electrons.\nThe size is basically determined by that of β-cage, but\nthese additional cations can extend slightly the effective\nsize of the confinement potential. The origin of the dif-\nferent excitation energies of β-cage clusters around 2.5\neV in Figs. 5, 6 and 7 is assigned to the difference in the\nnumber of cations and the kind of cations.\nIn an Na-K alloy system, a stronger cohesion effect for\nNa atoms makes Na-rich clusters more stable [12]. Na\nclusters in Na 12-LSX are nonmagnetic, because of a large\nS[8]. Atn <6.5 in K n/Na4K8-LSX, Na-rich clusters are\nexpected to be stabilized at β-cages as small bipolarons\nat the condition of U < Sin Fig. 17(b). The 2.3 eV\nreflection band in Fig. 6 are assigned to such Na-rich\nsmall bipolarons. The candidate of magnetic clusters of\nsmall polarons is K-rich ones. At 6 .5< n <8.5, K-\nrich small polarons are expected to be stabilized at the\ncondition of Eq. (7) for U > Sin Fig. 17(a), and are\nobserved at 2.8 eV reflection band in Fig. 6, in addition\nto Na-rich small bipolarons at 2.3 eV. At n >8.5, K-\nrich small bipolarons are stabilized at the condition of\nEq. (8).\nAt higher K-loading densities by the pressure loading\nin Kn/Na4K8-LSX, a new ferrimagnetism has been ob-\nservedattheloadingpressureof ≈0.5GPa[7]. TheCurie\nconstant is ≈3.5×10−4Kemu/cm3which is assigned to\nthe contribution of magnetic sublattices of β-cage clus-\nters and supercageones. The spontaneousmagnetization\nis much smaller than that expected from the Curie con-12\nstant, because of the cancellation of magnetizations by\nthe antiferromagnetic interaction between two magnetic\nsublattices in ferrimagnetism. The magnetic moments of\nβ-cage clusters under the pressure loading are assigned\nto small polarons at 1 pstates.\nIn Kn/Na7.3K4.7-LSX, the increase in localized mag-\nnetic moments have been observed clearly at 8 .2< n <\n9.7 in the increase in the Curie constant, and a nearly\npure ferromagnetism has been observed at 8 .4< n <9.7\nin the insulating phase [12]. Simultaneously, a reflection\nband of β-cage clusters at 2.8 eV has been observed at\nn >8. The origin of the magnetism is assigned to the\nferromagnetic superexchange coupling between magnetic\nmoments of β-cage clusters (small polarons) through sp3\nclosed-shell clusters in supercages. In reflection spec-\ntra,β-cage clusters are observed at 2.4 eV for n/greaterorapproxeql4\n[12]. These clusters are nonmagnetic and and assigned\nto the cace of U < Sshown in Fig. 17(b), where Na-rich\nclusters are preferentially stabilized. Clusters observed\nat 2.8 eV at 8 < n/lessorapproxeql9.7 are assigned to K-rich ones\n(small polarons) with magnetic moments at β-cages for\nU > S, and they become nonmagnetic (small bipolarons)\natn/greaterorapproxeql9.7, as illustrated in Fig. 17(a).\nIn Kn/K12-LSX, pure K clusters in β-cages can be\nmagnetic (small polarons) at large n. A ferrimagnetism\nby the antiferromagnetic interaction between localized\nmoments of β-cage clustersand the itinerant electron fer-\nromagnetism of supercage clusters has been observed at\nn≈9[1,9]. Thisferrimagnetismdisappearsat n≈11by\nthe pressure loading at ≈0.3 GPa, because of the gener-\nation of nonmagnetic β-cage clusters (small bipolarons)\n[14].\nC. Supercage clusters and their interaction with\nβ-cage clusters\nIn zeolite sodalite (SOD frameworkstructure), β-cages\nare arrayed in a body centered cubic structure by the\nsharing of 6Rs with eight adjoining β-cages. An an-\ntiferromagnetism of clusters in β-cages of sodalite has\nbeen observed clearly by the antiferromagnetic interac-\ntion through 6Rs [1, 41–55]. In zeolite LSX, each β-cage\nshares 6Rs with four adjoining supercages. The antifer-\nromagnetic interaction between β-cage clusters and su-\npercageonesoccursthrough6Rs, whereoneup-spinin β-\ncagearrangesdown-spinsin fouradjoiningsupercages, as\nillustrated in Fig. 19. These four supercages with a com-\nmon adjoining β-cage are the second nearest neighbors\nwith each other. Each supercage shares 12Rs with four\nadjoining supercages, and electrons in supercage clusters\nitinerate over many supercages as large polarons. If the\nnumber densityofmagnetic β-cageclustersincreases, the\nlong range magnetic ordering of an itinerant electron fer-\nromagnetism at supercage clusters is assisted geometri-\ncally by the antiferromagnetic interaction with the mag-\nnetic moments of β-cage clusters. At the same time, the\nmagnetic moments of β-cage clusters are ordered in theferrimagnetism, although the direct interaction between\nβ-cage clusters is absent. The hybridization effect of β-\ncage clusters with itinerant electrons of supercage clus-\nters is expected to play an important role in electrical\nproperties, if many β-cages are filled with small polarons\nwith magnetic moments.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nβ-cage supercage \nsupercage -supercage \nsupercage -β-cage \nFIG. 19. (Color online) Schematic illustration of cluster n et-\nworks in zeolite LSX. Clusters at supercages have an interac -\ntion network of a diamond structure. Each clusters at β-cages\nhas an interaction with clusters at four adjoining supercag es.\nThe direct interaction between β-cage clusters is absent. See\ntext in detail.\nIn the Kondo system, localized electron spins of mag-\nnetic atoms dilutely distributed in metal have an inter-\naction with conduction electron spins, and an electri-\ncal resistivity gradually increases at very low tempera-\ntures. Because of the Coulomb repulsion between local-\nized electrons at the magnetic atom, up-spins and down-\nspins of conduction electrons near the Fermi energy con-\ntribute equivalently to the localized electronic state, and\nthe Kondo singlet state is formed at very low tempera-\ntures. In the Kondo lattice system, an array of magnetic\natoms provide a remarkable increase in resistivity at low\ntemperatures, as observed in a typical Kondo insulator\nYbB12[68,69]. Theresistivitydecreasesunderhighmag-\nnetic fields up to ≈50×104Oe in YbB 12[70].\nIn Kn/Na4K8-LSX, a remarkable increase in resistiv-\nity is observed at low temperatures in Fig. 13. This re-\nsult resembles the Kondo insulator YbB 12. A similar\nincrease has been observed in K n/K12-LSX [1, 9]. The\nactivation energy indicated in Fig. 14 is temperature de-\npendent as observed in YbB 12[68, 69]. However, there is\nan essential difference between the Kondo insulator and\nKn/NaxK12−x-LSX in magnetism. The metallic narrow\nband at supercage clusters in K n/NaxK12−x-LSX is fer-\nromagnetic at low temperatures both by the intraband\nelectron-electron interaction and by the antiferromag-\nnetic interaction with magnetic clusters at β-cages. A\nenergy gap model of the ferrimagnetism is schematically\nillustrated in Fig. 20. Electrons in magnetic clusters at\nβ-cages have an antiferromagnetic interaction with itin-\nerant electrons at supercages, and the energy gap opens\nat the Fermi energy EFatlow temperatures. Achangeof13\nresistivity, however,is not observedundermagneticfields\nup to 13 ×104Oe in K n/Na4K8-LSX within the experi-\nmental accuracy, indicating that the gap in the itinerant\nelectron ferromagnetism seems to be kept under these\nmagnetic fields. A detailed theory is needed to explain\nthese results in the future.\nantiferromagnetic \nDOS \n(β-cage)E\n(supercage)EF\nDOS E\n(supercage)EFU−STC>T TC T)\nand paramagnetism ( TC< T) in K n/NaxK12−x-LSX. Local-\nizedelectrons in β-cageshaveanantiferromagnetic interaction\nwith itinerant electrons of narrow energy band of supercage\nclusters. The gap is opened at the Fermi energy EFat the\nferrimagnetism.\nIn Figs. 9 and 11, the Weiss temperature at the fer-\nrimagnetic region is positive and negative at lower and\nhigher values of n, respectively. The Weiss temperature\nTWin the meanfield theoryoflocalizedmomentsis given\nby Eq. (A18) in Appendix A, where the intra-sublattice\nmean field coefficient of β-cage clusters, λββ, is assumed\nto be zero. The asymmetry of TWcan not be explained\nbythen-dependenceofthe Curieconstantof β-cageclus-\nters,Cβ, which is defined by Eq. (A11), because the\nnumber density of magnetic clusters in β-cages,Nβ, is\nsymmetric for ferrimagnetism according to the model il-\nlustrated in Fig. 17(a). According to Eq. (A20), a neg-\native value of the Weiss temperature is expected at the\ncondition Csλss<2Cβλsβ, where λssandλsβare the\nintra-sublattice mean field coefficient of supercage clus-\nters and the inter-sublattice mean field coefficient be-\ntween supercage clusters and β-cage ones, respectively.\nCshere is the Curie constant of supercage clusters in the\nlocalized moment model and is given by Eq. (A10). The\nmain reason of the asymmetry of TWis expected to be\nthe increase in λsβwithn. According to the model at\nU > Sillustrated in Fig. 17(a), λsβincreases with n,\nbecause the Fermi energy EFincreases with nand then\nthe hybridization between electrons of supercage clusters\nand the localized electrons at β-cage clusters increases\nwithn.D. Magnetization process of ferrimagnetism\nThe magnetization process at 1.3 K in K n/NaxK12−x-\nLSX shown in Fig. 12 displays curves rounded out. The\nmagnetization process of ferrimagnetism at T= 0 is il-\nlustrated schematically in Fig. 21. In an ordinary fer-\nrimagnetism of classical magnetic moments, a constant\nmagnetization is observed up to a spin-flop field, and\na constant increase in magnetization up to the satura-\ntion field, as indicated by black lines. In the ferrimag-\nnetism in K n/NaxK12−x-LSX, the magnetic sublattice at\nsupercages is an itinerant electron ferromagnetism, and\nthe magnetization, Ms, increases with the applied mag-\nnetic field, because of the suppression of magnetization\nby the dynamical spin fluctuation [14, 71–74]. At low\nfields, the dominant magnetization of the magnetic sub-\nlattice is oriented to the applied magnetic field. For ex-\nample, the dominant magnetization in the N´ eel’s N-type\nferrimagnetismisthemagneticsublatticeat β-cages,Mβ,\nbelow the compensation temperature. According to the\nmean field theory, the effective field from MβtoMsis op-\nposite to the external field, and the total magnetization\nis given by Mβ−Ms. With the increase in the external\nfield,Msdecreasesand the total magnetizationincreases.\nAbove the spin-flop field, the angle between MsandMβ\ndecreases and Msincreases with the external field. The\ntotal magnetization increases up to the saturation value\nMβ+Ms(max), as indicated by red curves in Fig. 21. In\nK7.7/Na4K8-LSX,Mβ−Ms≈0.3 G at low fields, and\nMβ+Ms(max)≈2.7 G in Fig. 12. If we assign Mβ≈0.7\nG in the sudden increase in the Curie constant in Fig. 9,\nMsat low fields and Ms(max) are estimated to be ≈0.4\nand≈2.0 G, respectively.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nM\nMβMβ\nMsMs(max)\nHspin-flop \nFIG. 21. (Color online) Schematic illustration of the magne -\ntization process of ferrimagnetism up to high magnetic field s.\nMβandMsare magnetizations of magnetic sublattices at β-\ncages and supercages, respectively. See text in detail.14\nV. SUMMARY\nWe measured electronic properties in detail for\nKn/NaxK12−x-LSX mainly for x= 4. Ferrimag-\nnetic properties are observed in K n/Na4K8-LSX and\nKn/Na1.5K10.5-LSX. At the same time, the Curie con-\nstantincreases,andareflectionbandof β-cageclustersat\n2.8eVisobservedinaccordancewiththeferrimagnetism.\nAn electrical resistivity indicates metallic value at n/greaterorapproxeql6\nin Kn/Na4K8-LSX. The ferrimagnetism is explained by\nthe antiferromagnetic interaction between the magnetic\nsublattice of localized moments at β-cage clusters and\nthat of itinerant electron ferromagnetism at supercage\nclusters. The electrical resistivity increases extraordinar-\nily at low temperatures in ferrimagnetic samples. We try\nto explain the anomaly in the electrical resistivity by the\nanalogyof the Kondo insulator, where itinerant electrons\nof supercage clusters interact with localized electrons of\nβ-cage clusters. However, itinerant electrons of the nar-\nrow energy band of supercage clusters are ferromagnetic,\ndifferently from nonmagnetic electrons of the ordinary\nenergy band in the Kondo insulator.\nACKNOWLEDGMENTS\nWe are deeply grateful to Profs. R. Arita, K. Naka-\nmura, and H. Aoki for theoretical studies and discus-\nsions. We also thank Mr. S. Tamiya (Osaka Univer-\nsity) for chemical analysis. This work was supported\nby Grant-in-Aid for Scientific Research on Priority Ar-\neas (No. JP19051009), Grant-in-Aid for Scientific Re-\nsearch (A) (No. JP24244059 and No. JP13304027) and\n(C) (No. JP26400334), Grant-in-Aid for Creative Sci-\nentific Research “New Phases of Matter in Multidisci-\nplinary Approaches” (No. JP15GS0213), Global COE\nProgram “Core Research and Engineering of Advanced\nMaterials-Interdisciplinary Education Center for Materi-\nals Science” (G10), the 21st Century COE Program “To-\nwards a new basic science: depth and synthesis” (G17),\nMEXT Japan.\nAppendix A: Ferrimagnetism\nWe calculate a ferrimagnetism by the use of the\nmean field (molecular field) theory. We assume two\nnonequivalent magnetic sublattices of localized moments\ncorresponding to supercage clusters and β-cage ones\nin Kn/NaxK12−x-LSX. The geometrical arrangement\nshown in Fig. 19 and the itinerant electron ferromag-\nnetism of supercage clusters are not considered.\nWedefinemeanfieldsforsupercageclustersand β-cage\nclusters, HmsandHmβ, respectively, as\nHms=λssMs−λsβMβ, (A1)\nHmβ=λββMβ−λsβMs, (A2)whereMsandMβare magnetizations of respective mag-\nnetic sublattices for the ferrimagnetism, and λss,λββand\nλsβtheintra-sublatticemeanfieldcoefficientofsupercage\nclusters, that of β-cage clusters and the inter-sublattice\nmean field coefficient between supercage clusters and β-\ncage ones, respectively. The minus sign of the second\nterm in the right hand side of above equations means\nan antiferromagnetic interaction between two magnetic\nsublattices.\nThe magnetizations of both sublattices under the ex-\nternal magnetic field Hat the temperature Tare given\nas\nMs=NsgsµBBJs/parenleftbigg\ngsµBJsH+Hms\nkBT/parenrightbigg\n,(A3)\nMβ=NβgβµBBJβ/parenleftbigg\ngβµBJβH+Hmβ\nkBT/parenrightbigg\n,(A4)\nwhereNsandNβare the number densities of supercage\nclustersand β-cageones,respectively, gsandgβthegval-\nues ofrespectiveclusters, and JsandJβthe total angular\nmomentum quantum numbers of respective clusters. kB\nis the Boltzmann constant. BJ(y) is the Brillouin func-\ntion, and is given for |y| ≪1 as\nBJ(y) =J+1\n3Jy. (A5)\nAt sufficiently high temperatures of paramagnetism, fol-\nlowing conditions are satisfied:\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglegsµBJsH+Hms\nkBT/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1, (A6)\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglegβµBJβH+Hmβ\nkBT/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1. (A7)\nThen, we obtain following magnetizations by using\nEqs. (A1) and (A2) as\nMs=Cs\nT(H+λssMs−λsβMβ),(A8)\nMβ=Cβ\nT(H+λββMβ−λsβMs),(A9)\nwhere the Curie constants of supercage clusters and β-\ncage ones, CsandCβ, respectively, are given as\nCs=Nsg2\nsµ2\nBJs(Js+1)\n3kB, (A10)\nCβ=Nβg2\nβµ2\nBJβ(Jβ+1)\n3kB. (A11)\nFrom Eqs. (A8) and (A9), the total magnetic suscep-\ntibilityχis given by\nχ=Ms\nH+Mβ\nH\n=T(Cs+Cβ)−CsCβ(2λsβ+λββ+λss)\nT2−T(Csλss+Cβλββ)+CsCβ/parenleftbig\nλssλββ−λsβ2/parenrightbig.(A12)15\nThe Curie temperature TCis obtained from Eq. (A12)\nby the divergence condition at the higher temperature as\nTC=Csλss+Cβλββ\n2\n+/radicalBig\n(Csλss−Cβλββ)2+4CsCβλsβ2\n2.(A13)\nAt sufficiently high temperatures, χis expected to ap-\nproache the Curie-Weiss law\nχ≈Cs+Cβ\nT−TW, (A14)\nwhereCs+CβandTWare the total Curie constant and\nthe Weiss temperature, respectively. We obtain the re-\nlation at sufficiently high temperatures from Eq. (A12)as\nCs+Cβ\nχ≈T+CsCβ(2λsβ+λββ+λss)\nCs+Cβ\n−(Csλss+Cβλββ).(A15)\nFinally, we extract TWfrom Eqs. (A14) and (A15) as\nTW=−CsCβ(2λsβ+λββ+λss)\nCs+Cβ\n+Csλss+Cβλββ(A16)\nIf we assume no intra-sublattice interaction of β-cage\nclusters as λββ= 0, we obtain TCandTWas\nTC=Csλss\n2\n1+/radicalBigg\n1+4Cβλ2\nsβ\nCsλ2ss\n,(A17)\nTW=Cs\nCs+Cβ(Csλss−2Cβλsβ).(A18)\nThe positive and negative values of TWare obtained as\nTW>0 atCsλss>2Cβλsβ, (A19)\nTW<0 atCsλss<2Cβλsβ.(A20)\n[1] T. Nakano and Y. Nozue, Adv. 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Ser. 868, 012002 (2017)." }, { "title": "2304.14009v1.Effective_Tight_Binding_Model_of_Compensated_Ferrimagnetic_Weyl_Semimetal_with_Spontaneous_Orbital_Magnetization.pdf", "content": "E\u000bective Tight-Binding Model of Compensated Ferrimagnetic Weyl Semimetal with\nSpontaneous Orbital Magnetization\nTomonari Meguro1,\u0003Akihiro Ozawa2,yKoji Kobayashi1,zand Kentaro Nomura1x\n1Department of Physics, Kyushu University, Fukuoka 819-0395, Japan and\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\nThe e\u000bective tight-binding model with compensated ferrimagnetic inverse-Heusler lattice\nTi2MnAl, candidate material of magnetic Weyl semimetal, is proposed. The energy spectrum near\nthe Fermi level, the con\fgurations of the Weyl points, and the anomalous Hall conductivity are\ncalculated. We found that the orbital magnetization is \fnite, while the total spin magnetization\nvanishes, at the energy of the Weyl points. The magnetic moments at each site are correlated with\nthe orbital magnetization, and can be controlled by the external magnetic \feld.\nI. INTRODUCTION\nTopological semimetals are new classes of materi-\nals characterized by the topologically-nontrivial gapless\nnodes. One of the representative systems is the Weyl\nsemimetal (WSM), a gapless semiconductor with non-\ndegenerate point nodes called Weyl points [1{3]. In mo-\nmentum space, these nodes behave as a source or sink\nof a \fctitious magnetic \feld (Berry curvature), which is\ndistinguished by the sign degrees of freedom, so-called\nchirality [4, 5]. The Weyl points with the opposite chi-\nrality must appear in pairs [6, 7]. Originating from the\nWeyl points, distinctive magnetoelectric e\u000bects, such as\nthe chiral magnetic e\u000bect [8{11], arise. Generally, to re-\nalize the WSM, one has to break either the inversion\nor time-reversal symmetry of Dirac semimetal [12{14],\nwhich has degenerate gapless linear dispersions. As early\ntheoretical works, the WSM phases in an antiferromag-\nnetic pyrochlore structure [2] and a topological insula-\ntor multilayer [3] were proposed. Especially, in Ref. 3,\nBurkov et al. discussed the anomalous Hall e\u000bect (AHE)\nin the WSMs with broken time-reversal symmetry, so-\ncalled magnetic Weyl semimetals (MWSMs). It was\nshown that the anomalous Hall conductivity (AHC) is\nproportional to the distance between the pair of Weyl\npoints with the opposite chirality.\nAfter these theoretical predictions, exploring for the\nWSM phases in speci\fc materials has been demonstrated.\nIn the early stage of the experimental studies, non-\nmagnetic WSMs with the broken inversion symmetry,\nsuch as TaAs, were examined [15, 16]. On the other\nhand, a great deal of e\u000bort has been devoted to realize\nthe MWSMs. After then, both theoretical and experi-\nmental studies have succeeded in discovering the MWSM\nphases in some systems, such as layered-kagome [17{40]\n\u0003meguro.tomonari@mbp.phys.phys.kyushu-u.ac.jp\nyPresent address: Institute for Solid State Physics, Univer-\nsity of Tokyo, Kashiwa 277-8581, Japan; akihiroozawa@issp.u-\ntokyo.ac.jp\nzPresent address: Physics Division, Sophia University, Chiyoda-\nku, Tokyo 102-8554, Japan; k-koji@sophia.ac.jp\nxnomura.kentaro@phys.kyushu-u.ac.jpand Heusler systems [41{47]. The layered-kagome sys-\ntems attract much attention from viewpoints of anoma-\nlous transport phenomena [18{26, 31, 32, 38] and various\nmagnetic orderings [17, 34, 36, 39, 40]. Antiferromag-\nnetic Mn3Sn shows the AHE even without net magneti-\nzation [18{24]. Ferromagnetic Co3Sn2S2shows the giant\nAHE and small longitudinal conductivity, resulting in the\nlarge anomalous Hall angle reaching 20% [26{30, 33, 35,\n37, 39]. As other promising candidates, recent studies\nreported ferromagnetic Heusler systems with relatively\nhigh Curie temperature TCcompared to those of the\nlayered-kagome systems. For example, Co2MnGa [41{\n43] withTC\u0019694 K [44] and Co2MnAl [45, 46] with\nTC\u0019724 K [47] are also studied. In addition to the\nlayered-kagome and Heusler systems, other systems such\nas EuCd2As2[48{50] and LnAlPn[51, 52] (Ln= lan-\nthanides,Pn= Ge, Si) are also reported as candidate\nmaterials of MWSMs. A wide variety of candidates for\nMWSMs has been explored and has in\ruenced on both\nthe \feld of topological materials and magnetism.\nThe magnetic Weyl semimetal phase has been re-\nported in compensated ferrimagnetic inverse Heusler al-\nloy Ti2MnAl [53] by \frst-principles calculations. In\nTi2MnAl, magnetic moments at two Ti sites are anti-\nparallel to that at the Mn site, showing zero net magne-\ntization. The transition temperature is determined to be\n650 K [54], which is comparable to those of other Heusler\nWeyl systems, such as Co2MnGa. Besides, Ti2MnAl\nexhibits a large AHE despite its vanishing total spin\nmagnetization, similar to Mn3Sn. However, in contrast\nto Mn3Sn, the density of states at the Weyl points in\nTi2MnAl is negligibly small, resulting in the manifesta-\ntion of a large anomalous Hall angle compared to other\nMWSMs. These properties indicate that the unique elec-\ntronic and magnetic structures of Ti2MnAl provide dis-\ntinctive magnetoelectric response and spintronic func-\ntionalities compared to conventional materials.\nIn order to study the magnetoelectric responses spe-\nci\fc to Ti2MnAl, a quantitative analysis is necessary.\nFirst-principles calculations are widely used to precisely\ncalculate the electronic structure, considering all the elec-\ntron orbitals. However, with the \frst-principles calcula-\ntions, it is generally di\u000ecult to study the magnetoelectric\nresponses related to the complicated spin texture, sucharXiv:2304.14009v1 [cond-mat.mes-hall] 27 Apr 20232\n(a) Lattice structure(b) Primitive cell\n(c) Inter-sublattice hopping(d) Intra-sublattice hoppingzyx\nTi1\nMn\nAl\nTi2\nFIG. 1. (a) Conventional unit cell of Ti 2MnAl. Ti and Mn\nare responsible for the ferrimagnetic ordering. (b) Primitive\nunit cell of Ti 2MnAl. In our model, only Ti1 (A), Ti2 (B),\nand Mn (C) sublattices are taken into account. (c) Inter-\nsublattice nearest-neighbor hoppings. (d) Intra-sublattice\nnearest-neighbor hoppings. Every sublattice has the same\nhopping vectors.\nas the magnetic domain wall. This is because the ma-\ntrix of the Hamiltonian becomes huge due to the lack\nof translational symmetry. Therefore, a simple tight-\nbinding model describing the Weyl points of Ti2MnAl is\nbene\fcial to calculate these magnetoelectric responses.\nIn this paper, we construct an e\u000bective tight-binding\nmodel of Ti2MnAl using a few orbitals. We consider the\nsingle orbitals of two Ti and Mn, spin-orbit coupling, and\nexchange interaction between compensated ferrimagnetic\nordering and itinerant electron spin. Using our model,\nwe study the electronic structure, AHE, spin and orbital\nmagnetizations, and magnetic anisotropy. We show that\nour model describes the con\fguration of the Weyl points\nthat is consistent with the results obtained by the \frst-\nprinciples calculations. Also, we discuss the control of\ncompensated ferrimagnetic ordering by orbital magneti-\nzation.\nII. MODEL\nIn this section, we introduce a simple tight-binding\nmodel of the compensated ferrimagnetic Weyl semimetal\nTi2MnAl. The crystal structure of Ti2MnAl is shown in\nFig. 1(a). Each sublattice (Ti1, Ti2, Mn, and Al) forms\na face-centered-cubic (FCC) lattice, and thus the prim-\nitive unit cell is the FCC type [Fig. 1(b)]. By focusing\non pairs of the sublattices, Ti1-Al (orange and gray) and\nTi2-Mn (red and blue) form the rocksalt structure. The\nrest of the combinations, e.g., Ti1-Ti2 or Ti1-Mn, form\ndiamond or zincblende structures.Our model consists of a single orbital from each of\nTi1 (A), Ti2 (B), and Mn (C) atoms. We neglect Al or-\nbitals, which are not responsible for the magnetism, for\nsimplicity. We explain our model Hamiltonian Hby di-\nviding into three components, H=Ht+Hexc+HSOC,\nwhereHtrepresents the hopping, Hexcthe exchange cou-\npling, andHSOCthe spin-orbit coupling (SOC).\nThe hopping component Htreads\nHt=\u0000X\nhijis\u0010\ntABay\nisbjs+tBCby\niscjs+tCAcy\nisajs+ h:c:\u0011\n\u0000X\nhijis\u0010\ntAAay\nisajs+tBBby\nisbjs+tCCcy\niscjs\u0011\n+X\nis\u0010\n\u000fAay\nisais+\u000fBby\nisbis+\u000fCcy\niscis\u0011\n; (1)\nwhereais,bis, andcisare the annihilation operators for\nelectrons at Ti1 (A), Ti2 (B), and Mn (C) sites, respec-\ntively. The \frst line corresponds to the inter-sublattice\nnearest-neighbor hopping [Fig. 1(c)]. The second line cor-\nresponds to the intra-sublattice nearest-neighbor hopping\n[Fig. 1(d)]. The third line is the on-site energy.\nThe exchange component is\nHexc=\u0000X\nim\u0001(JAsA;i+JBsB;i\u0000JCsC;i);(2)\nwheresA;i=ay\nis(\u001b)ss0ais0is the itinerant spin operator\nof A site, and the same applies to sBandsC.mis the\nunit vector that is parallel to the magnetic moment of Ti\n(A and B) and antiparallel to that of Mn (C). J\u000b(\u000b=\nA;B;C) are the coupling strength.\nThe SOCHSOCoriginates from the broken inversion\nsymmetry of the crystal structure. The dominant sym-\nmetry breaking comes from the imbalance between TI1\nand Al sublattices. We assume the amplitudes of the\nSOC terms for the Ti2-Ti2 and Mn-Mn hoppings are the\nsame, for simplicity. Since the atomic number of Ti and\nMn are close to each other, compared with Al, we neglect\nthe SOC for the Ti1-Ti1 hopping. The SOC term can be\ndescribed in a Fu-Kane-Mele-like form [55],\nHSOC=\u0000i8\u0015SOCp\n2a2X\nhijih\nby\nis(dBij\n1\u0002dBij\n2)\u0001(\u001b)ss0bjs0\n+cy\nis(dCij\n1\u0002dCij\n2)\u0001(\u001b)ss0cjs0i\n:(3)\nHere\u0015SOCis the strength of the SOC. d\u000bij\n1;2are the two\nnearest-neighbor hopping vectors from the site itojof\nthe sublattice \u000b. Note that dBij\n1;2=nlijanddCij\n1;2=\n\u0000nlij(lij= 1;2;3;4) withn1=a\n4(1;\u00001;\u00001),n2=\na\n4(\u00001;1;\u00001),n3=a\n4(\u00001;\u00001;1), andn4=a\n4(1;1;1).\nWe set the lattice constant a= 1 for simplicity. The\nhopping parameters are set to tAB= 1:1t0,tBC= 0:4t0,\ntCA= 1:2t0,tAA= 0:05t0,tBB= 0:85t0, andtCC=\n\u00000:05t0. On-site energies \u000fA=\u000fB=\u000fC=\u00002:15t0. The\nstrengths of the exchange coupling JA=JB= 0:7t0,3\nJC= 1:7t0. The strength of the SOC \u0015SOC =\u00000:2t0.\nWe set the energy unit t0= 0:33 eV. The parameters\nare set so that the energy bands, density of states, and\nAHCs become consistent with \frst-principles calculations\nas discussed later.III. ELECTRONIC STRUCTURE\nNow, we study the electronic structure of this model.\nThe momentum representation of the Hamiltonian is ex-\nplicitly expressed as,\nH(k) =0\n@\u0000tAAf(k)\u0000JAm\u0001\u001b\u0000tABg(k) \u0000tCAg\u0003(k)\n\u0000tABg\u0003(k)\u0000tBBf(k) +\u0015SOCR(k)\u0001\u001b\u0000JBm\u0001\u001b\u0000tBCP\ni=x;y;zcoski\n2\n\u0000tCAg(k)\u0000tBCP\ni=x;y;zcoski\n2\u0000tCCf(k)\u0000\u0015SOCR(k)\u0001\u001b+JCm\u0001\u001b1\nA:(4)\nHere,f(k) andg(k) correspond to the FCC and diamond\nhoppings, respectively, and are de\fned as,\nf(k)=4\u0012\ncoskx\n2cosky\n2+cosky\n2coskz\n2+coskz\n2coskx\n2\u0013\n;(5)\ng(k) = exp \n\u0000i4X\n\u0016=1n\u0016\u0001k!\n: (6)\nR(k) corresponds to the FCC hopping with SOC and is\nde\fned as,\nR(k) =0\nBB@sinkx\n2\u0010\ncosky\n2\u0000coskz\n2\u0011\nsinky\n2\u0000\ncoskz\n2\u0000coskx\n2\u0001\nsinkz\n2\u0010\ncoskx\n2\u0000cosky\n2\u00111\nCCA: (7)\nWe use the magnetic moment pointing in the out-of-\nplane direction as m= (0;0;1). By solving the eigen-\nvalue equationH(k)jn;ki=Enkjn;ki, the eigenvalues\nEnkand eigenstatesjunkiare obtained. Here, n=\n1;2;:::;6 is the band index labeled from the bottom. Fig-\nures 2(a) and 2(b) show the band structure along the\nhigh-symmetry lines and the density of states (DOS) as\na function of energy, respectively. The high-symmetry\nlines are shown in Fig. 2(d). We assume that the Fermi\nlevelEFis the energy which is satis\fed 4 =6 \flling, and\nis being set as E=t 0= 0. At the energy EF, we have\nthe Weyl points as discussed later. In Fig. 2(a), red and\nblue lines indicate up and down spin band, respectively,\nin the absence of SOC. Green lines indicate those in the\npresence of SOC. Here, we focus on the spin up bands\nnearE=t 0= 0:0. As shown in Fig. 2(b), the spin up\nbands show the local minimum of the DOS near the Fermi\nlevel (E=t 0= 0:0). This minimum is consistent with the\nresult obtained by the \frst-principles calculations and\nmay be signi\fcant to the large anomalous Hall angle [53].\nAs shown in the insets of Fig. 2(b), the energy spec-\ntrum of these majority spin bands show the gapless lin-\near dispersions, corresponding to the Weyl points, as dis-\ncussed later. On the other hand, down-spin state (blue)\nis gapped at the Fermi level ( E=t 0= 0:0).Next we study the Weyl points structure in Brillouin\nzone. We show that our model describes the Weyl points\nstructure similarly located to the result obtained by the\n\frst-principles calculations. From our model Hamilto-\nnianH(k) 24 gapless nodes (Weyl points) are obtained\nbetweenn= 4 band and n= 5 band. The kx-kzplane\nhas the Weyl points as shown in Fig. 2(e). Each Weyl\npoints with chirality + ( \u0000) is labeled by W+\n\u000b(W\u0000\n\u000b).\u000b\ndistinguish the pairs of the Weyl points. To characterize\nthese nodes by chirality, we compute the Berry curva-\nturebnk=rk\u0002ankof then= 4 band. Here, ankis\nthe Berry connection, de\fned as ank=\u0000ihn;kjrkjn;ki.\nFigure 2(f) shows the Berry curvature distribution on kx-\nkzplane. The strength of the shade of the arrows is the\namplitude of b(k). The red and blue circles indicate the\nsources (+) and sinks ( \u0000) of the Berry curvature. The\ncon\fguration of the Weyl points is consistent with those\nobtained by the \frst-principles calculations [53].\nIV. ANOMALOUS HALL EFFECT\nLet us study the AHE in this section. Using the Kubo\nformula [56], the anomalous Hall conductivities (AHCs)\n\u001bij(i6=j) can be expressed as follows,\n\u001bij=\u0000ie2\n~Z\nBZd3k\n(2\u0019)3X\nm6=nf(Enk)\u0000f(Emk)\nEnk\u0000Emk+i\u0011\n\u0002hn;kj~vijm;kihm;kj~vjjn;ki\nEnk\u0000Emk:(8)\nHere,f(E) = 1=(e\f(E\u0000EF)+ 1) is the Fermi-Dirac distri-\nbution function and ~v=@H(k)\n@kis the velocity operator.\nFigure 2(c) shows \u001bxy,\u001byz, and\u001bzxas a function of en-\nergy. Near the energy of the Weyl points, the \u001bxyis\nmaximized , while \u001byzand\u001bzxvanish. The value of \u001bxy\nat the peak is \u001bxy\u00190:80 [e2=ah] = 496 S=cm, which rea-\nsonably agrees with the result obtained by \frst-principles\ncalculations, 550 S =cm [53].\nThen we study the relation between the AHCs and\nthe con\fguration of the Weyl points. As discussed in\nthe introduction, when the Fermi level is located at the4\n-6.-4.-2.0.2.4.6.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n����������|��-�-���(a)(b)(c)\n(d)(e)(f)E –EF[eV]—up—down—with SOC\nw/oSOCw/ SOCDOS [a.u.]AHC \t\t\t\t\t\t\t\t\n\t0\t1\t2\t3\t4\t5\t6\n\t0\t1\t2\t3\t4\t5\t6\t0\t0.2\t0.4\t0.6\t0.8\t1\n01234560123456\n!!\"!\"\"01234560123456024624610-1-210-1-20.00.51.0-1.0-0.510-1-2GXWKGLUWLK|UX—sxy syz—szxE –EF[eV]\nE –EF[eV]\n𝑊\"#𝑊\"$𝑊!\"𝑊!#𝑊%#𝑊%$𝑊&#𝑊&$𝑊'$𝑊'#kzkxkxkzahe2\nFIG. 2. (a) Band structure along the high symmetry lines. (b) The density of states for (red) up- and (blue) down-spin states\nand (c) anomalous Hall conductivities \u001bxy,\u001byx,\u001bzxas a function of energy. (d) The Brillouin zone and high symmetry lines\nof the system. (e) Con\fguration of the Weyl points with and without spin-orbit coupling and (f) the Berry curvature with\nspin-orbit coupling on the kx-kzplane.\nWeyl points, the AHCs can be calculated with the dis-\ntances between the Weyl points with opposite chirality\n\u0001Q\u000b= (K+\n\u000b\u0000K\u0000\n\u000b) [3],\n\u001bWeyl\nij =e2\n(2\u0019)2~12X\n\u000b=1X\nk\u000fijk\u0001Q\u000b\nk: (9)\nHereK+(\u0000)\n\u000b is the position of the Weyl points with chi-\nrality +(\u0000). This\u001bWeyl\nxy is being\u00190:75 [e2=ah], which\nis in good agreement with the maximized value obtained\nby the Kubo formula, shown in Fig. 2(c). Therefore,\nthe AHE in our model mainly originates from the Weyl\npoints.\nNext, we discuss the role of SOC for the Weyl points\nand the AHC. In the absence of SOC, the Weyl points\nare distributed symmetrically, as represented by empty\ncircles in Fig. 2(e). This can be anticipated by the\ncrystal symmetry of the system [53]. In this case, the\nsumP\n\u000b\u0001Q\u000bcancels, and thus AHC \u001bWeyl\nxy vanishes.\nOn the other hand, in the presence of SOC, the po-\nsitions of the Weyl points are shifted as represented\nby \flled circles in Fig. 2(e). For the pairs ( W+\n1,W\u0000\n1)\nand (W+\n2,W\u0000\n2), the distance between the Weyl points\u0001Q\u000b=1;2\nz becomes longer. Whereas, for the pairs of\n(W+\n3,W\u0000\n3) and (W+\n4,W\u0000\n4), \u0001Q\u000b=3;4\nz becomes shorter.\nTherefore, the cancellation of the sumP\n\u000b\u0001Q\u000bis bro-\nken, giving rise to the \fnite AHC \u001bWeyl\nxy. We note that\nthe Weyl points in other planes are shifted in a similar\nmanner to those in the kx-kzplane.\nV. SPIN AND ORBITAL MAGNETIZATIONS\nIn this section, we study spin and orbital magnetiza-\ntions in our model. The spin magnetization is calculated\nby the following equation,\nMspin\nz=\u0016BZEF\nE0[D\"(\u000f)\u0000D#(\u000f)]d\u000f: (10)\nHere,\u0016Bis the Bohr magneton, and E0is the lower band\nedge.D\"(\u000f) [D#(\u000f)] is the DOS for the up-spin (down-\nspin). The spin magnetization as a function of energy is\nshown in Fig. 3(a). Recall that the magnetic moments\nof Ti and Mn are parallel to the zaxis.Mspin\nzvanishes\natE=EF, indicating the compensated ferrimagnetic5\n-1.-0.50.0.51.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n𝑀!\"#$%𝜇&Å'𝐸−𝐸!eV(a)(b)\n𝑀()*$+\t×10,'𝜇&Å'𝐸−𝐸!eV\n1.00.50.0-0.5-1.0-2-101\n-2-101\n1.00.50.0-0.5-1.0𝑀!\"#$%&𝑀'\"#$%&𝑀(\"#$%&\nFIG. 3. (a) Spin magnetization Mspin\nzand (b) each compo-\nnent of orbital magnetization Morbitas a function of energy\nfor the magnetic moment mparallel to the zaxis.\nphase. Owing to this characteristic DOS, \fnite spin mag-\nnetization may be obtained by tuning EF, using a gate\nvoltage, for instance. When EFis increased (decreased),\n\fnite and positive (negative) spin magnetization might\nbe generated. This indicates that, in Ti2MnAl, one can\ninduce and switch the spin magnetization electrically.\nRecently, the electrical control of ferrimagnetic systems\nhas been studied from viewpoint of a functional magnetic\nmemory, for example in Ref. 57.\nNext we study the orbital magnetization [58{60],\nMorbit\nk =e\n2~Z\nBZd3k\n(2\u0019)3X\ni;j\u000fijk\nImX\nm6=nf(Enk)hn;kj~vijm;kihm;kj~vjjn;ki\n(Emk\u0000Enk)2\n\u0002(Enk+Emk\u00002EF):\n(11)\nFigure 3(b) shows each component of the orbital magne-\ntizationMorbitas a function of energy. We \fnd that\nMorbit\nz can be \fnite, while Morbit\nx andMorbit\ny vanish.\nAlthough orbital magnetization is usually much smaller\nthan spin magnetization, Morbit\nz is \fnite at EF, where\nMspin\nzis fully compensated. As discussed later, this small\nbut \fnite orbital magnetization might be used to switch\nthe directions of magnetic moments on Ti and Mn sites.\nVI. MAGNETIC ISOTROPY\nIn this section, we study the magnetic anisotropy and\nangular dependences of the AHCs and orbital magneti-\nzation. We start to study magnetic anisotropy. Figure 4\n04590135180225270315360-1.5-1.-0.50.0.51.1.5\n04590135180225270315360-0.06-0.04-0.020.0.020.040.06\n04590135180225270315360-2.45-2.4-2.35-2.3-2.25-2.2-2.15-2.1\n𝜃!\"[deg]𝜃!\"=𝜃#$𝜃!\"=0\n𝜃!\"𝜃#$\n𝜃#$04590135180225270315360-2.10-2.15-2.20-2.25-2.30-2.35-2.40-2.45Total Energy eVFIG. 4. The magnetization-angle dependence of the total\nenergy of electron, Eq. (12), when one rotates the magnetiza-\ntions of Mn and Ti uniformly (solid red line) or the magneti-\nzation of Mn solely (dotted blue line).\nshows the total energy of electrons, computed as\nEe=1\nNk6X\nn=1X\nkEnkf(Enk); (12)\nwhereNkis the number of k-mesh. Here, we consider the\ntwo tilted con\fgurations in which the magnetic moments\nof (a) Mn only or (b) both Mn and Ti are tilted in x-\nzplane. Their tilting angles are denoted by the angles\n(a)\u0012Mnand (b)\u0012Mn=\u0012Ti. The schematic \fgures for\nthese situations are shown in the insets of Fig. 4. In case\n(a), as the blue dotted line indicates, the electron energy\nis maximized at \u0012Mn= 180\u000e, indicating the stability of\nthe compensated ferrimagnetic ordering. On the other\nhand, in case (b), the electron energy is independent of\n\u0012Mn=\u0012Tias the red solid line indicates. Therefore, in\nour model, the compensated ferrimagnetic ordering does\nnot show the easy-axis magnetic anisotropy. With these\nresults, the ferrimagnetic interaction between Ti and Mn\ncan be estimated as 0 :1t0\u001933 meV, when t0= 0.33 eV\nis assumed.\nThen, we study a correlation between magnetic mo-\nments and orbital magnetization. In the previous sec-\ntion, we showed that, at the energy of the Weyl points,\nour model shows compensated spin magnetization and\n\fnite orbital magnetization. In the following, we tilt the\nmagnetic moments on both Ti and Mn sites in the x-z\nplane, as shown in the inset of Fig. 5(a). Figure 5(a)\nshows each component of the orbital magnetization as\na function of \u0012. We \fnd that the orbital magnetization\nfollows the direction of the magnetic moments. This fea-\nture can be used for the control of the magnetic mo-\nments, as similarly discussed in Refs. 24 and 40. Under\nan external magnetic \feld, magnetic moments on each\nsite and orbital magnetization couple with the \feld via\nZeeman interaction. However, as discussed in the pre-\nvious paragraph, the ferrimagnetic coupling between the\nmagnetic moments on Ti and Mn is stronger than the\ntypical strength of the Zeeman interaction ( \u00190:15 meV\nat 1 T). Thus, the Zeeman interaction of the magnetic6\n04590135180225270315360-1.5-1.-0.50.0.51.1.5\n04590135180225270315360-0.06-0.04-0.020.0.020.040.06\n04590135180225270315360-2.45-2.4-2.35-2.3-2.25-2.2-2.15-2.104590135180225270315360-1.5-1.-0.50.0.51.1.5\n04590135180225270315360-0.06-0.04-0.020.0.020.040.06\n04590135180225270315360-2.45-2.4-2.35-2.3-2.25-2.2-2.15-2.1\n𝜃[deg]AHC !!\"#𝜎!\"𝜎#!𝜎\"#𝜃[deg]𝑀$%&'(\t𝜇)Å*𝑀#$%&'(𝑀!$%&'(𝑀\"$%&'((a)\n(b)\n𝜃𝜃04590135180225270315360\n045901351802252703153600.060.040.020.00-0.02-0.04-0.061.51.00.50.0-0.5-1.0-1.5\nFIG. 5. The magnetization-angle dependence of (a) orbital\nmagnetization and (b) AHCs at E=EFwhen one changes\nthe direction of magnetic moment uniformly in the x-zplane.\nmoments on both sites cancels each other. This cancella-\ntion implies that only the orbital magnetization couples\nwith the external magnetic \feld. In addition, owing to\nSOC, the directions of the magnetic moments are lockedwith the orbital magnetization, as shown in Fig. 5(a).\nConsequently, the direction of the magnetic moments can\nbe controlled by an external magnetic \feld, even with-\nout net magnetization. The changes in the direction of\nthe magnetic moments may be probed by the AHE. Fig-\nure 5(b) shows the \u0012dependence of the AHCs. We \fnd\nthe relation \u001bij/\u0000P\nk\u000fijkMorbit\nk. This relation indi-\ncates that the directions of the magnetic moments are\nexperimentally determined by measuring the AHCs.\nVII. CONCLUSION\nIn this paper, we constructed an e\u000bective model of\ncompensated ferrimagnetic Weyl semimetal Ti2MnAl.\nThe Weyl points con\fguration in our model reasonably\nagrees with those obtained by the \frst-principles calcu-\nlations. The \fnite AHC in the presence of SOC can be\nunderstood by the shifting of the Weyl points. 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Belotelov2,3,4 \n1 Moscow Institute of Physics and Technology (MIPT), 141700 Dolgoprudny, Russia \n2 Russian Quantum Center, 121353 Moscow, Russia \n3 Physics and Technology Institute, Vernadsky Crimean Federal University, 295007 Simferopol, Russia \n4 Photonic and Quantum Technologies School, Lomonosov Moscow State University, 119991 Moscow, Russia \n5 Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991, Moscow, Russia \n \nFerrimagnets containing several partially compensated magnetic sublattices are \nconsidered the most promising materials for all-optical data storage and for ultrafast \ncommunications based on spin waves. There are two magnetic phases of the ferrimagnets: \ncollinear and non-collinear ones. Up to now spin dynamics in ferrimagnets has been studied \nmostly in the collinear state without paying much attention to the kind of the magnetic phase. \nHere we investigate laser induced ultrafast spin dynamics in a rare-earth iron garnet film in the \nnoncollinear phase as well. We identify a crucial influence of the magnetic phase on the excited \nspin modes which allowed us to discover several prominent effects previously overlooked. In \nparticular, the non-collinearity makes the quasi-antiferromagnetic mode sensitive to the \nexternal magnetic field and brings its frequency close to the frequency of the quasi-\nferromagnetic mode. The latter maximizes near the magnetization compensation point and \nvanishes towards the collinear phase. Spectacularly, at the phase transition the quasi-\nferromagnetic mode becomes soft and its amplitude significantly increases reaching 7°. This \nopens new opportunities for the ultrafast control of spins in ferrimagnets for nonthermal data \nstorage and data processing. \n1. INTRODUCTION \nUltrafast magnetic phenomena driven by femtosecond light pulses are emerging topics of \nmodern material science. The growth in the number of studies in this field is stimulated by \ntechnological demands and fundamental puzzles ranging from ultrafast all-optical data storage1–\n5 and spin transport 6,7 to quantum computing8,9. \nFerrimagnets are unique among ordered magnetic materials as they combine properties \nof both ferro- and antiferromagnets. Being composed of several antiferromagnetically coupled \nmagnetic ion sublattices, these materials sustain low frequency quasi-ferromagnetic (q-FM) and \nhigh frequency quasi-antiferromagnetic (q-AFM), so-called “exchange” modes10. The latter is not \nattainable by GHz conventional microwave stimuli and can be excited by femtosecond laser \npulses due to either thermal or optomagnonic effects11. Ultrafast spin dynamics along with \nultrafast switching, domain wall motion, spin waves, and skyrmion formation were extensively \nstudied in ferrimagnetic metals, such as GdFeCo5,12–16 and CoGd17–19. In contrast to metals and \nalloys, insulating oxides are almost lossless at optical frequencies. It makes them perfect \nmaterials for essential applications such as all-optical non-thermal magnetic recording1,2 and \ncontrol spin waves in all-dielectric nanostructures20. \nAnother important feature of ferrimagnets is possibility for full compensation of their \nsublattices magnetic moments which takes place at some temperature 𝑇ெ called magnetization \ncompensation point. The theory of magnetic phase transitions in compensated ferrimagnets was previously extensively studied21–24. In the H-T phase diagram of a ferrimagnet two main phases \nare possible: collinear and non-collinear ones. The collinear phase is accompanied with collinear \nalignment of magnetizations of both sublattices and external magnetic field. The non-collinear \nphase is characterized by bevel of sublattices magnetizations combined with a net magnetization \nslant to the external magnetic field. \nPrevious experiments on excitation of spin dynamics in rare-earth iron garnets (RIG) by \nfemtosecond laser pulses were primarily concerned on the magnetic states in the collinear \nphase10,25–28 and the kind of magnetic phase was not identified. An optical excitation of the \nexchange resonance between rare-earth and transition metal sublattices far from the \ncompensation point was studied in10 for an in-plane configuration of external magnetic field. The \nresults were found to be in an agreement with the conventional Kaplan–Kittel theory29,30 which \ndescribes q-AFM mode far from 𝑇ெ. In 28,31 a comprehensive experimental study of RIG in an in-\nplane magnetic field was made in a wide temperatures range including magnetization \ncompensation point, and both an exchange and ferromagnetic resonances were investigated. \nNevertheless, magnetic phases were not identified and the features of spin dynamics were not \nobserved. As a result, a simplified description of the mode frequencies behavior in terms of \nferromagnetic resonance and Kaplan-Kittel theories satisfied the experimental data. \nImportantly, the role of the magnetic phase on spin dynamics was not discussed \npreviously and, in particular, the non-collinear phase was not addressed in this aspect. At the \nsame time, nowadays, the room-temperature non-collinear phase, actively studied in various \nmagnetic materials32–34, is a potential candidate for chiral spin textures based magnetic non-\nvolatile memory35, spintronic devices36 and even qubits for quantum computing. Hence, the \nunderstanding of spin dynamics in this phase is of urgent fundamental and practical demand. \nIn the current work we confront magnetic phase of a ferrimagnet with its ultrafast spin \ndynamics and highlight a crucial role of the ground state in the character of spin oscillations. Spin \ndynamics in the non-collinear phase of RIG in the vicinity of the magnetization compensation \ntemperature point is studied comprehensively. We scrutinized magnetic phase transitions in the \nmaterial via magneto-optical measurements and distinguished the temperature-dependent \nfeatures of the spin modes by the magneto-optical pump-probe technique. The experimental \nresults are well described by the quasi-antiferromagnetic theoretical approach. \n2. RESULTS \nUnique magnetic properties of RIG are governed by its complex crystal structure. Fe3+ ions \noccupy sites in tetrahedral and octahedral sublattices of the cubic unit cell, with the total \nmagnetization of the Fe3+ ions in the tetrahedral positions being greater than in the octahedral \nones. Uncompensated magnetic moment of Fe3+ ions gives rise to net magnetization 𝐌ୣ which \nis antiparallelly ordered to magnetization 𝐌ୖ of rare-earth element occupying the third \n(dodecahedral) sublattice. Due to a huge exchange field between Fe3+ ions in tetrahedral and \noctahedral sublattices they can be treated as a single one with magnetization 𝐌ୣ in this case. \nSpins of the rare-earth sublattice are ordered by an exchange field produced by iron ions. The \nexistence of magnetic moment compensation point ( 𝑇ெ) is mainly due to a strong temperature \ndependance of 𝑀ୖ(𝑇), while 𝑀ୣ is hardly affected22. \nThe magnetization state of a RIG film can be defined using the sublattice magnetizations \n𝐌ୣ and 𝐌ୖ forming Néel vector 𝐋=𝐌ୣ−𝐌ୖ and magnetization vector 𝐌=𝐌ୣ+𝐌ୖ (Fig. 1a). The angle 𝜃=ఏಷିఏ\nଶ characterizes the deviation of the Néel vector from the film plane, \nwhile the angle 𝜖=ఏಷାఏ\nଶ characterizes a bevel of the sublattices magnetization from a collinear \nantiparallel orientation. Here 𝜃ி and 𝜃ୖ are the angles between the film plane and 𝐌ୣ and 𝐌ୖ, \nrespectively. For vectors in the upper half-space of the x-z plane 𝜃> 0 and for the bottom half-\nspace 𝜃< 0. Wherein, in statics MFe,R always lie in the plane formed by the magnetic field and \nanisotropy axis (Supplementary, Section II). \n \nFigure 1. (a) Configuration of the magnetic sublattices of the sample in the non-collinear and \ncollinear phases. (b) The hysteresis loops for different temperatures of the sample (inset shows \nexperimental configuration). (c) Experimental (bright) and theoretical (blurred) phase \ndiagrams of the RIG film. Black curves (dashed and continuous) represent phase transition \nboundary between non-collinear and collinear phases. The dotted straight line represents the \nslice at 4 kOe made for (d). (d) Temperature dependence of 𝜃 for the applied in-plane 4 kOe \nmagnetic field. Black points represent experimental data, whereas red line is theoretical \napproximation. The arrows show sublattices magnetization vectors orientation with respect to \nexternal magnetic field for collinear (blue filling), non-collinear (green filling) and compensated \n(white line) states. Vertical (dashed and continuous) lines represent boundaries between non-\ncollinear and collinear phases. \nHere we investigate а RIG film with Gd rare-earth ions (see Methods). The film has a \nuniaxial magnetic anisotropy with the axis perpendicular to the film. We consider the case when \nthe external magnetic field is in-plane and therefore perpendicular to the anisotropy axis. The \nanalysis of the system’s potential energy minimum reveals that in this case two magnetic phases \nare possible, depending on the temperature and external magnetic field: collinear and \nnoncollinear (Supplementary, Section III). In the collinear phase the sublattice magnetic \nmoments are antiparallel ( 𝜀= 0) and vectors 𝐌 and 𝐋 are aligned with the in-plane external \nmagnetic field H (𝜃= 0). The non-collinear phase provides the deflection of vectors 𝐌 and 𝐋 \nfrom H ( 𝜃≠ 0). This deflection is accompanied with noncollinearity of the sublattices magnetic \nmoment ( 𝜀≠ 0). The boarder between two phases is determined by the condition 𝑚=𝑚 \n(Supplementary, Section III). Here 𝑚 is a difference between sublattice magnetizations \n𝑚=𝑀ி(𝑇)−𝑀ோ(𝑇), while 𝑚 is a kind of critical magnetization defined by: \n𝑚=𝜒ୄ𝐻+ଶ\nு, (1) \nwhere χୄ=(ெూ(்)ାெೃ(்))మ\nଶஃெూ(்)ெ(்) is magnetic susceptibility (Supplementary, Section III), Λ is an \nexchange parameter between Fe and rare-earth sublattices, 𝐾=𝐾ୣ+𝐾ୖ, 𝐾ୣ and 𝐾ୖ are \nanisotropy constants of each sublattice (numerical values are given in Supplementary, Section I). \nThe collinear phase exists for |𝑚|>𝑚ୡ୰ and the non-collinear one establishes if |𝑚|<𝑚. \nThus, a second type phase transition from non-collinear to collinear phases occurs at 𝑚=𝑚. \nThis transition is directly related to the critical temperature 𝑇 at which 𝑚=𝑚. \nTo experimentally determine the magnetic phase diagram 𝜃(𝑇,𝐻) we measured the \nmagneto-optical hysteresis loops at various temperatures. Some of the loops are presented in \nFig.1b. The experiment was carried out in an almost in-plane external magnetic field (the field’s \ntilt angle was around 1 deg.) for the normal light incidence. Therefore, the measured Faraday \nrotation angle was proportional to the out-of-plane component of the sample gyration vector. \nSince the gyration in RIG is mostly provided by Fe3+ ions37, the observed Faraday rotation \nconforms to 𝐌ୣ. Furthermore, since the tilt between Fe and Gd sublattices given by angle 𝜖 is \nsmall (Supplementary, Section II), such measurement provides information regarding deflection \nof the Néel vector ( 𝜃≈𝜃ୣ). At zero magnetic field 𝐌ୣ is directed normally to the surface ( 𝜃ୣ=\n𝜋/2) due to the uniaxial magnetic anisotropy. A magnetic field applied in-plane deflects 𝐌ୣ from \nthe normal. Consequently, 𝜃 is found as follows: \n𝜃(𝐻,𝑇)≈𝜃ୣ(𝐻,𝑇)= asinቆ𝐹𝑅(𝐻,𝑇)\n|𝐹𝑅(0,𝑇)|ቇ, (2) \nwhere 𝐹𝑅(𝐻,𝑇) and 𝐹𝑅(0,𝑇) are angles of the Faraday rotation (FR) under applied magnetic \nfield H and zero magnetic field, correspondingly. \nThe experimentally measured magnetic phase diagram 𝜃(𝑇,𝐻) and its crossection at 𝐻=\n4 kOe are given in Fig. 1c,d, respectively. A distinct jump of derivative 𝜕𝜃 𝜕𝑇⁄ takes place at 𝑚=\n𝑚 (solid black curve in Fig. 1c), e.g. at 𝑇ୡ୰~383 K for 𝐻= 4 kOe (Fig. 1d), indicating the second \nkind phase transition. The other phase transition of the second kind appears for smaller \ntemperatures when 𝑚< 0 and |𝑚| =𝑚 (dashed black curve in Fig. 1c) which corresponds to \n𝑇ᇱ~284 K at H=4 kOe (Fig. 1d). For a fixed H the noncollinear phase ( 𝜃≠ 0) appears for the \ntemperature interval 𝑇ᇱ(𝐻) <𝑇<𝑇(𝐻). This temperature range decreases for larger \nmagnetic fields (Fig 1c) which is due to the Zeeman energy of the sample. Notably, in the non-\ncollinear phase theory predicts bistability of the system due the degeneracy of +𝜃 and –𝜃 states \n(see Supplementary, Section III, Fig. S3). However, in our experiments a small out-of-plane magnetic field component exists which lifts this degeneracy and selects the direction of M௭ \nprojection parallel to this small out-of-plane field component. Hence L flips while crossing 𝑇ெ due \nto a change of sing of 𝑚 at the temperature 𝑇ெ of the magnetization compensation, which \nindicates that for the studied sample 𝑇ெ= 336 K. \nUltrafast spin dynamics in the sample was excited by fs-laser pump pulses at 787 nm and \nobserved by the Faraday rotation (FR) of the fs-laser probe pulses at 515 nm delayed with respect \nto the pump pulse in the configuration shown in Fig. 2a (Methods). The pump-probe experiment \nwas conducted in a wide temperature range which allowed us to observe features of the non-\ncollinear states below and above TM. Moreover, we resolved the spin dynamics at the phase \ntransition ( Tcr=383 K at 𝐻= 4 kOe) and in the collinear phase, for larger temperatures (Fig.2b). \nFigure 2b,c demonstrate dynamics of the Néel vector in terms of the time-resolved Faraday \npolarization rotation (TRFR) at 𝐻= 4 kOe. Two kinds of oscillations are clearly observed: low \nfrequency ones at the time scale up to 350 ps (Fig.2b) and high frequency ones at the time scale \nof 50 ps (Fig. 2b and zoomed view in Fig. 2c). Notably, the high-frequency component in the TRFR \nsignal was not observed for the temperatures above TM. Data on the frequency and magnitude \nof these modes were extracted from gauged TRFR signals and compared with the theoretical \nvalues (Fig.3 a,b). \n \nFigure 2. Configuration of the pump-probe experiment (a). Ultrafast magnetization dynamics \nrepresented by the probe Faraday transients in the range of 350 ps for different temperatures \nfrom 303 K (below T M) to 393 K (above T M): (b) general view for the range up to 350 fs and (c) \nzoomed view for the range of 50 ps demonstrating the higher frequency mode in detail. The \ndata in (c) are presented after subtracting the lower-frequency mode. The amplitude of the \noscillations in (c) for 308-323 K are scaled (the scale factor is presented on the right side of the \nfigure). \nTheoretical analysis of the spin dynamics in the current ferrimagnetic system is performed \nbased on the Euler-Lagrange equations of motion (Supplementary, Section II). In our calculations \nwe assumed that gyromagnetic ratio of Fe and Gd ions are equal ( 𝛾ி=𝛾ீௗ=𝛾)38. In the case \nof non-collinear phase ( |𝑚| <𝑚 or 𝜖≠ 0 ) the mode frequencies are given by (Supplementary, \nSection IV): \n𝜔୯ି,୯ି=ቆΩଵଶ+Ωଶଶ+𝜔ଶ ±ට(Ωଵଶ+Ωଶଶ)ଶ+2𝜔ଶΩଶቇଵ\nଶ\n, (3) \nwhere Ωଵଶ=ఠಹమ\nଶ−𝜔ு𝜔cos𝜃, Ωଶଶ=ଵ\nଶ(2𝜔ுcos𝜃−𝜔)ଶ, 𝜔ு=𝛾𝐻, 𝜔=𝛾||\nఞ. In Eq. (3) \nthe sign “+” corresponds to the higher frequency (q-AFM) while the sign “-“ corresponds to the \nlower frequency(q-FM) (inset in Fig. 3a). Noteworthy, the term “quasi” is typically used for canted \nAFMs, such as FeBO 3, owing uncompensated magnetic moment39 which is similar to the \nferrimagnetic iron garnets. Frequencies of the modes for collinear state are presented in \nSupplementary, section IV. The frequencies calculated using Eq. (3) (solid curves in Fig.3a,b) agree \nwell with the experimental findings (dots in Fig. 3a,b). \nThe features of the q-AFM and q-FM modes in the non-collinear state differ drastically \nfrom those in the collinear phase far from 𝑇ெ. In the latter case, the q-AFM mode has the \ncharacter of the Kaplan-Kittel exchange mode, whose frequency is unaffected by the external \nmagnetic field and is primarily influenced by the exchange magnetic field 10,25,28. However, in the \nnon-collinear state, the situation dramatically changes: the q-AFM mode frequency becomes \nmagnetic field-dependent (Fig.3a, brown symbols). This provides an important and convenient \ntool for its control that was previously unavailable in the collinear states far from 𝑇ெ. \nThe behavior of the q-FM mode is nearly the opposite. Typically, the frequency of q-FM \nmode depends on the applied magnetic field in the collinear state (e.g. see ref. 28). However, in \nthe non-collinear state this turns upside-down: the q-FM mode becomes hardly sensitive to the \nmagnetic field, as shown in Fig.3a by blue symbols. \nMoreover, our study shows (Fig.3b) that near the compensation point 𝑓୯ି and 𝑓୯ି \nbecome close to each other: 𝑓୯ି decreases and 𝑓୯ି increases for the temperature increase \ntowards TM so that both modes have extremums at this temperature. There is only a small \nfrequency gap at TM: Δ𝜔=𝛾൬ටଶ\n+Hଶ−ටଶ\n൰, which can be controlled by external magnetic \nfield. This situation is also quite unusual since, conventionally, at the temperatures far from 𝑇ெ, \nthe frequencies of the two modes have several orders difference. For example, frequencies of 3 \nGHz and 410 GHz were observed for the q-FM and q-AFM modes for the films without \ncompensation point at similar experimental conditions25. \nWe broadened the view of modes frequencies behavior and calculated frequencies of q-\nFM and q-AFM modes for different values of the external magnetic field and temperature using \nEq. 3 (Fig. 3 c-d). A boundary between the collinear and non-collinear phases are shown in Fig. \n3c,d by dashed white line. A peculiar non-monotonous dependence of 𝑓୯ି on the magnetic \nfield is observed in the non-collinear phase near the phase transition temperature 𝑇: it \ndecreases with the increase of the magnetic field (Fig.3с) which is in a high contrast with what \nhappens in the collinear phase and in the ferromagnetic materials. Notably, the theory predicts \nthat at 𝑇 𝑓୯ି tend to zero (Fig. 3a,c). \n \n \nFigure 3. (a,b) Experimentally found frequencies of the q-FM (blue dots) and q-AFM (brown \ndots) modes compared to theoretical curves calculated from Eq.(3): (a) versus magnetic field \nat T=313 K and (b) versus temperature for 𝑯=𝟒 kOe. The inset in (a) shows complex trajectory \nof 𝑴𝑭𝒆 and 𝑴𝑹 vectors during laser-induced simultaneous excitation of the q-FM and q-AFM \nmodes. The trajectory is shown for the time interval equal to one period of the q-FM mode. Six \nperiods of the q-AFM mode are seen. Color of the trajectory curve denotes temporal \ncoordinate, so that the dynamics starts at the black point and proceeds to the light green color. \n(c-d) Calculated oscillation frequencies of the q-AFM (c) and q-FM modes (d) as a function of \nexternal magnetic field and temperature. White dashed lines at (c) and (d) indicate the \nboundary of phase transition between non-collinear and collinear phases. (e) Amplitude of the \nspin dynamics in terms of the magnetization deflection angle from the ground state versus \ntemperature for the q-FM (blue dots) and q-AFM modes (brown dots) at 𝑯=𝟒 kOe. \nLet’s now consider spin dynamics at around second kind phase transition between the \nnon-colinear and colinear phases, where the frequency curve derivative 𝜕𝑓 𝜕𝑇⁄ breaks (at around \n𝑇ୡ୰ = 383 K, 𝐻= 4 kOe). At the boundary between the non-collinear and collinear phases, an \nintriguing feature appears. The q-FM mode's amplitude increases pronouncedly and reaches a \nmaximum at 𝑇ୡ୰ (Fig. 3e). At this point, its frequency drops down to 7 GHz (Fig. 3a), which \nindicates a soft mode character 24. This phenomenon is explained by a significant increase of the \nmagnetic susceptibility which takes place at the magnetic phase transition of the second kind. \nWe should emphasize that here we demonstrate optical excitation of the soft mode in the \nferrimagnetic dielectric at room temperature which is quite important for advanced ultrafast spin \ncontrol. The excitation efficiency of spin dynamics at the soft mode is enhanced by around 4 \ntimes if compared to the collinear phase near the transition temperature (for instance, at 413 K) \nand by 10 times in comparison to the non-collinear phase (at 343 K). \n3. CONCLUSION \nTo conclude, in this study, we identify an importance of the magnetic phase of a \nferrimagnet for its ultrafast spin behavior. A rare-earth iron garnet near magnetization \ncompensation temperature was considered. We demonstrated several crucial peculiarities of \nspin dynamics in a non-collinear state that contrast sharply with the usually observed spin \ndynamics of the exchange and ferromagnetic modes in a collinear state far from the \ncompensation point. In particular, when temperature approaches the compensation point the \nfrequencies of q-AFM and q-FM modes behave oppositely: the former decreases, while the latter \none grows. The situation changes after crossing the compensation point for higher temperatures. \nWe also discovered that transition from the non-collinear phase to the collinear one is \naccompanied with softening of the q-FM mode which leads to a huge increase of the excitation \nefficiency and amplitude. The amplitude of the soft mode becomes more than 4 times larger than \nfor the collinear state (at 413 K) and up to 10 times higher than for the non-collinear phase (at \n343 K). As the deflection angle of the soft mode was found to reach ~7°, it can be potentially \ninteresting for nonlinear magnonic phenomena such as Bose-Einstein condensation40,41 and \nsuperfluidity as well as for all-optical switching. \nWe also found that, in contrast to the conventional case of the collinear phase where the \nq-AFM mode is field-independent and q-FM mode strongly depends on field. In the noncolinear \nphase the behavior becomes upside-down: q-AFM mode turns to a field-dependent character, \nwhile the q-FM mode gets almost field independent which is in agreement with our theoretical \npredictions based on q-AFM approximation of Euler-Lagrange equations of motion. The \ndescribed methodology allows temperature control of magnetization states of RIG for magnonic \nand spintronic devices. The approach described in current study is universal and can be applied \nfor RIG with various rare-earth ions as well as for other ferrimagnets with uniaxial anisotropy. \n4. MATERIALS AND METHODS \nA. Sample \nThe sample is a 2.2 μm thick RIG film of composition (Bi 0.6Gd2.4)(Fe4.28Ga0.57Ge0.15)O12 \nwhere rare-earth ions is gadolinium (Gd). The film is grown on (111) (CaGd) 3(GaZrMg) 5O12 \nsubstrate and has a uniaxial magnetic anisotropy with the axis perpendicular to the film. The Bi3+ ions have a large ionic radius, therefore, for the synthesis of Bi substituted RIG \nfilms the (CaGd) 3(GaZrMg) 5O12 substrate with the large lattice parameter as = 1.2494 nm was \nchosen. The RIG film was grown by liquid phase epitaxy method from an overcooled solution-\nmelt on a horizontally fixed (111) substrate at the isothermal conditions. The Bi 2O3–B2O3 -PbO \noxides were used as a solvent. Sample growth temperature Tg was 742.5 C. \nA mismatch between the crystal lattice of the substrate and the magnetic film, Δ a= - \n0.007 A was determined by the X-ray diffraction method. The film thickness h=2.2 µm were found \nfrom optical transmittance spectra. A magnetopolarimeter was used to measure the magnitude \nof the Faraday effect wavelength of 515 nm 4.54 deg/µm. The compensation temperature \nTM=336K was determined by the sign change of the Faraday effect. \nB. Static and time-resolved magneto-optical measurements \nFor static magneto-optical (MO) measurements of hysteresis loops the sample was placed \ninto in-plane external magnetic field of electromagnet (AMT&C Troitsk). The beam with ~250 fs \npulse duration was produced from a tunable optical parametric amplifier (Avesta PARUS), which \nwas pumped by a 1 kHz high-energy Yb regenerative amplifier (Avesta TETA). Linearly polarized \n(E vector directed along external magnetic field) normally incident 515 nm light pulses were \nutilized for polarization rotation measurements. The light was modulated by optical chopper \n(Thorlabs) at 500 Hz. Passed through the sample light pulses were detected using balanced \nphotodetector (Newport Nirvana 2007). Electrical signal from photodetector was detected using \nlock-in detector (Zurich instruments MFLI). \nFor time-resolved magneto-optical measurements the sample was placed into constant \nin-plane external magnetic field of electromagnet (AMT&C Troitsk). 787 nm circularly polarized \npulses pumped the sample at ~10° of polar angle. Linearly polarized along external magnetic field \nprobe pulses of 515 nm hit the sample at normal incidence. The pump and probe pulses were \nfocused on the sample to a 100 μm and 40 μm spots correspondingly. The pump and probe \nbeams fluence were 30 mJ/cm2 and 0.3 mJ/cm2 correspondingly. Temporal overlap between \npump and probe pulses was varied by using a 600 mm motorized translation stage (Thorlabs) \nwith a retroreflector in the control beam optical path. The pump pulses were modulated by \noptical chopper (Thorlabs) at 500 Hz. Passed through the sample probe pulses were detected \nusing balanced photodetector (Newport Nirvana 2007). Electrical signal from photodetector was \ndetected using lock-in detector (Zurich instruments MFLI). The sample was heated by Peltier \nelement electrically controlled with current stabilization. Temperature of the sample was \ncontrolled by thermistor. \nC. Theoretical description \nThe RIG ferrimagnetic film was theoretically analyzed using a two-sublattice model. \nQuasi-antiferromagnetic approximation was used to describe the magnetization behavior near \nthe compensation point, which means the two sublattice magnetizations were considered nearly \nantiparallel expect for a small bevel angle. To obtain the static ground state of the ferrimagnet, \nwe calculated minima of the potential energy using Lagrange and Hamiltonian functions. \nDynamics of the system was analyzed using Euler-Lagrange equations of motion. 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Belotelov2,3,4 \n1 Moscow Institute of Physics and Technology (MIPT), 141700 Dolgoprudny, Russia \n2 Russian Quantum Center, 121353 Moscow, Russia \n3 Physics and Technology Institute, Vernadsky Crimean Federal University, 295007 Simferopol, Russia \n4 Photonic and Quantum Technologies School, Lomonosov Moscow State University, 119991 Moscow, Russia \n5 Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991, Moscow, Russia \n \nContents \nSection I. Determination of the magnetic characteristics of the sample \nSection II. The Lagrangian of a two-sublattice system in the quasi-antiferromagnetic approximation \nSection III. The potential energy and equilibrium states of the system \nSection IV. Spin dynamics and mode frequencies \nSection I. Determination of the magnetic characteristics of the sample \nThe comparison of theory with experimental data becomes valuable only when one uses the \ninformation of the magnetic values of the sample, needed for calculations, such as the saturation \nmagnetization Ms, uniaxial anisotropy constant K, etc. For the proposed study, it is necessary to \nknow the value of the saturation magnetization depending on the temperature Ms(T) in a range in \nthe vicinity of the compensation temperature point. Although one can easily find temperature curve \nMs of iron-garnet [1,2], it may vary depending on the composition of the material. For this reason \nwe calculate a temperature magnetization curve using simulation molecular field theory (Fig. S1). \nThe curve corresponds to the composition (Bi 0.6Gd2.4)(Fe4.28Ga0.57Ge0.15)O12, which is used in our \nexperimental studies, is presented in Figure S1. As it can be seen, the compensation temperature \nis 336 K, which is in full agreement with our experimental results. At the same time, in the vicinity \nof the compensation point, the magnetization lies in the range of 0–5 emu/cm3, which is also in \nagreement with the experimental data from the literature for the samples of other \ncomposition [1,2]. The dependence in Fig. S1 allows one to match the theoretical and experimental \ncurves due to connection between the temperature T (which is recorded in the experimental study) \nand the magnetization M (which the theory operates on). \nUniaxial anisotropy constant K and the parameter (see below) were estimated as 13500 \nerg/cm3 and 3.7 · 10-4 respectively on the base of the experimental data presented in the paper. \n \n \n \n \n \n (a) \n (b) \n \nFigure S1. Temperature dependence of the sublattices magnetization (a) and net magnetization \n(b) \n \n \nSection II. The Lagrangian of a two-sublattice system in the quasi-antiferromagnetic \napproximation \nThe ferrimagnetic rear earth iron garnet (like the one in our studies) can be represented \ntheoretically as the two-sublattice system. One of the sublattices is represented by Fe ions, and the \nother one - by rare-earth Gd ions which are coupled by the exchange interaction. The external \nmagnetic field applied to the system can also be taken into account. Here we use the assumption \nthat the applied external field is much smaller than the Hilbert exchange field. \nLagrangian method can be applied to describe the magnetization vectors of sublattices for \nobtaining the precession equations (oscillation equations or) of the antiferromagnetic vector. \nThe two-sublattice system is characterized by MR and MFe magnetization vectors \ncorresponding to the rare-earth and iron sublattices, respectively, and having MR and MFe values \nand θR, φR, and θFe, φFe, angles in the spherical coordinate system (see Fig.1 in the main text). \nUsing the well-known relation for the kinetic energy in micromagnetism [3–5], the Lagrange \nfunction L of the system can be written as \n L sin sinFe Fe R R\nFe R\nFe RM M\nt t , (1) \nwhere γR and γFe are the gyromagnetic ratios, Φ is the potential energy, and t is time. Lagrangian \nEq. (1) is a function of 3 spatial coordinates and time, therefore a general analytical solution cannot \nbe obtained in many cases. For the two-sublattice system, a so-called quasi-antiferromagnetic \napproximation considering nearly anti-parallel alignment of the sublattices can be used to simplify \nthe description [3–5]. \nTherefore, antiferromagnetic vector L= MFe-MR can be introduced and characterized by the \nangle set θ and φ, and: \n \nFe R, ,\n, Fe R \n \n (2) \nwhere the parameters ε≪1 and β≪1 characterize the noncollinearity of the magnetization vectors \nof the sublattices. If ε=β=0, according to Eq. (2) the Fe R and R Fe , and that means \nthe sublattice magnetization vectors are collinear and antiparallel. Therefore, for sublattice \nmagnetization vectors one can obtain the following coordinates in Cartesian system: \n \n \n \n \n \n cos sin cos sin\ncos sin sin cos\nsin cos\ncos sin cos sin\ncos sin cos sin\ncos sinFe Fe\nR RM\nM \n \n \n \n \n \n \n\n \n \nM\nM (3) \nBy substituting the expressions for the sublattice magnetization vectors Eq. (3) into Eq. (1), \nneglecting the terms due to the smallness, and taking into account the property of Lagrangian \nfunction to be invariant under the addition of the full derivative of an arbitrary function of \ncoordinates and time, one can obtain the Lagrangian of the two-sublattice system as a function of \nthe angles θ and φ, parameters ε and β, and their time derivatives: \n sin cosm ML , (4) \nWhere M is the sum of the magnetization moduli of the sublattices, M = MFe+MR, the \nR Fe is the gyromagnetic ratio and the m=MFe-MR is the difference between the \nmagnetization moduli of the sublattices. Lagrangian function determined by Eq. (4) depends on \nthe potential energy Φ, which is determined by the material properties (the type of exchange \ninteraction and magnetic anisotropy, shape and size that determine demagnetization energy, etc.) \nand the configuration of the applied magnetic field. \n \nFigure S2. A ferrimagnetic film with uniaxial magnetic anisotropy in the external magnetic field \nH: the easy axis is denoted by the unit vector n, L is the aniferromagnetic vector. The θ angle is \nmeasured from the projection of the vector L onto the XOY plane. \n \nWe choose the coordinate system in the following way. XY plane coincides with the film \nplane; Z-axis is collinear to the film normal (Fig. S2). Uniform stationary external magnetic field \nH is applied along the X axis ( H={H,0,0}, here and below we assume that H>0). The easy axis of \nthe uniaxial effective magnetic anisotropy n={0,0,1} is directed along the film normal away from \nthe substrate (the Z axis). The demagnetization energy for the thin film is proportional to \n 22 ,Fe RM M n and is much smaller than the Zeeman and anisotropy energies in the vicinity \nof the compensation point, therefore it could be neglected. Thus, the potential energy of such \nsystem can be written as: \n 2 2\n2 2 Fe R\nFe RK KM M Fe R\nFe R Fe RM n M nM M H M M (5) \nIn Eq. (5) KFe>0 and KR>0 are the magnetic anisotropy constants of each sublattice, Λ >0 is the \nintersublattice exchange constant and the “+” sign of the second term represents the \nantiferromagnetic character of the exchange interaction between the sublattices. Using Eq. (3) and \ntaking into account H and n directions, one can rewrite (5) as: \n2 2 2 2cos cos cos sin sin cos ( cos 1) sin2mH H H K M M (6) \nIn Eq. (6) 2Fe RM M , K is the effective magnetic anisotropy constant, K=KFe+KR. \nConsidering ε and β as O(1/Λ) the terms proportional to higher than 1/Λ powers are neglected to \nobtain Eq. (6). Thus, terms proportional to ε2·β2 were neglected for the exchange term in (6), the \nterms proportional to ε·β were neglected for the Zeeman term, and all terms proportional to ε or β \nwere neglected for the anisotropy energy term. \nNext, substituting the relation for the potential energy (6) into the Lagrange function (4) and \nconsidering ( ε,β) as a generalized coordinates set, with respect to them one can calculate the \nLagrange equations: \n 0\n0d\ndt\nd\ndt \n \n \nL L\nL L \nwhich gives the relation between two sets of coordinates, ( ε,β) and (φ,θ): \n \n \n \n \ncos sin cos\ncos sinH\nHM M\nM (7) \nSubstituting Eq. (7) into Eq. (4) with the potential energy Eq. (5), one can exclude the \nnoncollinearity parameters ε and β and reduce the Lagrangian of the two-sublattice system to a \nfunction of only two angles φ and θ of the antiferromagnetic vector L and their time derivatives: 2 2\n2 cos sin cos sin sin2\ncos cos sineffmH H\nmH K \n \n L\n (8) \nwhere 2M . Lagrangian in form Eq. (8), in contrast to Eq. (4), is the function of two \nvariables, which greatly simplifies the system analysis. \nSection III. The potential energy and equilibrium states of the system \nThe equilibrium states of the two-sublattice system are found as the minimum of the effective \npotential energy obtained using the known relationship between the Lagrange and the Hamiltonian \nfunctions for the stationary case =0 and =0: \n 2 2 2 2 2 2sin cos sin cos cos sin2 2effU H H mH K (9) \nEq. (9) shows that antiferromagnetic vector L in the equilibrium state characterized by \n𝜃,𝜑 angles always lie in the XZ plane ( 𝜑=0 for m>0 or 𝜑=𝜋 for m<0). This can be explained \nusing the first term in Eq. (5) that provides minima if the scalar product ( M,H) of the ferromagnetic \nvector M=MFe+MR and external magnetic field H is positive: ( M,H)>0. On the other hand, L and \nM vectors are nearly parallel for m>0 and nearly antiparallel for m<0. \nThe values of 𝜃 corresponding to the equilibrium state depend on the system parameters, as \nshown in Table 1. \n \n \n Table S1. Equilibrium states of a two-sublattice magnetic system in the external field \n \n System parameters Stationary value of angle \nθ Stationary \nvalue of angle \nφ Scheme of the equilibrium state \nI 22 mH H K 00 00 \n \nII 20 2mH H K 02 2 \n0 2cos2mH\nH K \nIII m=0 02 Not defined \n \nIV 20 2mH H K , \nm<0 02 2 , \n0 2cos2mH\nH K 0 \n \nV 22 mH H K , \nm<0 00 \n \n \nCases (I) and (V), the so-called collinear phase, corresponding to 22 m H H K with \nθ0=0 describe antiferromagnetic vector L collinear to H. This is similar to the ferromagnetic state, \nas the ferromagnetic vector M=MFe+MR is parallel to H, while the parallel or antiparallel \nalignments of L and H are determined by the mutual orientation of L and M vectors that depends \non sign of m. \nCases (II) and (IV), the so-called non-collinear phase, appear for rather small values of m \nless than critical one 2cm m H K H . The direction of L in the equilibrium state is \ndetermined as: \n 0 2cos2m H\nH K. (10) \nIt is important, that Eq. (10) has two solutions, so it means that there are two equilibrium \nstates with +𝜃 and −𝜃. Moreover, Eq.(9) shows that these states are degenerate as they have the \nsame potential energy 𝑈ୣ. Therefore, for the same temperature that determines m, and the same \nexternal magnetic field there are two equivalent equilibrium positions described by +𝜃 and −𝜃 \nangles. \nIn the third case (III) describing the magnetization compensation point m=0, the \nantiferromagnetic vector L in the equilibrium state is directed strictly along the easy axis of the \neffective anisotropy n, and perpendicular to the external field H, so θ0 = ±π/2. \nUsing Eq. (10), one can clarify the role of the . By the calculation of the dependence \nMFe+MR = M(H), which can be performed using formulas (3), (7) and (10), one can easy confirm \nthat the equals the magnetic susceptibility of the system at m=0 for the case Heasy axis: \nxM H . \nIn addition, according to Eq. (10) for | m|=mc θ0=0, and this is the second-order phase \ntransition from the collinear to non-collinear state at this point. The magnetic susceptibility tensor \nelement zx z xdM dH equals zero for the collinear phase (| m|>mc) and in the non-collinear one \n(|m| 0 and 𝜃< 0 states have the same energy according to Eq.(9). However, \npractically it is very complicated to apply the magnetic field strictly in-plane with a good precision, \nso that small out-of-plane magnetic field component always exists. The rigorous theoretical \nanalysis of such a configuration is out of the scope of this manuscript. However, it is obvious that \nthe presence of 𝐻௭ component lifts the degeneracy between the 𝜃> 0 and 𝜃< 0 states with +𝑀௭ \nand −𝑀௭ projections. According to Eq. (5), the state with 𝑀௭𝐻௭> 0 has lower energy than with \n𝑀௭𝐻௭< 0. Thus, for the configurations where 𝐻௭ is not zero, the state with 𝑀௭ component parallel \nto this small out-of-plane field is preferable. As L=MR-MFe is nearly parallel to MR, 𝐿௭ and 𝑀௭ \nhave the same signs for m>0 and the opposite signs for m<0. This results in the L flip in a vicinity \nof the compensation point due to a change of the balance between almost oppositely directed MFe \nand MR vectors, i.e. to a change of sing of m while crossing the compensation point (Fig. S5). \n \nFigure S5. 𝜃(𝐻,𝑇) diagram in the presence of the small out-of-plane magnetic field. \n \n \nSection IV. Spin dynamics and mode frequencies \nIn order to calculate the Lagrange equations for the angles θ and φ of the antiferromagnetic \nvector L \n0\n0eff eff\neff effd\ndt\nd\ndt \n \n \n R L L\nR L L \nit is necessary to know the form of the Rayleigh function of the micromagnetic system. For the \ntwo-sublattice case, it can be written as follows [3–5]: \n 2 2 2 2 2 2Rcos cos2Fe\nR R R Fe Fe Fe\nR FeM M \n R (12) \nwhere α is the Hilbert damping parameter. Substituting Eq. (2) into Eq. (12), and neglecting the \nterms containing ε and β in powers higher than 1st, one can calculate the derivatives of R with \nrespect to generalized velocities and . Lagrange equations for angles φ and θ can be obtained \nusing Eq.(8) as: \n \n \n \n 2 2 2 2\n2 2\n2 2cos sin cos sin cos cos\n cos cos2 cos sin cos sin\n HH\nmH mH KM M M\nM (13) \n \n \n \n 2\n2\n2 2\n2 2sin2 cos2 cos sin cos sin cos cos\n sin cos sin sin cos sin sin cos\n mH H\nH H H HM\nM M\n 2 cos sin cos mHM. (14) \nThe further analysis of the behavior of the angles of the antiferromagnetic vector and the \ncalculation of the frequencies of the corresponding oscillations require the linearization of \nequations (13)-(14) near the equilibrium states considered in the previous section. Substituting \nφ=φ0+φl and θ=θ0+θl, where φ0 and θ0 correspond to the equilibrium angular coordinates of the \nvector L, φl<<1 and θl<<1 are small deviations, so that higher than the first orders of these values \nare neglected, one can obtain the following equations. For the non-collinear phase ( θ0 ≠ 0): \n \n \n \n \n \n \n \n \n \n \n 22 2\n0 0 0\n22\n2\n02\n2\n2 2\n0 0 0 02(1 ) 2 0\n22 0cos cos cos\ncos cos cos cos cosl l l l\nl l l lK mH H\nK mHM\nM (15) \nFor the collinear phase ( θ0 = 0): \n \n \n \n \n 2 2\n2 2\n2\n2 222 0\n2 0l l l l\nl l l lm K mH H H\nm mH H HM\nM (16) \nThe first set of signs in each equation corresponds to the case m>0 θ0>1 or m<0 θ0<1, while the \nsecond one to the to the other cases. \nEqs. (15) and (16) have harmonic solutions θl=θAexp(iωt) and φl=φAexp(iωt) with the \nfrequencies 𝜔 determined as the zeros of Eq. (15),(16) determinant: \n 1\n2 22 2 2 2 2 2\n, 1 2 0 1 2 0 2 2q AFM q FM (17) \nfor 0 2m H K H ; and \n 1\n2 2 2 4 2 2\n, 3 4 0 3 41162q AFM q FM (18) \nfor 2 m H K H . In (17) and (18) 02K , KKm , HH , 2 2\n1 0 2 cos 2H H KK \n, 2 2\n2 02 cos 2H KK , 3 2H KK , 2 2\n4 0 2 2K . The choice of the sign \nbefore the root corresponds to the choice of the higher or lower frequency branch, the quasi-\nferromagnetic and quasi-antiferromagnetic modes, respectively. The frequency relations are \ninvariant to sign inversion of m. Notice that in case of significant difference of the gyromagnetic \nratios of the sublattices, e.g. R Fe , the frequencies of quasi-FM and quasi-AFM modes differ \nbelow and above compensation point ( ) ( )j jm m , while the equilibrium states θ0(m)= θ0(-\nm) are the same. Detailed consideration of these cases is out of the scope of the present study. \nEq. (18) remains valid for the magnetization compensation point m=0 and cos θ0=0: \n 2 2 2\n, 0 2 2q AFM q FM H H (19) \nand can be obtained directly from the linearized Lagrange equations for the other coordinate \nsystem choice with Y axis directed along the easy axis of the effective anisotropy n which helps \nto avoid uncertainty of 𝜑 for m=0, and to avoid consequent zeroing of the precession equation, \nunlike Eq. (16). \n[1] S. Geller, J. P. Remeika, R. C. Sherwood, H. J. Williams, and G. P. Espinosa, Magnetic \nStudy of the Heavier Rare-Earth Iron Garnets , Physical Review 137, A1034 (1965). \n[2] M. S. Lataifeh and A. Al-sharif, Magnetization Measurements on Some Rare-Earth Iron \nGarnets, Appl Phys A Mater Sci Process 61, 415 (1995). \n[3] Zvezdin A. K., Dynamics of Domain Walls in Weak Ferromagnets , ZhETF Pisma \nRedaktsiiu 29, 605 (1979). \n[4] M. D. Davydova, K. A. Zvezdin, A. v Kimel, and A. K. Zvezdin, Ultrafast Spin Dynamics \nin Ferrimagnets with Compensation Point , Journal of Physics: Condensed Matter 32, \n01LT01 (2020). \n[5] T. G. H. Blank, K. A. Grishunin, E. A. Mashkovich, M. v. Logunov, A. K. Zvezdin, and A. \nv. Kimel, THz-Scale Field-Induced Spin Dynamics in Ferrimagnetic Iron Garnets , Phys \nRev Lett 127, 37203 (2021). \n " }, { "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": "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. 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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": "2312.16553v1.Sublattice_selective_inverse_Faraday_effect_in_ferrimagnetic_rare_earth_iron_garnet.pdf", "content": "Sublattice-selective inverse Faraday effect in ferrimagnetic rare-earth iron garnet\nToshiki Hiraoka, Ryo Kainuma, Keita Matsumoto, and Kihiro T. 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": "2009.12073v2.Temperature_dependence_of_the_damping_parameter_in_the_ferrimagnet_Gd__3_Fe__5_O___12__.pdf", "content": " Temperature dependence of the damping parameter in the ferrimagnet \nGd 3Fe5O12 \nIsaac Ng,1,2 a) Ruizi Liu1,3 a), Zheyu Ren1,3, Se Kwon Kim,4 and Qiming Shao 1,2,3 b) \n1Department of Electronic and Computer Engineering Department, Hong Kong University of \nScience and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China \n2Department of Physics, Hong Kong University of Science and Technology , Clear Water Bay, \nKowloon, Hong Kong SAR, China \n3Guangdong -Hong Kong -Macao Joint Laboratory for Intelligent Micro -Nano Optoelectronic \nTechnology, The Hong Kong University of Science and Technology, Hong Kong, China \n4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, \nRepublic of Korea \na) Contributed equally b) Email: eeqshao@ust.hk \n \nAbstract \nThe damping parameter 𝛼FM in ferrimagnets defined according to the conventional practice for \nferromagnets is known to be strongly temperature dependent and diverge at the angular \nmomentum compensation temperature, where the net angular momentum vanishes. However, \nrecent theoretical and experimental developments on ferrimagnetic metals suggest that the \ndamping parameter can be defined in such a way, which we denote by 𝛼FiM, that it is free of \nthe diverging anomaly at the angular momentum compensation point and is little dependent o n \ntemperature. To further understand the temperature dependence of the damping parameter in \nferrimagnets, we analyze several data sets from literature for a ferrimagnetic insulator, \ngadolinium iron garnet , by using the two different definitions of the damping parameter. Using \ntwo methods to e stimate the individual sublattice magnetization s, which yield results consistent \nwith each other, we found that in all the used data sets, the damping parameter 𝛼FiM does not \nincrease at the angular compensation temperature and shows no anomaly whereas th e \nconventionally defined 𝛼FM is strongly dependent on the temperature. \n \n \n \n Antiferromagnets have been one important focus in spintronics due to their properties distinct from \nmore conventional ferromagnets including the zero stray field, ultrafast dynamics, and immunity \nto external field1,2. Recently, antiferromagnetically coupled ferrimagnets have emerged as a new \nmaterial platform to study antiferromagnetic dynamics as suggested by the recent discoveries of \ncurrent -driven magnetization switching near magnetization compensation point3,4, where the net \nmagnetization vanishes, and fas t domain -wall dynamics at the angular momentum compensation \ntemperature4,5,6,7, where the net angular momentum vanishes. However, magnetic resonance and \ndynamics of ferrimagnets have not been fully understood par tly due to the involvement of multiple \nmagnetic sublattices and the resultant internal complexity. The dissipation rate of angular \nmomentum in magnetic material is manifested as the linewidth in resonance spectrum . One \nquantity of particular importance in the dissipative dynamics of ferrimagnets is the damping \nparameter, which is a characteristic of the magnetic material that determines the Gilber t-like \ndamping of angular momentum and is usually denoted by the dimensionless number 𝛼. Early \nliterature sugge sted that the effective damping parameter αFM for ferrimagnets defined by a value \nthat is proportional to the line width of the resonance response is strongly temperature -dependent \nand increases anomalously near the angular momentum compensation temperature ( TA).8 Recent \nstudies have provided a new interpretation: the damping parameter can be defined in such a way \nthat it is independent of temperature near the TA while the temperature depend ence of the \nferromagnetic resonance (FMR) is attributed to the temperature dependence of the net angular \nmomentum .9,10,11 A. Kamra et al. have theoretically demonstrated this new perspective by \naccounting the Rayleigh dissipation function in a two -sublattice magnetic system, and the resu ltant \nGilbert damping parameter is independent of temperature near the TA.11 This damping parameter \ndenoted by αFiM is defined as follows: \n αFiM=|snet\nstotal|αFM, (1) \nwhere the snet and stotal are the net and total angular momentum, respectively . The snet is calculated \nby the differenc e of the angular momentum between two sublattice s (snet=|𝑠1−𝑠2 |) and stotal is \ncalculated by the total magnitude of the angular momentum (stotal=|𝑠1|+|𝑠2|). D.-H. Kim et al. \nhave experimentally studied the current -driven domain wall motion in ferrimagnetic metal alloy \nGdFeCo and revealed that the damping parameter αFiM is indeed independent of temperature near \nthe TA.10 Furthermore, T. Okuno et al. has reported that αFiM of the GdFeCo is temperature independent when the FMR measurement temperature is approaching the TA.6 The FMR of \nferrimagnetic thin films below the TA is difficult to achieve because of much enhanced \nperpendicular magnetic anisotropy at lower temperatures . It would be desirable that a full \ntemperature range o f FMR can be investigated for ferrimagnets. \nThe divergence of the conventionally defined damping parameter αFM at TA can be understood \neasily by considering the energy dissipation rate given by 𝑃=αFM𝑠net 𝒎̇2 (which is twice the \nRayleigh dissipation function) , where 𝒎 is the unit magnetization vector. For the given power 𝑃 \nthat is pumped into the ferrimagnet by e.g., applying microwave for FMR , as the temperature \napproaches TA, the net spin density 𝑠net decreases and thus αFM increases. Exactly at TA, the net \nspin density vanishes, making αFM diverge and thus ill -defined. Note that the divergence of αFM at \nTA is due to the appearance of the net spin density 𝑠net in the dissipation rate and should not be \ninterpreted to indicate the divergence of the dissipation rate, which is always finite. In terms of the \nalternative damping parameter αFiM, the energy dissipation rate is given by 𝑃=αFiM𝑠tot 𝒎̇2. The \ntotal spin density 𝑠tot is always finite and has weak temperature dependence, and thus αFiM is well -\ndefined at all temperatures with possibly we ak temperature dependence. This suggests that αFiM, \nwhich is well -defined at all temperatures, might be more useful to describe the damping of \nferrimagnetic dynamics, particular ly in the vicinity of TA, than the more conventional αFM which \ndiverges and thus ill -defined at TA. One way to appreciate the physical meaning of αFiM is to \nconsider a special model, where the energy dissipation of a ferrimagnet occurs independently \nthrough the dynamics of each sublattice and all the sublattice s have the same damping parameter . \nIn this case, αFiM is nothing but the damping parameter of the sublattices. So far, the discussion of \nferrimagnetic damping is limited to ferrimagnetic metals, while ferrimagnetic insulators have \nshown the potential for u ltralow -power spintronics .12,13,14,15,16 \nIn this paper, we investigate the temperature dependence of damp ing parameters in ferrimagnetic \ninsulator, gadolinium iron garnet ( Gd3Fe5O12, GdIG) , by surveying the literature of studies on the \ntemperature dependence of FMR. Since the stotal is usually not given in the literature, we adopt \ntwo different methods to cal culate the individual sublattice magnetization ( MFe and MGd) and then \nevaluate stotal. The first method is to use the magnetization of yttrium iron garnet ( Y3Fe5O12, YIG) \nas the MFe as done in Ref.17, where nuclear magnetic resonance experiments show that the \nmagnetization contribution from iron is similar in YIG and GdIG since yttrium does not cont ribute \nthe magnetization in YIG, and then obtai n MGd from the net magnetization and MFe. The second method uses Brillouin -like function to simulate the temperature dependence of GdIG \nmagnetization , the angular momentum of each individual sublattice can be calculated with the \nBrillouin function . We found consistent results between these two different methods that the \ndamping parameter αFiM is almost temperature -independent near the TA, unlike the conventionally \ndefined αFM which is strongly temperature -dependent and diverge at TA. \nThe FMR linewidth ( ΔH) of GdIG is utilized to find the conventional damping parameter αFM: \n ΔH=αFM\ngeffμB/ℏfres+ΔH0 , (2) \nwhere geff is the effective Landé g -factor, μB is the Bohr magneton, ℏ is the reduced Planck \nconstant , ΔH0 is the frequency -independent inhomogeneous broadening linewidth, and fres is the \nresonance frequency. Then, to convert the αFM to the αFiM, we need to find the ratio snet\nstotal. Note that \nαFM diverges as the temperature approaches TA, meaning that Eq. (2) can be used only when it is \nsufficiently far away from the TA. Therefore, we will only employ data sufficiently far away from \nTA in this perspective. The net spin density snet is calculated from the difference between the \nangular momentum of Fe and Gd: \n sFe=MFe\ngFeμB/ℏ , \nsGd=MGd\ngGdμB/ℏ , \nsnet=|sFe-sGd|=Mnet\ngeffμB/ℏ , (3) \nwhere the Mnet is the net magnetization, gFe and gGd is the Landé g-factor of the iron and \ngadolinium sublattice, respectively. The net magnetization is given by \n Mnet=|MFe-MGd| , (4) \nwhich is normally measured by a superconducting quantum interference device or a vibrating -\nsample magnetometer and provided in the literature.18,19,20 \n \nMETHOD 1 \nWe can use the magnetization of YIG as an approx imation for the MFe to calculat e the MFe and \nMGd from GdIG net magnetization, as yttrium does not contribute to the magnetization of YIG , \nwhich we refer to as Method 1. Experimentally, Boyd et al.17 used the nuclear ferromagnetic resonance technique to determine temperature -dependent MFe in YIG and GdIG and found that \nthey are very similar. Thi s approximation has been used in previous literature and has produced \nreasonable results.21 The magnetization of YIG is obtained from Ref.18. With MFe and MGd \nknown, we can determine the angular momentum of each sublattice with it s respective g -factor . \nThe g factors of Fe and Gd are very similar , the g -factor of iron in measured from YIG and is \ndetermined as 𝑔𝐹𝑒=2.0047 .22 The g -factor of Gd sublattice is 𝑔𝐺𝑑=1.994 and is determined by \nmeasurement of GdIG 23. The TM and TA will be very close to each other, with TA slightly higher \nthan TM. We can calculate the total spin density stotal using \nstotal=sFe+sGd . (5) \nThe net spin density snet can be calculated using Eq. (3) and we can obtain effective g -factor \nmeanwhile. Finally, we can calculate the αFM using Eq. (2) and the αFiM using Eq. (1). \n \nMETHOD 2 \nThe second method is to use the Brillouin -like function to simulate the temperature dependence of \nmagnetization.24 Due to the weak coupling of the Gd -Gd interaction, the gadolinium magnetic \nmoments follow a paramagnetic behavior and increase drastically at low temperatures. The net \nmagnetization in GdIG can be describe d by the sum of the three sublattices with a and d sublattice s \ncorrespond ing to Fe an d c sublattice correspond ing to Gd: \nMnet=|Ma+Mc−Md|. (6) \nThe individual magnetization component can be simulate d by the Brillouin function 𝐵𝑆𝑖(𝑥𝑖) \nMi(T)=Mi(0)BSi(xi) . (7) \nThe 𝑀𝑖(0) is the individual m agnetization at 0 K. \nMd(0)=3nmFe=3ngdSdμB , \nMa(0)=2nmFe=2ngaSaμB , \nMc(0)=3nmGd=3ngcScμB , (8) \n𝑛 is the number of GdIG formula unit per unit volume, it can be calculated using 𝑁𝐴/(𝜌𝑀𝑟), where \n𝑁𝐴 is the Avogadro’s number, 𝜌 and 𝑀𝑟 are the density ( 6.45 𝑔𝑐𝑚−3 25) and molar mass (942.97) of GdIG respectively . 𝑆𝑖 is the electron spin of the respective sublattice . For GdIG, 𝑆𝑑 and 𝑆𝑎 are \n5/2 and 𝑆𝑐 is 7/2. 𝑔𝑖 is the individual g factor and 𝑥𝑖 is defined as: \nxd=(μ0SdgdμB\nkBT)(nddMd+ndaMa+ndcMc) , \nxa=(μ0SagaμB\nkBT)(nadMd+naaMa+nacMc) , \nxc=(μ0ScgcμB\nkBT)(ncdMd+ncaMa+nccMc) , (9) \n𝑛𝑖𝑗 are the Weiss coefficients between two sublattice s, which account for the intersublattice \nmolecular field coupling ( 𝑖≠𝑗) or intrasublattice molecular field interactions ( 𝑖=𝑗).24 𝜇0 is \npermeability of vacuum . \n \nTo determine the snet and stotal from the magnetization fitting will require the sublattice g -factor \ngGd and gFe. gFe in a and d sublattice can be experimentally measured from YIG and is \ndetermined as 𝑔𝐹𝑒,𝑑=2.0047 ,𝑔𝐹𝑒,𝑎=2.003.22 The g -factor of Gd c sublattice has the same value \nas the one in Method 1, 𝑔𝐺𝑑=1.994.23 With the value of the individual sublattice g -factor, the \nangular momentum of each sublattice can be calculate d from Eq. (3). Then we can calculate the \neffective gyromagnetic ratio and effective g -factor with the sublattice magnetization and angular \nmomentum . \nγeff=MFe,d−MFe,a−MGd,c \nsFe,d−sFe,a−sGd,c , \n(11) \ngeff=γeffℏ\nμB , \nThe ratio snet/stotal can be calculated where snet=|sFe,d−sFe,a−sGd,c| and stotal =sFe,d+\nsFe,a+sGd,c, with both the 𝑔eff and angular momentum known . Eventually, the value of αFiM is \nobtained from Eq. (1). \n \nFigure 1. The a nalysis of GdIG data from Rodrigue et al.23 and Dionne et al.18 (a) Calculated \nindividual magnetization as a function of temperature using Method 1. (b) The Magnetization \ncurve of GdIG using Brillouin fitting method (Method 2) compare d to the magnetization from \nDionne et al.18 (c) The geff of GdIG calculated from Method 1 as the green cross and from Method \n2 as the red line compared to the grey dot geff from Rodrigue et al. ([100] direction) .23 (d) (e) (f) \nComparing the damping parameter αFM (red dot) to αFiM based on Method 1 (green cross ) and αFiM \nbased on Method 2 (grey dot) for three directions ([100], [110] and [111]) . \n \nFigure 2. The a nalysis of GdIG data from Flaig et al.26 (a) Calculated individual magnetization as \na function of temperature using Method 1. (b) The Magnetization curve of GdIG using Brillouin \nfitting method (Method 2) compare d to the magnet ization from Flaig et al.26 (c) The geff of GdIG \ncalculated from Method 1 as the red cross and from Method 2 as the green line compared to the \nblue dot geff from Flaig et al.26 (d) Comparing the damping parameter αFM (red dot) to αFiM based \non Method 1 (green cross ) and αFiM based on Method 2 (blue dot). \n \nFigure 3. The a nalysis of GdIG data from Calhoun et al19,20. (a) Calculated individual \nmagnetization as a function of temperature using Method 1. (b) The Magnetization curv e of GdIG \nusing Brillouin fitting method (Method 2) compare d to the magnetization from Calhoun et al.20 (c) \nThe 𝑔eff of GdIG calculated from Method 1 as the green cross and from Method 2 as the grey line \ncompared to the yellow dot 𝑔eff from Calhoun et al.19 (d) Comparing the damping parameter αFM \n(red dot) to αFiM based on Method 1 ( green cross and αFiM based on Method 2 ( blue dot). \n \nRESULTS AND DISCUSSION S \nWe analyze three datasets and evaluate the validity of Method 1 and M ethod 2 using the formula \nprovided above. The first dataset is from Rodrigue et al.23, where the ΔH and geff in three \ndirections [100], [110], [111] are provided. Note that the value of geff is calculated using the Kittel \nequation in Rodrigue ’s paper . The Mnet is obtained from Dionne et al.18 where the GdIG has a \nsimilar compensation temperature to Rodrigue et al.23. fres=9.165GHz and we assume that ΔH0 is \nzero since the GdIG is a polished sphere . We analyze t he data using Method 1 and Method 2 and \nplot the results in Fig. 1. For Method 1, w e can observe that the calculat ed temperature dependence \nof the MGd (see Fig. 1a) and the obtained g -factor (see Fig. 1c) are reasonable . Using Method 2, \nwe get the fitting curves for magnetization from each sublattice and g -factor, which fit accurately \nto the experimental data. \n \nThe second dataset of the temperature dependence of FMR below the TA is from Maier -Flaig et \nal.,26 where the g -factor, ΔH, and Mnet are also provided. Again, we can see that the magnetization \nas a function of temperature from two methods are in accordanc e with Flaig’s data (see Fig. 2). \ngeff calculated dots from Method 1 and fitting curves from Method 2 are highly consistent with \nthe data, which illustrates that both two methods are well established. \n \nThe third set of data is from B. A. Calhoun et al. ,19,20 where fres= 9.479GHz. Similar results to the \nabove two datasets are obtained as shown in Fig. 3. \n \nTo directly compare the above two methods, the ferrimagnetic damping parameter αFiM calculated \nfrom the se two methods are plotted against each other in Fig. 1, 2 and 3, using the data from \nRodrigue et al.23, Flaig et al.26 and Calhoun et al.19. For all datasets , two different methods all give \nconsistent results and have similar values: the newly defined damping parameter αFiM of a \nferrimagnetic material is not divergent near the TA and has much lower value than αFM. The αFiM \nin all three datasets is at low value, revealing the achievability of fast domain -wall dynamics in \nferrimagnetic insulator at the angular momentum compensation temperature . \n \nCONCLUSION \nIn this work, we survey the literature dataset of FMR studies on the fe rrimagnet ic insulator GdIG \nand find that the ferrimagnetic damping parameter αFiM does not increase when the temperature \napproaches the TA, differing from the conventionally defined αFM that shows divergence near the \nTA. This validates the recently developed theory about damping in the ferrimagnetic systems and \nreveals that the damping parameter, when it is appropriately defined with no divergence at all \ntemperatures, is not as high as previously thought. Our work suggests that analyzing the dynamics \nof ferrimagnets needs extra caution, that is not required for ferromagnets, in particular in the vicinity of the TA to avoid unphysical divergences . Besides, potentially lower damping in \ninsulators suggests that ferrimagnetic insulators are promising for future ultrafast and ultralow -\npower spintronic applications. \n \nACKNOWLEDGEMENT \nThe authors at HKUST were supported by the Hong Kong Research Grants Council -Early Career \nScheme (Grant No. 26200520 ) and the Research Fund of Guangdong -Hong Kong -Macao Joint \nLaboratory for Intelligent Micro -Nano Optoelectronic Technology (Grant No. \n2020B1212030010) . S.K.K. was supported by Brain Pool Plus Program through the National \nResearch Foundation of Korea funded by the Ministry of Science and ICT (Grant No. NRF -\n2020H1D3A2A03099291) and by the National Research Foundation of Korea funded by the \nKorea Government via the SRC Center for Quantum Coherence in Condensed Matter (Grant No. \nNRF -2016R1A5A1008184). \n \nREFERENCES \n1. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. \nNature Nanotechnology vol. 11 231 –241 (2016). \n2. 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C., Williams, H. J. & Espinosa, G. P. Magnetic \nStudy of the Heavie r Rare -Earth Iron Garnets. Phys. Rev. 137, A1034 –A1038 (1965). \n22. Geschwind, S. Sign of the ground -state cubic crystal field splitting parameter in Fe3+. \nPhys. Rev. Lett. 3, 207 –209 (1959). 23. Rodrigue, G. P., Meyer, H. & Jones, R. V. Resonance measureme nts in magnetic garnets. \nJ. Appl. Phys. (1960) doi:10.1063/1.1984756. \n24. Coey, J. M. D. Magnetism and magnetic materials . Magnetism and Magnetic Materials \nvol. 9780521816144 (Cambridge University Press, 2010). \n25. Espinosa, G. P. Crystal chemical study of the rare -earth iron garnets. J. Chem. Phys. 37, \n2344 –2347 (1962). \n26. Maier -Flaig, H. et al. Perpendicular magnetic anisotropy in insulating ferrimagnetic \ngadolinium iron garnet thin films. (2017). \n27. Caretta, L. et al. Relativistic kinematics of a magne tic soliton. Science (80 -. ). 370, 1438 –\n1442 (2020). \n28. Zhou, H. -A. et al. Compensated magnetic insulators for extremely fast spin -orbitronics. 1 –\n17 (2019). \n " }, { "title": "1707.01712v1.Switching_from_pyroelectric_to_ferroelectric_order_in_Ni_doped_CaBaCo4O7.pdf", "content": "1 \n Switching from pyroelectric to f erroelectric order in Ni doped CaBaCo 4O7 \nC.Dhanasekhar1, 5, A. K Das1, Ripandeep Singh2, A. Das2, G. Giovannetti3, D. Khomskii4, A.Venimadhav5* \n1,5Department of physics, Indian Institute of Technology, Kharagpur -721302, India \n2Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India \n3Consiglio Nazionale delle Ricerche-Istituto Nazional e per la Fisica della Materia (CNR-INFM), CASTI \nRegional Laboratory, 67100 L’Aquila, Italy \n4Physikalisches Institut, Universitätzu Köln, ZülpicherStrasse 77, 50937 Köln, Germany \n5Cryogenic Engineering Centre, Indian Institute of Technology, Kharagpur -721302, India \n \n \nAbstract \nWe report ferroelectric ordering in Ni substituted CaBaCo 4O7. Magnetization showed ferrimagnetic \ntransition at 60 K and an additional transition is found ~ 82 K , further, enhanced antiferromagnetic \ninteractions and decrease in sat uration magnetization are notic ed with Ni substitution. The dielectric \nand pyroelectric measurements illustrate a strong coupling betw een spin and charge degrees of \nfreedom; ferroelectric behavior is confirmed with enhanced orde ring temperature (~82 K) and \nsaturation polarization (250 μC/m2). Neutron diffraction has revealed an increase in c-lattice \nparameter in Ni sample and all the Co/Ni moments are reoriented in a- direction; evidently a non-\ncollinear ferrimagnetic to collinear ferrimagnetic spin order i s observed. The coupling between the \ntriangular and Kagome layers weakens and leads to ↑↑↓↓ AFM orde ring in the Kagoma layer. This \ncan be viewed as a 2D-collinear layer with unequal bond distanc es and most likely responsible for the \nswitching of electric polarization. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 \n \n \nMultiferroics having simultaneous presence of magnetic and elec tric dipolar ordering having the \nability to couple electric and magnetic polarization are potent ial materials for important novel \napplications. Though simultaneous presence of ferromagnetism an d ferroelectricity is rather \nuncommon, since the pioneering works on BiFeO 3 and TbMnO 3 a number of magnetically induced \nferroelectric materials were discovered [1-4]. In such systems, the symmetry breaking necessary for \nthe dipolar ordering is facilitated by non-collinear spin order ing through inverse Dzyaloshinskii-\nMoriya (DM) or spin orbit coupling [5-7]. Alternately, colline ar spin structure has shown promising \nmagnetoelectric coupling through symmetric exchange striction m echanisms in RMnO 3, RMn 2O5 and \nIsing spin chain Ca 3CoMnO 6 [8-10]. In recent years the Swedenborgite (SB) minerals having general \nformula of ABaCo 4O7 (A= Rare earth, Ca) have gained large attention due to their po lar crystal \nstructure [11-14]. More interestingly, they contain geometrical ly frustrated alternating triangular (T) \nand kagome (K) layers with Co tetrahedra, with Co ions in mixed valence states of 2+ and 3+. The \nratios of these valences can be changed from 3: 1 to 1:1 by rep lacing the rare earth by calcium. In \nCaBaCo 4O7 (CBCO), an orthorhombic structure (with Pbn2 1) is realized due to strong buckling of the \nCoO 4 tetrahedra in the kagome layers which partially releases the f rustration. In contrast to \nRBaCo4O7, in CBCO a charge ordering of Co2+ and Co3+ is realized [14, 15]. A ferrimagnetic \nordering with net moment along b-direction is suggested from th e neutron diffraction studies [15]. \n In CBCO, an exceptionally large electric polarization (17000 µ C/m2) is found along the polar axis, \nbelow the ferrimagnetic ordering. However, due to non-switchabl e electric polarization this material \nis classified not as ferroelectric (FE), but as pyroelectric. J ohnson et al. have theoretically confirmed \nthe pyroelectric nature of CBCO in both magnetically ordered an d disordered phases [12]. On the \nother hand Fishman et al. have presented a slightly differe nt magnetic structure for the CBCO by \ncombining the neutron diffraction, magnetization with terahertz measurements, and the giant electric \npolarization is mainly attributed to the set of bond in the bit etrahedral Co chains along c-axis [14]. 3 \n The distinction of pyroelectri cty and ferroelectricity is signi ficant for many applications, because the \nenergy barrier between the two oppositely polarized states is i nfinite for pyroelectric materials, but \nfor most applications the possibility of switching of electric polarization is crucial. In order to take \nadvantage of multiferroicity, ferroelectric ground state is des irable. Below we report the \ntransformation of pyroelectric to ferroelectric state with enha nced polarization and enhanced ordering \ntemperature in Ni-substituted CBCO. A ferrimagnetic ordering wi th resulting moment in a- direction \nis found instead of b-direction as in the case of CBCO. Here, t he kagome layers have \nantiferromagnetic ordering with ↑↑↓↓ spin structure as revealed by neutron diffraction study. The \nunequal bond distances of ↑↑ and ↑↓ neighbours suggest magnetoe lectric coupling through \nmagnetostriction [26]. \nThe CaBaCo 4-xNixO7 (x=0 (CBCO) and 0.10 (CBCNO)) samples were prepared by solid s tate \nreaction method [16, 22,] and the details of synthesis and char acterization is given in the \nsupplementary material (SM) [23]. Temperature dependence of ac magnetization for CBCO and \nCBCNO samples is shown in Fig.1 (a and b). Both CBCO and CBCNO sample show the \nferrimagnetic ordering at 60 K, while CBCNO shows an additional transition at 82 K. In contrast to \nthe CBCO sample, the ac magnetization of the CBCNO increases be low 40 K. Temperature \ndependence of dc magnetization for CBCNO samples also shows two magnetic transitions at 60 K \nand 82 K as shown in Fig.1 (b). Further, dc magnetization incre ase below 40 K, consistent with the ac \nmagnetization behavior. Though there is no clear understanding of the 82 K transition, its presence \nwas reported in CBCO with Sr2+ doping at Ca site and Zn2+, Fe3+ doping at Co site [17-19]. \nM-H measured at 5 and 45 K for the CBCO and CBCNO samples is sh own in the Fig.1 (c-f). \nCBCNO shows the decrease of coercive field and saturation magne tic moment at 5 K. More \nimportantly, butterfly shape M-H loops is noticed in CBCNO, In fact this has been notices below 60 \nK and it hints the increased AFM interaction in CBNCO. The resu lts suggest that the magnetic \nbehaviour of CBNCO is more complex than that of CBCO. 4 \n The temperature dependent dielectric permittivity of the CBCO a nd CBCNO samples is shown in the \nFig. 2 (a) and (c). CBCO shows a peak at 60 K that matches with the magnetic ordering temperature. \nThe CBCNO samples show two peaks at 60 K and 82 K, which matche s with the magnetic \ntransitions. Interestingly, the dielectric permittivity is high er in CBCNO. \nInterestingly, varying natures of electric polarization is repo rted in CBCO, from the reversal of P in \nbulk sample (ferroelectric ) to a non-switchable P (pyroelctric) in single crystals along polar axis (c-\naxis) and along b axis [11,20-21]. Hence, it is necessary to re veal the dipolar ground state of our bulk \nCBCO. Pyrocurrent ( Ip) measurement can distinguish between pyroelectric and ferroele ctric samples \ndepending on the poling field ( Ep). Fig. 2 (b) summarizes the pyroelectric behaviour of CBCO \nsample; only positive pyrocurrent + Ip is observed for both positive and negative poling fields ± Ep \n(=150 kV/m); corresponding + P is shown in Fig. 2(b). Further, to confirm the pyroelectric \nbehaviour, we have measured the Ip in heating and cooling paths in absence of Ep, and the results are \nshown in the inset of Fig. 2(b). The sign of the Ip is opposite for cooling and heating paths, and this \nbehaviour is typical for pyroelectric materials (see the SM Fig .S1 [23]) [22]. \nThe inset of Fig.2 (d) shows the comparison of Ip peaks for the CBCO and CBCNO samples \nmeasured under a poling field of +150 kV/m. The Ip peak of CBCNO, becomes broader compared to \nCBCO and correlates with the magnetic and dielectric transition s. The most important result is \npresented in Fig. 2(d), where the P behaviour of CBCNO sample for Ep of ±150 kV/m is shown. The \nswitching of the P for the ± Ep suggests the ferroelectric nature of the CBCNO sample. The \npolarization magnitude of 100 μC/m2 is higher than that of the CBCO sample; also the absence of the \nshift in temperature of the Ip peak (Fig. 2 (h)) with different heating rates supports the fe rroelectric \nnature of CBCNO. At low temperatures, the P increases up to Ep ~ 3 00 kV/m ( P =250 μC/m2) \nwithout saturation, which suggests the need for higher poling f ield to saturate the electric dipoles. 5 \n In order to verify the intrinsic nature of ferroelectric behavi our we have conducted additional Ip \nmeasurements to obtain spontaneous ferroelectric polarization d ue to permanent dipoles [24, 25]. \nHere the Ep is applied only once above the transition temperature and cool ed down to low \ntemperature (~10 K). Then the sample short circuited and subjec ted to long rest (h sc 30 mins), later \nconducted subsequent cooling and heating cycles to obtain the s pontaneous polarization. Initially the \nsample was cooled under Ep (+300 kV/m) from 85-10 K. At 10 K the Ep was removed and the sample \nwas shorted for 30 minutes. Now, without Ep, the sample was heated (Fig.2 (f); path 1) with a ramp \nrate of 5 K/min up to 50 K (< the maximum Ip peak temperature of 65 K) and the heating was stopped \nfor a halt time (h t) of 5 mins until the Ip vanishes (path 1). After that the sample is cooled from 50 K-\n10 K (path 2); the Ip goes to negative and recovers to a stable current at low tempe rature as shown in \nFig. 2 (f). The negative Ip is expected as dT/dt i s negative. In the last step (Fig.2 (f); path-3), sample is \nagain heated from 10 K to 90 K with the same rate and obtained a peak at ~ 70 K. The experiment \nwas repeated for another h t of 30 minutes. The magnitude of Ip obtained is same for both the halt \ntimes confirms the spontaneous n ature of polarization (inset of F i g . 2 ( f ) ) . F u r t h e r , w e h a v e a l s o \nconfirmed the switching of the Ip by varying the temperature sinusoidally in between 40-42 K as \nshown in the Fig. 2 (g). Note that the temperature cycling was done after waiting at 50 K for 30 \nminutes (after path 1). The switching of Ip and phase shift of 900 between Ip a n d t e m p e r a t u r e i s \nreminiscent to ferroelectric samples. The above experiments exc lude extrinsic character of the \nobserved ferroelectric behavior, due e.g. thermally stimulated free charge carriers and charge \naccumulation at the grain boundaries. Ferroelectric behaviour i s not only found in CBCNO with 10% \nof Ni, but also found in samples with lower Ni (5 %) substituti ons (see the SM Fig.S2 [23]). \nThus, the magnetic and transport measurements suggest two impor tant differences of CBCNO from \nthat of the parent sample: (a) CBCNO is ferrimagnetic with decr eased magnetic moment and with \ndominant antiferromagnetic interactions. (b) CBCNO demonstrates the ferroelectric behavior instead 6 \n of the pyroelectric one in CBCO. Neutron diffraction can shed s ome light on the origin of these \nchanges. \nThe low temperature neutron powder diffraction (NPD) data of th e CBCO and CBCNO samples are \nshown in Fig. 3(a) & (b). The structural refinement on both the samples confirms the orthorhombic \nPbn2 1 symmetry. A minor impurity phase of CoO [~ 3 wt %] is identifi ed in both the samples. The \ndetailed structural parameters and average bong lengths for the CBCO and CBCNO samples at 300 K \nand 6 K are given in SM [23].The lattice parameters obtained at 6 K for CBCO are a= 6.2556(5) Å, b \n=11.0383(9) Å, c= 10.1563(8) Å and for CBCNO they are a= 6.2561 (6) Å, b=11.0246(9) Å \nand c= 10.1673(9) Å. CBCNO shows reduction in volume (see TABLE .II. SM [23]), consistent with \nthe smaller ionic radii of Ni2+ ion. And Ni occupied in charge ordered state of CBCO by Co2+, as \nexpected due to a stable valence of Ni2+. The lattice parameters ‘b’ found to be smaller in CBCNO, \nwhile the ‘c’-parameter has increased. Although the magnetic st ructure of the CBCO and CBCNO is \nrepresented with similar irreducib le representation, the intens ities of magnetic reflections are different \nin both samples (shown in right side of Figs.3 (a) and (b)). Th e temperature variation of the (012) \nmagnetic reflection [shown in inset of Fig.3 (b)] of the CBCNO confirms the ferrimagnetic transition \nat ~ 60 K; whereas in case of CBCO, (101) magnetic reflection g overns the ferrimagnetic ordering at \nthis temperature. \nThe spin structures of CBCO and CBCNO are shown in Fig.3(c) and (d). The magnetic structure for \nthe CBCO is in agreement with Caignaert et al . (shown in Fig.3(c)) where “Co2 Co3” zigzag \nferromagnetic chains running along b direction couple antiferro magnetically with Co4 and Co1 (in T \nlayer) spins [15]. The resultant ferrimagnetic moment is in b-d irection for CBCO. Recently, Fishman \net al. pointed out that the easy-plane and easy-axis anisotropies wit hin each K and T layers plays a \nmajor role to induce the non-collinear ferrimagnetic ordering i n CBCO [14]. In CBNCO, an increase \nin c-parameter compared to the CBCO suggests the weakening of i nteractions between the adjacent T \nand K-layers: this leads to the spin reorientation and the coll inear magnetic structure. The Co (Ni) 2 - 7 \n Co (Ni) 3 and Co (Ni) 2 –Co4 spins in the kagome layer form ↑↑↓ ↓ AFM spin order in the ab plane as \nshown in the Fig 3 (d). And Co1 spins of T-layer are antiferrom agnetically coupled with the ↑↑↓↓ \nspin structure (formed by Co (Ni) 2- Co (Ni) 3–Co4) in the K-la yer) results in an overall collinear \nferrimagnetic ordering with the net moment in a-direction. The overall decrease of saturation moment \nwith Ni supports the neutron spin structure [Table. IIa. SM [23 ]].The 82 K transition in CBCNO \nsamples may not be associated with the long range ordering, sin ce the main magnetic reflection (0 1 \n2) disappears above 65 K. We suppose that the short range corre lations arising from the K-layers can \nbe responsible for this magnetic anomaly. \nWhat could be the reasons for the change from the pyroelectric behavior of the undoped CBCO to \nferroelectric one observed in CBCNO? One natural candidate coul d be the change from noncollinear \nto a collinear ↑↑↓↓ magnetic structure discussed above. It is w ell documented that such ordering can \nnaturally produce (switchable) ferroelectric polarization, as i s observed in E-type manganites [8] or in \nCa3CoMnO 6 [10], and as is theoretically discussed e.g. in [26, 27]. The same mechanism (formation \nof inequivalent, short and long ↑↓ and ↑↑ bonds or respective s hifts of oxygens, could also be \nefficient in this case (see Fig.4). A switchable polarization in each kagome layer would then point in \nb-direction, not along the original pyroelectric c-axis. Howeve r, detailed analysis shows that in the \nsimple form this mechanism may not work here: polarization in t he neighbouring kagome layers \nseem to be opposite and compensate each other. Thus in this mod el CBCNO would be not \nferroelectric but rather antiferroelectric. \nOur ab intio calculations confirm this conclusion. The electric polarizatio n was calculation using the \nBerry phase method [28]. The outcome of our calculations is tha t indeed there appears some extra \npolarization in magnetically-ordered state, but it is directed along c, as in undoped case [12], \nhowever, the polarizations in b-direction, discussed above, app arently cancel in neighbouring kagome \nlayers. Thus this simple picture in its pure form does not expl ain our experimental observations. One \ncan argue that if by some reasons kagome layers would become in equivalent, e.g. if there is a 8 \n tendency of Ni ions to segregate in every second layer, this co mpensation would not work and the \nmaterial would indeed become ferroelectric. At the moment we do not have experimental indications \nthat it might be the case. But in any case, as discussed above, we believe that the FE behavior of \nCBCNO is intrinsic, though its detailed microscopic explanation is still missing. \nIn summary, Ni substitution in CaBaCo 4O7 (CBCO) produces ferroelectricity necessary for spintronic \napplications. The dielectric, magnetic and pyrocurrent measurem ents shows strong correlation \nbetween electric and magnetic degrees of freedom in CBCNO samp les. Ni substitution drastically \nmodifies the spin structure from non-collinear to a collinear ferrimagnetic order. Collinear spin \nstructure-driven multiferroics are not many, and CBCNO seems pa rticularly interesting having 2D \n↑↑↓↓ structure in the kagome layers. The exchange striction in the ↑↑↓↓ collinear structure could in \nprinciple be the driving force for the ferroelectric behavior, although at the moment we do not have \nfull explanation of the observed behavior. The 2D nature of the magnetic structure seems interesting \nfor designing new magnetoeletric materials and can be extended to artificial epitaxial thin films. \nAcknowledgements \nThe authors of IIT Kharagpur acknowledge DST, India for FIST pr oject and IIT Kharagpur funded \nVSM SQUID magnetometer. The work of D.Kh. was supported by the German project SFB 1238 and \nby the Koeln Univertsity via German Excellence initiative. \n \n* venimadhav@hijli.iitkgp.ernet.in\nREFERENCES \n[1] Daniel Khomskii, Physics 2, 20 (2009). \n[2] R. Ramesh & Nicola A. Spaldin, Nat. Mater. 6, 21 (2007). \n[3] Sang-Wook Cheong & Maxim Mostovoy, Nat. 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Matter 20, 434217 (2008). \n[27] Gianluca Giovannetti, Sanjeev Kumar, Daniel Khomskii, Silvia Pi cozzi, and Jeroen van den Brink, \nPhys. Rev. Lett. 103, 156401 (2009). \n[28] R. D. King-Smith and David Vanderbilt, Phys. Rev. B 47, 1651 (1993). Raffaele Resta, Rev. Mod. \nPhys. 66, 899 (1994). \n \n \n \n \n \n 11 \n Figures \n \nFIG.1. (Color online). (a) and (b) Real part of ac magnetization vs te mperature for CBCO \nand CBCNO samples measured under H ac=1Oe ; Inset of (b) shows the field-cooled dc \nmagnetization for CBCNO sample measured under 3000 Oe. (c-f) sh ow the M vs H \nmeasured at various temperatures for CBCO and CBCNO samples. \n \n \nFIG.2. (Color online). (a) Temperature dependent dielectric constant f or CBCO and (c) \nCBNCO samples; (b) P(T) of CBCO; inset of (b) shows Ip(T) of CBCO measured under \n12 \n zero Ep. (d) P(T) of CBCNO; (e) time vs. temperature of the measurement prot ocol; (f) \nIp(T) for CBCNO measured using protocol described in text. Inset of (f) shows Ip(T) for 30 \nmins halting at 50 K. (h) heating rate dependence of Ip(T) measured under Ep ±300 kV/m. (g) \nIp versus temperature cycling after 30 minutes of waiting. \n \n \nFIG.3. (Color online). Observed and calculated diffractions patterns and their difference at 6 \nK for CBCO (a) and CBCNO (b) samples. Red circles are experimen tal data points and black \ncurve is calculated pattern. Green line at the bottom shows the difference between the \nmeasured and calculated patterns.Inset of (b) shows the tempera ture variation of (012) peak \nintensity. (c) and (d) are spin structures of CBCO and CBCNO sa mples at 6 K view along \nthe [001] direction in the range 0 250 K. The peak in fr becomes more broader in a magnetic field of 0H = 0.2 \nT and the behavior of fr is similar to an empty coil. On the other hand, the current I in a \nzero magnetic field showed a gradual decrease with lowering temperature as like an \nempty coil, except a weak anom aly showed up very close to T C in the temperature \n 9interval of 200 K to 250 K which can be clearly s een in the inset. This is in contrast to the \nlarge decrease in I found at T C for La 0.67Ba0.33MnO 3.26 The weak anomaly present in the \nzero field I is completely suppressed in the magnetic fields of 0H = 60 mT and 0.2 T. \nThe fig. 6(a) shows the magnetic field dependence of fr at selected temperatures T \n= 10 K – 200 K. These data were taken by monitoring the f r while sweeping the magnetic \nfield from 0H = 0 T → +3 T → -3 T → +3 T. We have not shown the initial field sweep \n(0 T → +3 T) here, since it closely matches with the down field sweep (+3 T to 0 T). We \nhave also not shown here the field dependence of I, since there has been no considerable \nchange in I with the magnetic field sweep. The fr at T = 10 K shows a butterfly curve \nwithin the field interval of 0H = +1.5 T – -1.5 T and shows a peak on either side of the \norigin at 0H = ±0.75 T (i.e., HP, the field at which peak occurs) with a large hysteresis, \nbut saturates at the highest field 0H = ±3 T. The field dependent fr at temperatures above \nT = 10 K showed the similar behavior as for T = 10 K, except for T = 200 K where the \npeak is completely vanished with absolutely no hysteresis. The significant point here is \nthat the peak position shifts toward s the origin and the hysteresis in fr becomes narrower \nat higher temperatures ( T > 10 K) as like the hysteresis in M-H loops [fig. 2(b)]. This can \nbe clearly concluded from the figures (b) – (e). The magnetic field dependences of M and \nfr at T = 10 K are shown in the figu res (b) and (c), respectively. A large hysteresis is seen \nin both the cases. As descri bed above for fig. 6(a), the fr in the figure (c) showed a peak \nin the negative side at HP = -0.75 T while the field is swept from 0H = +3 T → -3 T. \nThis position of the peak ( HP) is closely matching with the coercive field ( HC), but it is \nslightly less in value. Interestingly, the temperature dependences of both HC and H P show \n 10a sim\nilar behavior i.e., the rapid in crease with lowering temperature below T = 200 K and \nare shown in the figures (d ) and (e), respectively. \n \nThe resonance frequency of an ICO, fr = 1/(2πLC) where L – the inductance of \nthe empty coil and C – the capacitance in the circuit, changes due to the change in the \nreal part of rf magnetic permeability ( ′) of the sample. This is unlike the case of a \nmetallic La 0.67Ba0.33MnO 3 which showed an extra contribution arising from the \nincomplete penetration of ac magnetic field due to the induced eddy current.26 Thus, the \nincrease in ′ upon transition from paramagnetic to ferrimagnetic while cooling leads to a \ndecrease in the fr around T C. An external applied magne tic field suppresses the spin \nfluctuations in the sample and leads to a considerable decrease in ′ which in turn \ndecreases the fr at T C. On the other hand, the dynamical magnetic and electrical losses in \nthe sample lead to rf power absorption in the sample th at changes the complex impedance \n(Z) of the tank circuit, which in turn lead s to a change in the current through the ICO \ncircuit. The expression for the impedance of the inductance coil can be modified for a \nhighly insulating materials (in which electro magnetic field completely penetrates the \nsample) as Z = ( R0+L0)+jL0′, where R 0 is the resistance, L0 is the inductance of \nthe empty coil, and ′ and are the real and imaginary parts of the permeability of the \nsample, respectively. Therefore, the effective resistance of the coil changes mainly due to \nthe magnetic loss characterized by the which reflects the electromagnetic power \nabsorption in the sample [P = ½ Hac2Re(Z)],26 and hence the current in the circuit shows \nan anomaly near the ferrimagnetic transiti on. The suppression of this peak in the \n 11m\nagnetic field of 0H = 60 mT and 0.2 T is due to the suppression of ′ and by the \nmagnetic field. \n \nThere is a close correlation between the temperature dependence of HC and HP \n[fig. 6 (d) and (e)]. The slight difference in the values of HC and HP is possibly because \nthe change in fr (which is related with th e dynamical magnetization i.e. M/H) is \nmeasured at fr ~ 1 MHz in our ICO experiment and the low frequency ( f <100 kHz) \nmeasurement might lead to a closer agreement between the HC and HP values. It is worth \nmentioning that the coercivity of GaFeO 3 is high, H C = 0.9 T at T = 5 K. Since \n where MS – saturation magnetization, it im plies a rapid increase in the \nanisotropic constant ( K1) with lowering temper ature. Indeed, Pinto et al .12/ CHK M S\n34 reported \nunusually large anisotropic magnetizati on in single crystal of orthorhombic \nGa0.92Fe1.08O3. The coercivity is expected to increas e if a Ga is replac ed by a rare earth \nion R. For example, H C = 1.5 T, 1.65 T and 1.8 T for R = Dy, Sm and Y, respectively in \nthe RFeO 3 series.35 \n \nIV. CONCLUSIONS \nWe have studied electrical, magnetic, magnetodielectric and magnetoabsorption \nproperties of GaFeO 3. The dielectric permittivity exhibited a weak anomaly at TC which \nis suppressed in 0H = 60 mT suggesting a ME coupling in GaFeO 3. The sample showed \nwell defined magnetic hysteresis loops belo w 230 K with a rapid increase in the \ncoercivity below 200 K from 0.1 T at 200 K to 0.9 T at 5 K. The resistivity of this \ncompound above 200 K obey the Arrhenius law with an activation energy of E a = 0.67 \n 12eV. The P-E loops suggested that the leakage cu rrent is drastically reduced below T = \n200 K. The m\nagnetoabsorption study showed an anomaly in both fr and I at T C and we \nfound that there is a close correlation between the temperature dependence of HC and HP, \nthe peak found in the fr versus magnetic field. Very recently, Tokunaga et al.36 showed \nthat GdFeO 3, one of the most orthodox perovskite oxides, possesses a ferroelectric \nground state in which ferroelectric polarization is generated by exchange interaction \nbetween Gd3+ and Fe3+ ions. 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Tokura, Nat. \nMater. 8, 558 (2009). \n \n \n 17 \nFigures : \n \n \n Fig. 1 (color online) (a) Observed (blue colo r) and Reitveld refinement (red color) of \nthe XRD pattern for the GaFeO\n3 compound with space group Pc21n at room \ntemperature, (b) temperature dependence of the resistivity ( ) in a narrow \ntemperature interval (265 K – 360 K) and the inset shows the linear behavior of \nArrhenius plot for the resistivity. 18 \n \n \n \n Fig. 2 (color online) (a) Temperatu re dependences of magnetization (M ) at different \nmagnetic fields (0.2 T – 5 T), (b) fi eld dependences of magnetization ( M) at different \ntemperatures (5 K – 300 K), (c) temp erature dependence of coercive field ( H\nC). \n 19 \n \n \nFig. 3 (color online) Temperature dependen ces of (a) the dielectric constant ( ) and \n(b) the dissipation factor ( D = tan) for f = 100 kHz, 500 kHz and 1 MHz. The insets \nshow the temperature dependences of and D (left scale) in 0H = 0 and 60 mT for f \n= 1 MHz. The inset in the figure (a) show s the magnetodielectric (MD) coefficient \n(right scale) as a function of temperature for f = 1 MHz. \n 20 \n \nFig. 4 (color online) (a) P-E loops at selected temper atures (frequency of the \nhysteresis cycle f = 1 kHz) (b) P-E loops at T = 150 K with different frequencies of \nthe hysteresis cycle ( f = 1 kHz – 50 Hz). \n 21 \n \n \nFig. 5 (color online) Temperature dependences of the resonance frequency ( fr) (right \nscale) and current ( I) (left scale) through the circu it for different strengths of dc \nmagnetic fields 0H = 0 T, 60 mT and 0.2 T. The da ta for the empty coil are also \nshown. The inset shows the temperature dependence of I in a narrow temperature \ninterval (190 K – 270 K). 22 \n \nFig. 6 (color online) The magnetic field dependences of the (a) resonance frequency \n(fr) at selected temperatures T = 10 K – 200 K, (b) magnetization ( M) at T = 10 K, (c) \nresonance frequency ( fr) at T = 10 K. Temperature dependences of the (d) coercive \nfield ( HC) and (e) position of the peak ( HP) which is observed in the fr versus field. \n " }, { "title": "2402.18816v1.Nano_Electromagnetic_Super_dephasing_in_Collective_Atom_Atom_Interactions.pdf", "content": "Nano-Electromagnetic Super-dephasing in Collective Atom-Atom Interactions\nWenbo Sun,1Adrian E. Rubio L´ opez,1and Zubin Jacob1,∗\n1Elmore Family School of Electrical and Computer Engineering,\nBirck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA\n(Dated: March 1, 2024)\nPure dephasing and spontaneous emission are two non-unitary processes of atoms or spins inter-\nacting with fluctuating electromagnetic (EM) modes. Collective spontaneous emission (e.g., super-\nradiance) originates from interactions with EM modes in resonance with atoms and has received\nconsiderable attention. Meanwhile, the analogous collective dephasing phenomena remain poorly\nunderstood. Here, we introduce the nano-EM super-dephasing phenomenon arising in the photonic\nenvironment near lossy material interfaces. We show that this effect is enhanced by over 10 or-\nders of magnitude compared to free space or photonic cavities due to the presence of long-range\ncorrelations in low-frequency evanescent EM fluctuations. We unravel the universality of nano-EM\nsuper-dephasing behaviors near ferrimagnets, metals, and superconductors and their dependence on\nlow-frequency material properties. We demonstrate that the scaling of nano-EM super-dephasing\nis independent of EM modes’ wavelengths and differs from the conventional N2scaling of superra-\ndiance by analyzing the decoherence of entangled states, including GHZ states. Finally, we show\nhow to experimentally isolate and control super-dephasing to open interesting frontiers for scalable\nquantum systems.\nIntroduction.– Pure dephasing and spontaneous emis-\nsion are two paradigms of non-unitary processes in inter-\nactions between light and atoms or spins. While spon-\ntaneous emission involves photon radiation and energy\ndecay, pure dephasing refers to the loss of phase coher-\nence without energy dissipation and is the main obsta-\ncle in current quantum information technologies. Recent\ninterest has focused on collective effects in interactions\nbetween light and ensembles of atoms or spins sharing\nthe same fluctuating electromagnetic (EM) modes [1–3].\nMultiple recent works have focused on collective spon-\ntaneous emission, which leads to superradiance effects,\nin ensembles of different sizes and dimensions in free-\nspace, cavities, and photonic crystals [4–12]. However,\nfor the other paradigm of non-unitary evolutions, collec-\ntive effects in pure dephasing due to coupling with the\nEM/photonic environment are much less understood de-\nspite their importance in multiqubit coherence decay [13–\n19] and quantum error corrections [20, 21].\nSuperradiance originates from collective interactions\nbetween fluctuating EM modes and two-level systems\n(TLSs) in resonance. In conventional photonic environ-\nments, such as free-space, cavities, and photonic crystals,\nthese resonant interactions can be made dominant by the\nenhancement/interference of propagating modes. In con-\ntrast, dephasing arises from broadband off-resonant, low-\nfrequency ( ≤MHz) environmental fluctuations. Here, we\nexplore a unique regime for collective light-matter inter-\nactions which can arise in the near-field of material inter-\nfaces, where fluctuations of low-frequency ( ≤MHz) EM\nmodes can be enhanced by over 20 orders of magnitude\ndue to low-frequency evanescent interface modes contri-\nbutions. This giant enhancement brings about a nano-\nEM environment where off-resonant collective interac-\ntions between quantum ensembles and low-frequencyfluctuating EM modes become dominant.\nIn this paper, we introduce the super-dephasing phe-\nnomenon that emerges in this unique regime of nano-\nEM interactions fundamentally different from resonant\neffects in cavities and photonic crystals. We find that\nnano-EM super-dephasing is enhanced by over 10 or-\nders of magnitude compared to free-space. This occurs\nfrom the enhancement of long-range correlations in low-\nfrequency evanescent EM fluctuations. In contrast, de-\nphasing is weak in free space because high-frequency EM\nfluctuations, which are the dominant contributors to the\neffect in vacuum, possess only short-range correlations\n(see section S3 in Supplemental Material [22]). Not sur-\nprisingly, the effect is weak even in macroscopic pho-\ntonic resonators, and only ultra-subwavelength cavities\ncan induce super-dephasing. Distinct from superradi-\nance, which accelerates energy emission and generates\nentanglement [5], nano-EM super-dephasing accelerates\ndisentanglement and decay of quantum coherence with-\nout energy dissipation. The observable outcome of su-\nperradiance is the photon emission burst [4, 8], while\nour phenomenon of nano-EM super-dephasing leads to\ndifferences in the decoherence time of different Bell\nstates. Additionally, since nano-EM super-dephasing is\nrelated to many-body collective interactions with near-\nfield EM modes associated with low-frequency ( ≤MHz)\nmaterial responses, it can exhibit material dependence\ndistinct from previously discussed resonant dipole-dipole\ninteractions (RDDI) [23–25], Casimir-Polder frequency\nshifts [26–30], spontaneous emission [31–35], and single-\nqubit dephasing [36] phenomena near material interfaces,\nwhich are either single-body effects or hinging on mate-\nrial properties at high (resonance) frequencies (see Ta-\nble S1 in Supplemental Material [22] for comparison of\nnano-EM super-dephasing with other phenomena). Un-arXiv:2402.18816v1 [quant-ph] 29 Feb 2024ii\nderstanding nano-EM super-dephasing will help the rapid\nscaling of atomic and spin systems, which are often in\nproximity to material interfaces in various quantum ap-\nplications [37–40].\nWe prove the universal behaviors of nano-EM super-\ndephasing through a comprehensive analysis of spin\nqubit arrays with different dimensions and materials with\ndistinct EM responses, including ferrimagnets, metals,\nand superconductors. We reveal that nano-EM super-\ndephasing hinges on important low-frequency material\nproperties (e.g., superconducting coherence factors) and\ndepends on a new figure of merit independent of EM\nmodes’ wavelengths, in stark contrast to superradiance\nwhich depends on EM wavelengths in free-space, cavities,\nand photonic crystals. We further demonstrate that the\nscaling of nano-EM super-dephasing in entangled states,\nincluding Greenberger–Horne–Zeilinger (GHZ) states, is\nbeyond the conventional N2scaling of superradiance.\nModel.– We consider a multi-spin-qubit system (mag-\nnetic TLS ensemble) interacting with quantized fluctu-\nating EM fields near material interfaces, as shown in\nFig. 1(a). We start with the interaction Hamiltonian\nˆHint=−/summationtext\nimiˆσz\ni·ˆB(ri) between spin qubits and the\nlongitudinal noise fields that induce dephasing effects.\nmi,ri,ˆσz\niare the spin magnetic moment along the quan-\ntization axis, position, and Pauli-z operator of the ith\nspin qubit. ˆB(ri) =/integraltext\ndωˆB(ri, ω) +h.c.is the fluctuat-\ning magnetic field operator at rifollowing macroscopic\nquantum electrodynamics quantization [41]. ˆB(ri, ω) is\nproportional to the magnetic dyadic Green’s function← →Gm(ω) =← →G0\nm(ω) +← →Gr\nm(ω).← →G0\nmrepresents the free-\nspace and the spin qubit substrate contributions.← →Gr\nmis\nthe reflected component determined by the properties of\nmaterial interfaces. Near field of lossy material interfaces\ngreatly enhances Im← →Gr\nm(ω) atω≤MHz by more than\n20 orders of magnitude compared to free-space, lead-\ning to dominant off-resonant couplings between quantum\nensembles and low-frequency ˆB(ri, ω). We employ the\npoint-dipole approximation in ˆHintsince the size of spin\nqubits is much smaller than low-frequency EM fluctua-\ntions wavelengths.\nThrough the time-convolutionless projection operator\ntechnique [42], we derive a time-local master equation\nfrom ˆHintconnecting off-resonant interactions with near-\nfield EM fluctuations to nano-EM super-dephasing dy-\nnamics (derivations in Supplemental Material [22]):\ndρ\ndt=−i[ˆHd, ρ] +/summationdisplay\ni1\n2dΦs(t;ri)\ndt/bracketleftig\nˆσz\niρˆσz\ni−ρ/bracketrightig\n+\n/summationdisplay\ni̸=j1\n2dΦc(t;ri,rj)\ndt/bracketleftig\nˆσz\niρˆσz\nj−1\n2ˆσz\niˆσz\njρ−1\n2ρˆσz\niˆσz\nj/bracketrightig\n,(1)\nwhere ρis the density matrix for the multi-spin-qubit sys-\ntem, and ˆHdis the atom-atom dipolar interaction Hamil-\ntonian governing the unitary evolution. The second termdescribes nano-EM single-qubit dephasing (SQD) with\ndephasing functions Φ s(t;ri) representing the time evo-\nlutions of SQD concerning the spin qubit at ri. The\nthird term depicts pairwise nano-EM collective dephas-\ning with dephasing functions Φ c(t;ri,rj) representing the\ntime evolutions of collective dephasing involving spin\nqubits at riandrj. The non-Markovianity of near-field\nEM bath can be reflected by the time dependence of\nnano-EM SQD rates dΦs(t;ri)/dtand collective dephas-\ning rates dΦc(t;ri,rj)/dt[43]. We focus on the dephasing\nprocesses since they are usually the dominant sources of\ndecoherence and delay discussions of unitary interactions\nto Supplemental Material [22]. Through the construc-\ntive or destructive interference of the SQD and collective\ndephasing, the multi-spin-qubit system can experience\nnano-EM super- or sub-dephasing depending on its ini-\ntial state.\nOur results for Φ c(t;ri,rj) and Φ s(t;ri) in Eq. (1) are:\nΦc(t;ri,rj) =/integraldisplayωc\n0dω F R(ω, t)/braceleftig4µ0ω2\nℏπc2cothℏω\n2kBT\n/bracketleftig\nmi·Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]\n2·mj/bracketrightig/bracerightig\n,(2)\nΦs(t;ri) = Φ c(t;ri,rj)|rj→ri, (3)\nwhere coth ( ℏω/2kBT) incorporates thermal fluctuations\nof the EM bath at temperature T, and← →G⊺\nmis the trans-\npose of← →Gm. The integrands can be decomposed into\nthe time-dependent Ramsey filter function FR(ω, t) =\n(1−cosωt)/ω2and noise (correlation) spectra. Distinct\nfrom collective decay, nano-EM collective dephasing func-\ntions are sensitive to pulse sequences, which can shift the\nfilter function by averaging out noise at certain frequen-\ncies [44]. We define the magnitude of nano-EM dephasing\ntime|tc(s)\nϕ|as|Φc(s)(|tc(s)\nϕ|;ri,rj)|= 1, and define the sign\noftc(s)\nϕas sgn tc(s)\nϕ= sgn Φ c(s)(|tc(s)\nϕ|;ri,rj).\nIn Eq. (2), the tensor related to the EM bath proper-\ntiesM(ri,rj) = Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]/2̸=\nIm← →Gm(ri,rj, ω) near nonreciprocal materials, manifest-\ning important differences from EM mode correlations lim-\nited to the reciprocal conditions [45, 46]. Mis real and\nsymmetric ( M(ri,rj) =M⊺(rj,ri)) under riandrjex-\nchange in both reciprocal and nonreciprocal conditions.\nThis, in conjunction with Φ s(t;ri)>0 guaranteed by the\npositive definiteness of M(ri,ri), is necessary for physi-\ncally valid (i.e., completely positive and trace-preserving)\nnano-EM super-dephasing dynamics near arbitrary ma-\nterial interfaces [22]. Additionally, we introduce a cutoff\nfrequency ωc, similar to Lamb shift calculations [47], for\navoiding the ultraviolet divergence in Eqs. (2, 3) caused\nby the free-space vacuum contributions Im← →G0\nm.ωccan\nbe selected as the frequency for which the point dipole\napproximation becomes invalid. Its choice will not signifi-\ncantly affect nano-EM super-dephasing dominant by low-\nfrequency contributions [22]. Furthermore, the Green’siii\nFIG. 1. Universal nano-EM super-dephasing near lossy material interfaces. (a) A schematic of a multi-spin-qubit system\ninteracting with off-resonant, low-frequency fluctuating EM fields near lossy material interfaces. The bottom box shows a\npair of spin qubits with interqubit distance Dand relative angle θat the same distance zfrom the interfaces. ts\nϕandtc\nϕ\ndenote the nano-EM single-qubit and pairwise collective dephasing time, respectively. (b, c) Temperature and (d, e) distance\ndependence of ts\nϕnear (b, d) gyromagnetic YIG, polycrystalline silver, and (c, e) superconducting niobium thin films of 600 nm\nthickness. (f, g) Dependence of ts\nϕ/tc\nϕonD/2zatz= 15 nm from (f) YIG, silver, and (g) niobium thin films. The shaded\nregion corresponding to ts\nϕ/tc\nϕ<0 represents anti-correlations of nano-EM collective dephasing and SQD. Distinct dephasing\nbehaviors originate from MHz evanescent interface modes associated with different low-frequency material loss.\nfunction formalism is key for implementing numerical\ntechniques previously developed for calculating Im← →Gm\nnear materials with arbitrary geometries [34]. Calcula-\ntions of Im← →Gmare provided in the Supplemental Mate-\nrial [22]. These jointly guarantee the broad applicability\nof our theoretical framework.\nIn the following, to prove the universal nature of\nnano-EM super-dephasing, we apply our theory to three\nclasses of materials important in quantum applications,\nincluding nonreciprocal gyromagnetic yttrium iron gar-\nnet (YIG) with permeability tensor← →µYIG, polycrys-\ntalline silver with permittivity εAg, and superconducting\nniobium in the Pippard regime with complex conductiv-\nityσNb. Details of material models and experimentally\naccessible parameters [48–52] are provided in the Supple-\nmental Material [22].\nMHz evanescent interface modes.– To elucidate rela-\ntions between nano-EM super-dephasing and material\nproperties, we start by analyzing dephasing in a pair\nof spin qubits at a distance zfrom the interface, as\nshown in Fig. 1(a). At distance z≪c/ω, near-field\nIm← →Gr\nm(ω) is dominated by contributions from evanes-\ncent interface modes (EIMs) strongly confined to the in-\nterfaces with momentum q∼1/z≫ω/c[22]. Therefore,\nfrom Eqs. (1-3), nano-EM super-dephasing near differ-\nent lossy interfaces ubiquitously hinges on EIMs asso-\nciated with MHz intrinsic material loss [22], which are\ndistinct from their resonant frequency counterparts. Dif-\nferent MHz loss mechanisms in materials lead to distinct\nnano-EM SQD and collective dephasing behaviors.\nFigure 1(b, d) demonstrate the distance and temper-\nature dependence of SQD time ts\nϕnear YIG and silver\nthin films when mis perpendicular to the interfaces. At\nlower temperatures, we find increasing ts\nϕnear YIG thinfilms due to the reduced low-frequency magnetic damping\nIm← →µYIG. In contrast, ts\nϕnear silver films exhibits little\ntemperature dependence because the silver dielectric loss\nImεAgincreases at lower temperatures and cancels de-\ncreasing thermal fluctuation contributions. Meanwhile,\nwe find ts\nϕ∼z3associated with MHz magnetic EIMs in\nYIG and ts\nϕ∼zrelated to metallic EIMs in silver.\nFigure 1(c, e) demonstrate spin-echo ts\nϕprofiles\nnear superconducting niobium thin films, which are\nobtained by replacing FR(ω, t) with FSE(ω, t) =\n8 sin4(ωt/4)/ω2[22]. Here, we find an abrupt dip in the\ntemperature dependence of ts\nϕright below the transition\ntemperature Tc. This is caused by the coherence peak in\nsuperconductor loss (type-II coherence factor [51]) origi-\nnating from the thermally excited quasiparticle response,\nwhich is only prominent in the low-frequency component\nof Re σNb. At T= 0.8Tc,ts\nϕexhibits more complicated\ndistance dependence near superconducting niobium com-\npared to silver and YIG. At small z, we find ts\nϕ∼zdue\nto superconducting EIMs with large q∼1/zdependent\non the MHz quasiparticle response dominant in Re σNb.\nIn contrast, ts\nϕ∼z4at larger zsince the Cooper pair\ncondensate’s response dominant in Im σNbsignificantly\naffects MHz superconducting EIMs with lower momen-\ntum [22].\nPairwise nano-EM collective dephasing.– Due to EIMs\ncontributions, nano-EM collective dephasing is sensitive\nto low-frequency material properties and relies on the ra-\ntioD/2zof interqubit distance Dand distance from the\ninterfaces z. This is in stark contrast to collective decay\nin free-space and cavities dependent on EM modes’ wave-\nlengths. Figure 1(f, g) demonstrate the ratio of nano-\nEM SQD and collective dephasing time ts\nϕ/tc\nϕnear the\nthree material thin films, which determines the behav-\niors of tc\nϕbuilding on previous discussions of ts\nϕ. Theiv\nFIG. 2. Scaling of nano-EM super- and sub-dephasing in the\nGHZ state |ψ⟩GHZ. (a) Schematics of Nspin qubits arranged\nin 1D and 2D arrays with lattice constant bat distance zfrom\nthe silver and YIG films. (b) The figure of merit r= 2z/bdis-\ntinguishes different regimes with qualitatively different nano-\nEM super-dephasing scaling laws concerning N. Dimension-\nality and material properties significantly affect the scaling of\nnano-EM super- or sub-dephasing in |ψ⟩GHZ in the interme-\ndiate regime.\nsign of ts\nϕ/tc\nϕindicates correlations ( ts\nϕ/tc\nϕ>0) and anti-\ncorrelations ( ts\nϕ/tc\nϕ<0) between collective dephasing and\nSQD. We find that ts\nϕ/tc\nϕdecays significantly slower con-\ncerning D/2znear silver ( ∼(D/2z)−1) compared with\nYIG ( ∼(D/2z)−3) and superconducting niobium, indi-\ncating longer-range nano-EM collective dephasing near\nsilver. We also observe that nano-EM collective dephas-\ning exhibits different correlations with SQD at different\nD/2znear YIG and superconducting niobium, distinct\nfrom collective dephasing near silver. These two aspects\nhave an important influence on the scaling of nano-EM\nsuper-dephasing effects.\nMeanwhile, nano-EM collective dephasing exhibits\nuniversal features near all three materials. We show ts\nϕ/tc\nϕ\nnear all three materials decays algebraically with D, man-\nifesting universal long-range behaviors. In contrast, col-\nlective effects from other sources, including nuclear spins\nand charge fluctuations, can be either negligible [53] or\ndecay exponentially with D[54]. This indicates the im-\nportance of nano-EM collective dephasing in large quan-\ntum ensembles. Furthermore, we find that nano-EM col-\nlective dephasing is ubiquitously sensitive to relative ori-\nentations θof spin qubits and exhibits non-monotonic\nbehaviors concerning Datθ̸= 0. Further details regard-\ning film thickness effects when t≤D, z are discussed in\nSupplemental Material [22].\nNano-EM super-dephasing.– We now demonstrate\nnano-EM super-dephasing phenomena due to interfer-\nence of pairwise collective dephasing and SQD. We con-\nsider multiqubit decoherence of Nspin qubits arranged\nin one-dimensional ( N= 1×n) and two-dimensional\n(N=n×n) arrays with lattice constant bat distance\nFIG. 3. Engineering nano-EM super-dephasing with hyper-\nbolic meta-structures. (a) A schematic of a 2D spin qubit ar-\nray at distance zfrom silver gratings with periodicity p < z .\n(b) Distinct scaling behaviors of ts\nϕ/tc\nϕalong the x and y di-\nrections. The hyperbolic meta-structure reduces the range of\nnano-EM super-dephasing along the x direction as opposed\nto its enhancement of the RDDI range [25].\nz= 60 nm from YIG and silver films of thickness 8 µm,\nas shown in Fig. 2(a). We expect similar nano-EM super-\ndephasing behaviors near superconducting niobium and\nYIG interfaces due to the resemblance of pairwise collec-\ntive dephasing. We consider nano-EM super-dephasing\nfor the GHZ state |ψ⟩GHZ, which is widely used in var-\nious quantum applications [55]. We focus on mof all\nspin qubits perpendicular to the interfaces, as in the case\nof Loss-DiVincenzo spin qubits with quantization axes\ndetermined by the external magnetic field [56].\nHere, we introduce a new figure of merit (FOM)\nr= 2z/bdepending on the array configurations, whose\nvariations lead to qualitatively different nano-EM super-\ndephasing scaling laws concerning N, as shown in\nFig. 2(b). We highlight that ris independent of EM field\nwavelength, in stark contrast to the critical distance scal-\ning with wavelength for superradiance [4]. We investigate\nthe decay time tGHZ of multiqubit coherence quantified\nthrough the l1norm measure [57] for |ψ⟩GHZ. We char-\nacterize nano-EM super-dephasing scaling behaviors by\nα=−∂lntGHZ/∂lnNand plot αas a function of rat\nn= 10 in Fig. 2(b).\nFor 2D arrays, we find nano-EM super-dephasing be-\nhaviors can be classified into three regimes depending on\nr. In the r≪1 regime, collective dephasing is negligible\nnear both material interfaces. SQD dominates the multi-\nqubit decoherence processes, leading to tGHZ∼N−1. In\nther≳nregime, collective dephasing has rates com-\nparable to SQD near both materials. Therefore, col-\nlective dephasing strongly accelerates the decoherence of\n|ψ⟩GHZ through constructive interference with SQD, re-\nsulting in nano-EM super-dephasing with tGHZ∼N−2.\nRemarkably, in the intermediate regime, nano-EM super-\ndephasing behaviors become sensitive to material prop-\nerties. Near YIG films, collective dephasing suppresses\nthe decoherence of |ψ⟩GHZ to nano-EM sub-dephasing\ntGHZ∼[√\nNlogN]−1through overall destructive inter-\nference with SQD, owing to its anti-correlations with\nSQD for spin qubits separated by a large distance. In\ncontrast, near silver films, although collective dephasingv\nis slower than SQD, it is still correlated with SQD and\nmoderately accelerates decoherence of |ψ⟩GHZwith nano-\nEM super-dephasing tGHZ∼N−1.5. Derivations of the\nabove scaling laws are provided in Supplemental Mate-\nrial [22].\nFurthermore, we reveal that dimensionality can signif-\nicantly affect nano-EM super-dephasing phenomena, as\nshown in Fig. 2(b). Contrary to the 2D arrays case, in\nthe intermediate regime, super-dephasing in 1D arrays\ndoes not exhibit substantially different scaling behaviors\nfrom other regimes. Anti-correlations of collective de-\nphasing and SQD near YIG films can not lead to obvious\nnano-EM sub-dephasing of |ψ⟩GHZin 1D arrays.\nFinally, we point out that the FOM similarly de-\ntermines different nano-EM sub-dephasing regimes for\nstates in the decoherence-free subspace (DFS). Further\nanalysis regarding DFS states is provided in Supplemen-\ntal Material [22].\nEngineering nano-EM super-dephasing.– We now intro-\nduce an engineering approach for controlling the range\nof nano-EM super-dephasing phenomena since they can\nbe detrimental to quantum applications [20]. We con-\nsider nanostructured hyperbolic metamaterials based on\nperiodic silver gratings with periodicity psmaller than\ndistance zbetween spin qubits and the interface, as\nshown in Fig. 3(a). We employ the effective medium\napproximation valid for the low-frequency EM response\nof this structure [58]. Different from resonance fre-\nquency properties of hyperbolic media leading to long-\nrange RDDI [25], we exploit the low-frequency nano-EM\nresponse of the hyperbolic media to reduce the range of\ncollective dephasing.\nIn Fig. 3(b), we demonstrate that, for two qubits with\nmperpendicular to the interface, the nano-EM collec-\ntive dephasing range is reduced along the grating pe-\nriodicity direction (x direction). At large D/2z, nano-\nEM collective dephasing along the y direction follows\nts\nϕ/tc\nϕ∼(D/2z)−1similar to the planar silver interfaces,\nwhile nano-EM collective dephasing along the x direc-\ntion is of a shorter range with ts\nϕ/tc\nϕ∼(D/2z)−2. This\nalignment dependence originates from the hyperbolic dis-\npersion of the grating structure, which exhibits dielectric\nproperties along the x direction and metallic properties\nalong the y direction. We provide an analytic analysis\nof the above scaling behaviors in Supplemental Mate-\nrial [22].\nConclusion.– We introduced the universal nano-EM\nsuper-dephasing phenomena near lossy nanophotonic in-\nterfaces and analyzed their unique material dependence,\nensemble dimensionality dependence, scaling behaviors,\nand engineering methods. Nano-EM super-dephasing\ncan become important for quantum applications with\natomic and spin systems close to lossy interfaces, such as\nquantum memories and hybrid quantum systems [37, 39].\nNano-EM super-dephasing can be immediately observed\nin state-of-the-art shallow solid-state spin defect sys-tems [59, 60], which have intrinsic individual decoherence\ntime exceeding 100 µs at depth around 10 nm and suffer\nnegligible collective effects from other sources [53, 55], by\nmeasuring the difference in decoherence time of different\nBell states near lossy interfaces.\nBeyond this, our work is also important for quantum\nsensing based on dephasing, i.e., quantum dephasometry.\nNano-EM super-dephasing phenomena can be employed\nto improve dephasometry sensitivity and probe nonrecip-\nrocal material response, such as Hall viscosity, which may\nnot have prominent effects in single-qubit dephasing [22].\nAcknowledgement.– This work was supported by the\nArmy Research Office under Grant No. W911NF-21-1-\n0287.\n∗zjacob@purdue.edu\n[1] M. 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Comparison of quantum phenomena in light-matter interactions near material interfaces 1\nS2. Derivations of nano-electromagnetic super-dephasing dynamics 2\nA. Completely positive and trace-preserving maps 5\nB. Non-Markovianity 5\nC. High frequency cutoff 6\nS3. Comparison of super-dephasing in different photonic environments 6\nS4. Magnetic dyadic Green’s function near arbitrary interfaces 7\nA. Planar material interfaces 7\nB. Arbitrary nano-structured material interfaces 8\nS5. Material models and experimentally accessible parameters 8\nA. Ferrimagnetic YIG 8\nB. Polycrystalline silver 9\nC. Superconducting niobium 9\nS6. MHz evanescent interface modes 10\nS7. Film thickness effects 11\nS8. Derivations of nano-EM super-dephasing scaling laws 11\nA. Nano-EM super-dephasing of the Greenberger–Horne–Zeilinger (GHZ) state 12\n1. Planar silver interfaces 12\n2. Planar YIG interfaces 13\nB. Nano-EM sub-dephasing of the decoherence-free subspace (DFS) state 14\nS9. Engineering nano-EM super-dephasing with meta-structures 15\nReferences 16\nS1. COMPARISON OF QUANTUM PHENOMENA IN LIGHT-MATTER INTERACTIONS NEAR\nMATERIAL INTERFACES\nIn table. S1, we compare nano-electromagnetic (nano-EM) super-dephasing with other quantum phenomena arising\nfrom light-atom interactions near nanophotonic interfaces. Expressions in table. S1 are for magnetic two-level systems\n(TLSs) (spin qubit systems) with spin magnetic moments malong the quantization axis and meg,mgeperpendicular to\nthe quantization axis. We present the general expression for each phenomenon in terms of the magnetic dyadic Green’s\nfunction← →Gmto manifest their common origin from near-field electromagnetic (EM) fluctuations. Expressions for\nelectric TLSs can be obtained by replacing spin magnetic moments and← →Gmwith electric dipole moments of atoms\nand electric dyadic Green’s function← →G, and multiplying the expressions by c2. Distinct from other phenomena,\n∗zjacob@purdue.eduarXiv:2402.18816v1 [quant-ph] 29 Feb 20242\nnano-EM super-dephasing is related to many-body collective interactions with near-field EM modes associated with\nlow-frequency ( ≤MHz) material responses.\nUniversal phenomena General expressions\nCasimir-Polder frequency shift [1] δω=−µ0\nπℏ/integraltextωc\n0dωP(1\nω−ω0)/parenleftbigω\nc/parenrightbig2/bracketleftig\nmeg\ni·Im← →Gm(ri,ri, ω)·mge\ni/bracketrightig\nResonant dipole-dipole interactions [2] Vdd=−µ0\nℏ(ω0\nc)2meg\ni·Re← →Gm(ri,rj, ω0)·mge\nj\nSpontaneous emission [3, 4] Γ =2µ0\nℏ(ω0\nc)2meg\ni·Im← →Gm(ri,ri, ω0)·mge\niUniversal quantum phenomena arising from light-matter interactions near material interfaces\nUnitary\nNon-unitaryNano-EM super-dephasing\n(reciprocal/nonreciprocal interfaces)aΦc(t;ri,rj) =2µ0\nℏπ/integraltextωc\n0dω F (ω, t) cothℏω\n2kBTω2\nc2/bracketleftig\nmi·Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]·mj/bracketrightig\naThis work.\nTABLE S1. Comparison of nano-EM super-dephasing and other phenomena arising from quantum light-matter interactions\nnear material interfaces. We present the general expressions for the Casimir-Polder frequency shift δω, the resonant dipole-\ndipole interactions (RDDI) Vdd, and the spontaneous emission rates Γ in terms of magnetic dyadic Green’s function← →Gmnear\nnanophotonic interfaces [1–4]. ω0is the resonance frequency of the magnetic TLS. F(ω, t) is the frequency filter function, which\nwill be discussed in section S2. Different from other effects, nano-EM super-dephasing is related to collective interactions with\nlow-frequency fluctuating EM modes near lossy material interfaces.\nS2. DERIVATIONS OF NANO-ELECTROMAGNETIC SUPER-DEPHASING DYNAMICS\nIn this section, we first derive the master equation describing the nano-EM super-dephasing dynamics of a multi-\nspin-qubit system (magnetic TLS ensemble) interacting with fluctuating EM fields near arbitrary nano-structured and\ndissipative material interfaces. Then, we discuss the important properties of the quantum dynamics, such as complete\npositivity and non-Markovianity. We follow the quantization framework of macroscopic quantum electrodynamics\n(QED) [5] to describe fluctuating EM fields near material interfaces. The total Hamiltonian ˆHis:\nˆH=ˆHq+ˆHf+ˆHint, (S1)\nˆHq=/summationdisplay\niℏωiˆσ+\niˆσ−\ni, (S2)\nˆHf=/integraldisplay\nd3r/integraldisplay∞\n0ℏωˆf†(r, ω)ˆf(r, ω), (S3)\nˆHint=−/summationdisplay\ni(meg\niˆσ+\ni+mge\niˆσ−\ni+miˆσz\ni)·ˆB(ri). (S4)\nHere, Hqis the Hamiltonian of the multi-spin-qubit system, Hfrepresents the Hamiltonian of the EM fields, and\nHintdescribes the interaction between spin qubits and the fluctuating magnetic fields. Nano-EM single-qubit and\ncollective dephasing are induced by the interactions with the longitudinal noise fields/summationtext\nimiˆσz\ni·ˆB(ri).ωiandri\nrepresent the resonance frequency and position of the i thspin qubit. ˆ σ+(−)\ni =|1⟩⟨0|(|0⟩⟨1|) and ˆ σz\ni=|1⟩⟨1|−|0⟩⟨0|are\nthe raising (lowering) operator and the Pauli-z matrix corresponding to the i thspin qubit, respectively. Spin magnetic\nmoments meg\ni,mge\ni, and midepends on the direction of the quantization axis of i thspin qubit (spin-1 /2). For qubits3\nwith quantization axes along the zdirection, we have meg\ni= [mge\ni]∗= [ℏγi/2,−iℏγi/2,0],mi= [0,0,ℏγi/2], where γi\nis the gyromagnetic ratio. ˆf†andˆfare photon creation and annihilation operators satisfying the following relation:\n[ˆf†\nα(r, ω),ˆfβ(r′, ω′)] =δαβδ(r−r′)δ(ω−ω′), (S5)\nwhere α, β=x, y, z . Magnetic field operator ˆB(r) can be expressed in terms of ˆf†,ˆf, and electric dyadic Green’s\nfunction← →G:\nˆB(r) =/integraldisplay∞\n0dω[ˆB(r, ω) +ˆB†(r, ω)], (S6)\nˆB(r, ω) = (iω)−1/integraldisplay\nd3r′∇r×← →G(r,r′, ω)·ˆf(r′, ω). (S7)\nIn the interaction picture with respect to H0=Hq+Hf, the interaction Hamiltonian ˆHI\nint(t) is (superscript I\ndenotes the interaction picture):\nˆHI\nint(t) =−/bracketleftigg/integraldisplay∞\n0dω(iω)−1/integraldisplay\nd3r′/summationdisplay\ni(meg\niˆσ+\nieiωit+mge\niˆσ−\nie−iωit+miˆσz\ni)·[∇ri×← →G(ri,r′, ω)]·ˆf(r′, ω)e−iωt+h.c./bracketrightigg\n.\n(S8)\nWe focus on the interactions between longitudinal fluctuating EM fields and magnetic TLSs that induce nano-EM\nsingle-qubit and collective dephasing. The other components related to interactions between transverse fluctuating\nfields and spin qubits result in relaxation dynamics of spin qubits derived in Ref. [4]. Therefore, Eq. (S8) becomes:\nˆHI\nint(t) =−/bracketleftigg/integraldisplay∞\n0dω(iω)−1/integraldisplay\nd3r′/summationdisplay\ni(miˆσz\ni)·[∇ri×← →G(ri,r′, ω)]·ˆf(r′, ω)e−iωt+\n/integraldisplay∞\n0dω(−iω)−1/integraldisplay\nd3r′/summationdisplay\ni(miˆσz\ni)·[∇ri×← →G∗(ri,r′, ω)]·ˆf†(r′, ω)eiωt/bracketrightigg\n.(S9)\nIn the interaction picture, the Liouville–von Neumann equation becomes:\ndρI\ntot(t)\ndt=1\niℏ[ˆHI\nint(t), ρI\ntot(t)]. (S10)\nSubstituting the total density matrix ρI\ntot(t) =ρI(t)⊗ρI\nf(t) with the integral form of Eq. (S10), we have:\ndρI\ntot(t)\ndt=1\niℏ[ˆHI\nint(0), ρI\ntot(0)]−1\nℏ2/integraldisplayt\n0dτ[ˆHI\nint(t),[ˆHI\nint(τ), ρI\ntot(τ)]], (S11)\nwhere ρI(t) and ρI\nf(t) are the density matrices of the multi-spin-qubit system and EM bath, respectively. ρI(t) can be\nobtained by tracing off the field part ρI\nf(t):\ndρI(t)\ndt=−1\nℏ2/integraldisplayt\n0dτTrf[ˆHI\nint(t),[ˆHI\nint(τ), ρI\ntot(τ)]]\n=−1\nℏ2/integraldisplayt\n0dτTrf[ˆHI\nint(t)ˆHI\nint(τ)ρI\ntot(τ)−ˆHI\nint(t)ρI\ntot(τ)ˆHI\nint(τ)−ˆHI\nint(τ)ρI\ntot(τ)ˆHI\nint(t) +ρI\ntot(τ)ˆHI\nint(τ)ˆHI\nint(t)].\n(S12)\nTo this end, our derivations do not involve any assumption about the electromagnetic bath. Eq. (S12) relates\nρI(t) toρI(τ) at all previous moments τand is difficult to solve. Here, we assume the weak-coupling condition and\nemploy the time-convolutionless projection operator (TCL) technique to obtain a time-local master equation [6]. The\nsecond-order TCL generator leads to:\ndρI(t)\ndt=−1\nℏ2/integraldisplayt\n0dτTrf[ˆHI\nint(t)ˆHI\nint(τ)ρI\ntot(t)−ˆHI\nint(t)ρI\ntot(t)ˆHI\nint(τ)\n−ˆHI\nint(τ)ρI\ntot(t)ˆHI\nint(t) +ρI\ntot(t)ˆHI\nint(τ)ˆHI\nint(t)].(S13)4\nThe real part of Eq. (S13) describes the nano-EM dephasing processes of the multi-spin-qubit system, and the\nimaginary part of Eq. (S13) determines the unitary evolution. Substituting Eq. (S9) into Eq. (S13), we obtain\n(assuming that the EM bath is in the vacuum state):\n−1\nℏ2/integraldisplayt\n0dτTrf[ˆHI\nint(t)ˆHI\nint(τ)ρI\ntot(t)] =−µ0\nℏπ/integraldisplay∞\n0dωωsinωt\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nˆσz\niˆσz\njρI(t)\n+iµ0\nℏπ/integraldisplay∞\n0dωω(1−cosωt)\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nˆσz\niˆσz\njρI(t),(S14)\n1\nℏ2/integraldisplayt\n0dτTrf[ˆHI\nint(t)ρI\ntot(t)ˆHI\nint(τ)] =µ0\nℏπ/integraldisplay∞\n0dωωsinωt\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nˆσz\niρI(t)ˆσz\nj\n+iµ0\nℏπ/integraldisplay∞\n0dωω(1−cosωt)\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nˆσz\niρI(t)ˆσz\nj,(S15)\n1\nℏ2/integraldisplayt\n0dτTrf[ˆHI\nint(τ)ρI\ntot(t)ˆHI\nint(t)] =µ0\nℏπ/integraldisplay∞\n0dωωsinωt\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nˆσz\niρI(t)ˆσz\nj\n−iµ0\nℏπ/integraldisplay∞\n0dωω(1−cosωt)\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nˆσz\niρI(t)ˆσz\nj,(S16)\n−1\nℏ2/integraldisplayt\n0dτTrf[ρI\ntot(t)ˆHI\nint(τ)ˆHI\nint(t)] =−µ0\nℏπ/integraldisplay∞\n0dωωsinωt\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nρI(t)ˆσz\niˆσz\nj\n−iµ0\nℏπ/integraldisplay∞\n0dωω(1−cosωt)\nc2/bracketleftig\nmi· I← →Gm(ri,rj, ω)·mj/bracketrightig\nρI(t)ˆσz\niˆσz\nj,(S17)\nwhere we define the magnetic dyadic Green’s function← →Gm(ri,rj, ω) and I← →Gm(ri,rj, ω) as:\n← →Gm(ri,rj, ω) =1\nk2\n0∇i×← →G(ri,rj, ω)× ∇ j, (S18)\nI← →Gm(ri,rj, ω) =← →Gm(ri,rj, ω)−← →G†\nm(rj,ri, ω)\n2i, (S19)\nandk0=ω/c. We employ the following relation [5] in the derivations:\n/integraldisplay\nds← →G(ri, s, ω)·← →G†(rj, s, ω) =ℏµ0ω2\nπI← →G(ri,rj, ω). (S20)\nHere, it is worth noting that, only in the absence of any nonreciprocal material, we have← →Gm(ri,rj, ω) =← →G⊺\nm(rj,ri, ω) and I← →Gm(ri,rj, ω) = Im← →Gm(ri,rj, ω), where Im[ ·] denotes the imaginary part of the tensor. In\ngeneral, I← →Gm(ri,rj, ω)̸= Im← →Gm(ri,rj, ω) and I← →Gm(ri,rj, ω) is a complex value that can be separated into a\nsymmetric real part and anti-symmetric imaginary part:\nI← →Gm(ri,rj, ω) =Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]\n2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nreal part,\nsymmetric under ri,rjexchange+iRe[−← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]\n2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nimaginary part,\nanti-symmetric under ri,rjexchange. (S21)\nThrough substituting Eq. (S21) into Eqs. (S14–S17) and summing Eqs. (S14–S17) to obtain Eq. (S13), we find that\nthe anti-symmetric part of I← →Gm(ri,rj, ω)does not contribute to the nano-EM super-dephasing processes (real part\nof Eq. (S13)) . According to Fubini’s theorem, we change the integration order regarding ωandtin Eq. (S13) and\nobtain the nano-EM super-dephasing dynamics in the Schr¨ odinger picture (Eq. (1) in the main text):\ndρ(t)\ndt=−i[ˆHd(t), ρ(t)]+/summationdisplay\niγϕ\ns(t;ri)/bracketleftig\nˆσz\niρ(t)ˆσz\ni−ρ(t)/bracketrightig\n+/summationdisplay\ni̸=jγϕ\nc(t;ri,rj)/bracketleftig\nˆσz\niρq(t)ˆσz\nj−1\n2ˆσz\niˆσz\njρ(t)−1\n2ρ(t)ˆσz\niˆσz\nj/bracketrightig\n,(S22)\nwhere the dipolar interaction Hamiltonian ˆHdgoverning the unitary evolution can be obtained from the imaginary\npart of the summation of Eqs. (S14–S17). The time-dependent single-qubit and collective dephasing rates γs\nϕ(t;ri)5\nandγc\nϕ(t;ri,rj) at time tare:\nγc\nϕ(t;ri,rj) =2µ0\nℏπ/integraldisplayωc\n0dωsinωt\nω(2N(ω) + 1)ω2\nc2/bracketleftig\nmi·Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]\n2·mj/bracketrightig\n, (S23)\nγs\nϕ(t;ri) =2µ0\nℏπ/integraldisplayωc\n0dωsinωt\nω(2N(ω) + 1)ω2\nc2/bracketleftig\nmi·Im[← →Gm(ri,ri, ω) +← →G⊺\nm(ri,ri, ω)]\n2·mi/bracketrightig\n, (S24)\nwhere we incorporate the effects of the EM bath thermal fluctuations at temperature Tinto Eqs. (S23, S24) through\n2N(ω)+1 = coth( ℏω/2kBT).N(ω) = 1 /(eℏω/kBT−1) is the mean photon number of the thermal EM bath. ωcis the\ncutoff frequency that will be discussed later. In the absence of external pulse sequences, the nano-EM single-qubit and\ncollective dephasing functions Φ c(t;ri,rj), Φ s(t;ri) induced by the EM fluctuations can be defined by the temporal\nintegration of Eqs. (S23, S24) (Eqs. (2, 3) in the main text):\nΦc(t;ri,rj) = 2/integraldisplayt\n0dt γc\nϕ(t;ri,rj)\n=4µ0\nℏπ/integraldisplayωc\n0dω1−cosωt\nω2cothℏω\n2kBTω2\nc2/bracketleftig\nmi·Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]\n2·mj/bracketrightig\n=/integraldisplayωc\n0dω F R(ω, t)Jc(ri,rj, ω).(S25)\nΦs(t;ri) = 2/integraldisplayt\n0dt γs\nϕ(t;ri)\n=4µ0\nℏπ/integraldisplayωc\n0dω1−cosωt\nω2cothℏω\n2kBTω2\nc2/bracketleftig\nmi·Im[← →Gm(ri,ri, ω) +← →G⊺\nm(ri,ri, ��)]\n2·mi/bracketrightig\n=/integraldisplayωc\n0dω F R(ω, t)Js(ri, ω),(S26)\nHere, we decompose the integrands in Eqs. (S26, S25) into the Ramsey fileter function FR(ω, t) = (1 −cosωt)/ω2and\nnoise (correlation) spectra Js(ri, ω) (Jc(ri,rj, ω)). The filter function can be shifted by the pulse sequence applied\nin the dephasing processes [7]. In this paper, we also consider the filter function FSE(ω, t) = 8 sin4(ωt/4)/ω2for the\nspin-echo dephasing time near superconducting materials. We emphasize that Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]/2̸=\nIm[← →Gm(ri,rj, ω)], manifesting important differences from previous discussions of EM bath’s correlation properties\nlimited to the reciprocal cases [8]. We define the magnitude of nano-EM collective (single-qubit) dephasing time |tc(s)\nϕ|\nas|Φc(s)(|tc(s)\nϕ|;ri,rj)|= 1, and define the sign of tc(s)\nϕas sgn tc(s)\nϕ= sgn Φ c(s)(|tc(s)\nϕ|;ri,rj).\nTo this end, we would like to discuss several important aspects of the nano-EM super-dephasing dynamics Eq. (S22)\nand dephasing functions Eqs. (S25, S26).\nA. Completely positive and trace-preserving maps\nPhysically valid quantum dynamics must be trace-preserving and completely positive. This requires the tensor φ,\ndefined by [ φ]ij= Φ c(t;ri,rj) and [ φ]ii= Φ s(t;ri), to be positive semi-definite [9, 10]. This requires Φ c(t;ri,rj) =\nΦc(t;rj,ri) and Φ s(t;ri)>0. The real and symmetric form of Im[← →Gm(ri,rj, ω) +← →G⊺\nm(rj,ri, ω)]/2 guarantees\nΦc(t;ri,rj) = Φ c(t;rj,ri) near arbitrary nano-structured and dissipative material interfaces, which can be reciprocal\nor nonreciprocal. Meanwhile, positive definiteness of Im[← →Gm(ri,ri, ω)+← →G⊺\nm(ri,ri, ω)]/2 at all ωensures Φ s(t;ri)>0.\nIm← →Gm(ri,ri, ω) = Im← →G0\nm(ri,ri, ω)+Im← →Gr\nm(ri,ri, ω) can be separated into the free-space component Im← →G0\nm(ri,ri, ω)\nand the reflected component Im← →Gr\nm(ri,ri, ω). It is worth noting that Im[← →Gr\nm(ri,ri, ω) +← →Gr⊺\nm(ri,ri, ω)]/2 alone may\nnot be positive semi-definite at all ω.\nB. Non-Markovianity\nEq. (S22) derived using the TCL technique is not limited to the Markovian EM bath. It is worth noting that\nEq. (S22) is not a time-dependent Markovian master equation [9] since the single-qubit and collective dephasing rates6\nγϕ\ns(t;ri) and γϕ\nc(t;ri,rj) explicitly depends on the initial time t0[9], which is chosen to be t0= 0 here. The memory\neffects can be reflected by the dependence of γϕ\ns(t;ri) and γϕ\nc(t;ri,rj) ontandt0[9].\nC. High frequency cutoff\nWithout the cutoff frequency ωc, Eqs. (S25, S26) will not converge (even with the filter function FR) due to the\nfree-space vacuum fluctuation contributions. For ω→ ∞ , we have:\nIm← →Gm(ri,ri, ω)≈Im← →G0\nm(ri,ri, ω) =ω\n6πc,\nwhere← →G0\nm(ω) is the magnetic dyadic Green’s function in free-space. This divergence is also encountered in Lamb shift\nand Casimir force calculations and is usually related to the point dipole approximation or poor microscopic model\nof materials. Here, we introduce the cutoff frequency ωcto regulate the high-frequency behavior of the integrand in\nEqs. (S25, S26). ωccan be selected as the frequency at which the point dipole approximation or materials models\nfail. We further verify that for ωc<7.73×1020Hz (corresponding to the Compton wavelength), the choice of ωcwill\nnot have a significant quantitative influence on nano-EM super-dephasing near lossy material interfaces.\nS3. COMPARISON OF SUPER-DEPHASING IN DIFFERENT PHOTONIC ENVIRONMENTS\nIn this section, we compare nano-EM super-dephasing and dephasing effects in other photonic environments.\nEqs. (S25, S26) are dominated by contributions from fluctuating EM modes of different frequencies in different\nphotonic environments. In the nano-EM environment near lossy material interfaces, fluctuations of low-frequency\n(≤MHz) EM modes can be enhanced by over 20 orders of magnitude compared to free-space due to low-frequency\nevanescent interface modes contributions. Thus, low-frequency components have dominant contributions to the inte-\ngral in Eqs. (S25, S26) in the near-field of lossy interfaces. In contrast, in conventional photonic environments, such as\nfree-space and photonic crystals, EM fluctuations are more dominant at high frequencies (e.g., Im← →G0\nm(ri,ri, ω)∝ω\nin free-space). Therefore, in these photonic environments, Eqs. (S25, S26) are determined by high-frequency contri-\nbutions and super-dephasing phenomena are negligible. In the following, we quantitatively compare the magnitude\nof dephasing functions due to coupling to different photonic environments.\nIn free-space, we have Φ c(t;ri,rj) due to fluctuating EM modes in free-space:\nΦc(t;ri,rj) =4µ0\nℏπ/integraldisplayωc\n0dω(1−cosωt)\nc2cothℏω\n2kBT/bracketleftig\nmi·Im← →Gm(ri,rj, ω)·mj/bracketrightig\n=4µ0m2\ni\nℏπc2/integraldisplayωc\n0dω(1−cosωt) cothℏω\n2kBTIm/bracketleftbigg\n−c2\n4πω2D3(1−iωD\nc−ω2D2\nc2)eiωD/c/bracketrightbigg\n=µ0m2\ni\nℏπ2c2D/integraldisplayωc\n0dω(1−cosωt) cothℏω\n2kBT/bracketleftbiggcos(ωD/c )\nωD/c+(ωD/c )2−1\n(ωD/c )2sin(ωD/c )/bracketrightbigg\n,(S27)\nwhere we assume mi=mjperpendicular to ri−rjand|ri−rj|=D. The integrand is approximately (1 −\ncosωt) sin(ωr/c) at high-frequency ω→ ∞ . Therefore, the integral in Eq. (S27) will not converge without the\nhigh-frequency cutoff ωc. Eq. (S27) has oscillatory behaviors dependent on ωc. For |ri−rj|= 10 nm, we find\n|Φc(t;ri,rj)|<10−11for all ωcatt= 100 µs.\nIn stark contrast, in the nano-EM environment near lossy interfaces, |Φc(t;ri,rj)|can be of the same order of ∼1\nfor|ri−rj|= 10 nm at t= 100 µs. This manifests that nano-EM super-dephasing phenomena can be enhanced more\nthan 10 orders near lossy material interfaces compared to free-space. This giant enhancement originates from the\nlong-range correlations inlow-frequency evanescent EMfluctuations, instark contrast totheshort-range correlations\ninhigh-frequency (≈ωc)EMmodes dominant forcollective dephasing infree-space. Furthermore, this indicates that\nnano-EM super-dephasing is not sensitive to the choice of ωcsince it is determined by the low-frequency component\nof the integral.\nFor Φ s(t;ri) in free-space, we can similarly find:\nΦs(t;ri) =4µ0\nℏπ/integraldisplayωc\n0dω(1−cosωt)\nc2cothℏω\n2kBT/bracketleftig\nmi·Im← →Gm(ri,ri, ω)·mi/bracketrightig\n=4µ0m2\ni\nℏπc2/integraldisplayωc\n0dω(1−cosωt) cothℏω\n2kBTω\n6πc.(S28)7\nThe integrand is approximately ω(1−cosωt) at high frequency ω→ ∞ . Therefore, the integral in Eq. (S28) will\ndiverge without the high-frequency cutoff ωc. For ωc= 3.56×1019Hz corresponding to the Bohr radius, we have\nΦs(t;ri)≈1.6×10−6att= 100 µs. For ωc= 7.73×1020Hz corresponding to the electron Compton wavelength λe,\nwe have Φ s(t;ri)≈7.7×10−4att= 100 µs.\nIn stark contrast, Φ s(t;ri)≳1 att= 100 µs in the nano-EM environment. This manifests the large enhancement\nof Φ s(t;ri) due to MHz evanescent interface waves. Additionally, single-qubit dephasing in the nano-EM environment\nis also determined by low-frequency components of the integral and is not sensitive to the choice of ωc.\nS4. MAGNETIC DYADIC GREEN’S FUNCTION NEAR ARBITRARY INTERFACES\nIn this section, we discuss the calculations of the magnetic dyadic Green’s function (MDGF)← →Gm(ri,rj, ω) near\narbitrary nano-structured, dissipative, reciprocal and nonreciprocal material interfaces. We start from the electric\ndyadic Green’s function← →G(ri,rj, ω) defined by:\n∇ × ∇ ×← →G(ri,rj, ω)−k2\n0← →G(ri,rj, ω) =← →I δ(ri−rj), (S29)\nwhere k0=ω/cis the free-space wave vector and← →Iis the 3 ×3 identity matrix. Here, we express the components\n[← →Gm]αβof the MDGF defined in Eq. (S18) in terms of the Levi-Civita symbols ϵαklandϵβnm:\n[← →Gm(ri,rj, ω)]αβ=1\nk2\n0ϵαklϵβnm∇k\ni∇n\nj[← →G(ri,rj, ω)]lm. (S30)\n← →Gm(ri,rj, ω) =← →G0\nm(ri,rj, ω)+← →Gr\nm(ri,rj, ω) can be decomposed into a free-space component← →G0\nm(ri,rj, ω) related\nto the free-space vacuum fluctuations and a reflected component← →Gr\nm(ri,rj, ω) determined by the material interface\nproperties. The free-space component← →G0\nm(ri,rj, ω) is:\n← →G0\nm(ri,rj, ω) =i\n8π2k2\n0/integraldisplaydq\nkzeiq(ri−rj)eikz|zi−zj|/parenleftbigg\nk2\n0−q2\nx−qxqy∓qxkz\n−qxqyk2\n0−q2\ny∓qykz\n∓qxkz∓qykzk2\n0−k2\nz\n/parenrightbigg\n, (S31)\nwhere kz=/radicalig\nk2\n0−q2x−q2yis the longitudinal component of the wave vector, q= (qx, qy) is the transverse component\nof the wave vector. For z > z 0andz < z 0, components of the matrix in Eq. (S31) take the upper and lower signs,\nrespectively. Im← →G0\nm(ri,rj, ω) is proportional to the quantum vacuum fluctuations in the free-space.\nTo this end, we discuss the effects of the spin qubit substrate permittivity εsub. The substrate has contributions\nto EM fluctuations comparable to the free-space contributions, but has negligible influence on the material interface\ncontributions to near-field EM fluctuations discussed in this Letter. Since the dephasing functions are dominated\nby← →Gr\nm(ri,rj, ω), effects of the substrate permittivity εsubare overall negligible to the nano-EM super-dephasing\ndiscussed in the main text.\nIn the following, we present the calculations of← →Gr\nm(ri,rj, ω). In the nano-EM environment, Im← →Gr\nm(ri,rj, ω) has\nmuch more prominent contributions to Im← →Gm(ri,rj, ω) and is closely related to the EM properties of the lossy\nmaterial interfaces.\nA. Planar material interfaces\nWe first discuss← →Gr\nm(ri,rj, ω) near generic planar material interfaces that can be nonreciprocal and anisotropic. In\nthis case,← →Gr\nm(ri,rj, ω) can be calculated from the reflection coefficients rss, rsp, rpp, rps. Assuming the interface is\nperpendicular to the ˆzdirection, we have:\n← →Gr\nm(ri,rj, ω) =i\n8π2/integraldisplaydq\nkzeiq(ri−rj)eikz(zi+zj)/parenleftbiggrpp\nq2\nq2\ny−qxqy0\n−qxqyq2\nx0\n0 0 0\n+rss\nk2\n0q2\n−q2\nxk2\nz−qxqyk2\nz−qxkzq2\n−qxqyk2\nz−q2\nyk2\nz−qykzq2\nq2qxkzq2qykz q4\n\n+rps\nk0q2\nqxqykzq2\nykz qyq2\n−q2\nxkz−qyqxkz−qxq2\n0 0 0\n+rsp\nk0q2\n−qxqykzq2\nxkz0\n−q2\nykzqyqxkz0\nqyq2−qxq20\n/parenrightbigg\n,\n(S32)8\nwhere q=qxˆ x+qyˆ yis the in-plane momentum component and zi=ri·ˆz, zj=rj·ˆzare the zcomponents of spin\nqubit positions.\nFor general planar interfaces with the dielectric response tensor← →εand the magnetic response tensor← →µ, the four\nreflection coefficients rss, rsp, rpp, rpsdepend on qxandqy. The reflection coefficients can be obtained by first finding\nk′\nzfor modes with given qxandqyinside the materials through [11]:\ndet/parenleftbigg/bracketleftbigg← →ε0\n0← →µ/bracketrightbigg\n+/bracketleftbigg\n0← →κ← →κ0/bracketrightbigg/parenrightbigg\n= 0,← →κ=\n0−k′\nzqy\nk′\nz0−qx\n−qyqx0\n, (S33)\nderived from the Maxwell equations in materials. For a given pair of ( qx, qy), there exist four (two) k′\nzsolutions of\nEq. (S33) for anisotropic (isotropic) materials. The four reflection coefficients can then be solved by matching the\nboundary conditions for EM fields at the interface [11]. For planar material thin films, boundary conditions for EM\nfields at both sides of the film need to be considered. In this paper, the above processes are performed numerically\nfor obtaining← →Gr\nm(ri,rj, ω).\nB. Arbitrary nano-structured material interfaces\nThe above analytic approach is limited to material interfaces of planar geometries. For an arbitrary nano-structured\nmaterial interface, the reflection coefficients are usually not well-defined. Perturbation-based methods for solving\nMDGF can fail to converge in the presence of a high-contrast ratio due to low-frequency material response. In this\ncase, the MDGF can be simulated using the computational electromagnetics method introduced in Ref. [4]. The\nMDGF← →Gm(ri,rj, ω) relates a point magnetic dipole with magnetic moment mto the Hfield through:\nH(ri, ω) =k2\n0← →Gm(ri,rj, ω)·m(rj, ω), (S34)\nand the Hfield can be solved using the volume-integral-equations (VIEs) method [4]. Through this technique, the\nMDGF← →Gm(ri,rj, ω) near arbitrary nano-structured material interfaces can be obtained with high precision.\nS5. MATERIAL MODELS AND EXPERIMENTALLY ACCESSIBLE PARAMETERS\nIn this section, we present details of material models and experimentally accessible parameters for gyromagnetic\nyttrium iron garnet (YIG), polycrystalline silver, and superconducting niobium considered in this paper.\nA. Ferrimagnetic YIG\nWe employ the Landau–Lifshitz–Gilbert formula for the magnetic permeability tensor← →µYIGof the nonreciprocal\ngyrotropic ferrimagnetic YIG. In the main text, we consider a biasing magnetic field H0in the ˆzdirection, and the\nYIG thin film has a gyromagnetic response:\n← →µYIG(ω) =µ0\n1 +χxx χxy 0\nχyx 1 +χyy0\n0 0 1\n, (S35)\nwhere χxx, χxy, χyx, χyyare the components of the susceptibility tensor satisfying:\nχxx=χyy=χ′\nxx+iχ′′\nxx, (S36)\nχxy=−χyx=χ′′\nxy−iχ′\nxy, (S37)9\nand\nχ′\nxx=ω0ωm(ω2\n0−ω2) +ω0ωmω2α2\n[ω2\n0−ω2(1 +α2)]2+ 4ω2\n0ω2α2, (S38)\nχ′′\nxx=αωω m[ω2\n0+ω2(1 +α2)]\n[ω2\n0−ω2(1 +α2)]2+ 4ω2\n0ω2α2, (S39)\nχ′\nxy=ωωm[ω2\n0−ω2(1 +α2)]\n[ω2\n0−ω2(1 +α2)]2+ 4ω2\n0ω2α2, (S40)\nχ′′\nxy=2ω0ωmω2α\n[ω2\n0−ω2(1 +α2)]2+ 4ω2\n0ω2α2. (S41)\nHere, the Larmor frequency ω0=γµ0H0depends on the gyromagnetic ratio γand the bias field H0.ωm=γµ0Ms\ndepends on the saturation magnetization Ms. The Gilbert damping factor αdetermines the magnetic loss. In the\nmain text, we consider H0= 800 G, α= 1.7×10−3,Ms= 2160 G for the YIG thin film of 600 nm thickness\natT= 100 K [12, 13]. Temperature dependence of αandMsranging from 100 K to 300 K is obtained from the\nRefs. [12, 13].\nB. Polycrystalline silver\nWe consider the Drude model for the low-frequency dielectric response← →εAg=εAg(ω)← →Iof polycrystalline silver:\nεAg(ω) = 1−ω2\npτ2\nω2τ2+ 1+iσ0\nε0ω(1 +ω2τ2), (S42)\nwhere ωpis the plasma frequency, 1 /τis the electron mean collision rate, σ0is the direct current (DC) conductivity.\nIn the main text, we employ ωp= 2π×2.15×1015Hz, τ= 3.71×10−14s−1, σ0(100 K) = 1 .4×108S/m for the\npolycrystalline silver thin film of 600 nm thickness at T= 100 K [14]. Temperature dependence σ0is obtained from\nthe Ref. [14]. We focus on the local EM response of silver since the nonlocal effects are not prominent for polycrystalline\nsilver [15].\nC. Superconducting niobium\nIn this paper, we focus on pure niobium in the Pippard regime [16], where the nonlocal effects are not prominent\nfor the niobium EM response. In this regime, the niobium complex conductivity σNb=σ1+iσ2at temperature T\ncan be described by the Mattis-Bardeen theory [16]:\nσNb(ω, T)\nσn=σ1(ω, T) +iσ2(ω, T)\nσn=/integraldisplay∞\n∆(T)−ℏωdx\nℏωtanh(x+ℏω\n2kBT)g(x)−/integraldisplay∞\n∆(T)dx\nℏωtanh(x\n2kBT)g(x), (S43)\nwhere σnis the conductivity of the normal state, ∆( T) is the BCS gap satisfying:\nln/braceleftigℏωD+/radicalbig\n(ℏωD)2+ ∆2(0)\n∆(0)/bracerightig\n=/integraldisplayℏωD\n0dx/radicalbig\nx2+ ∆2(T)tanh/bracketleftigg/radicalbig\nx2+ ∆2(T)\n2kBT/bracketrightigg\n, (S44)\nwhere ωDis the phonon Debye frequency. g(x) is given by:\ng(x) =x2+ ∆2(T) +ℏωx\nu1u2, (S45)\nu1=/braceleftigg/radicalbig\nx2−∆2(T), ifx >∆(T)\n−i/radicalbig\n∆2(T)−x2,ifx <∆(T)(S46)\nu2=/radicalbig\n(x+ℏω)2−∆2(T). (S47)\nAs shown in Ref. [16], the complex conductivity of niobium predicted by the Mattis-Bardeen theory matches well\nwith experimental measurements.10\nForT > T c(Tc= 9.2 K is the critical temperature for niobium), the niobium EM response follows the behaviors of\nnormal conductors described by the Drude model. For T < T c, the permittivity← →ε=εNb(ω, T)← →Ican be written as:\nεNb(ω, T) = 1 +i\nε0ωσNb(ω, T) = 1−σ2(ω, T)\nε0ω+iσ1(ω, T)\nε0ω. (S48)\nIn the main text, we employ ωD= 2π×6.045 THz, σn= 5×108S/m, and ∆(0) = 1 .4 meV [17] for the niobium\nthin film of 600 nm thickness in the Pippard regime. Temperature dependence of εNb(ω, T) can be calculated from\nEq. (S43). It is worth noting that there exists a coherence peak in σ1below Tcdue to the constructive interference\nof quasiparticle absorption rates [16] (type-II coherence factor), as described by Eq. (S43).\nS6. MHZ EVANESCENT INTERFACE MODES\nIn this section, we provide a universal perspective of the origin of distinct nano-EM super-dephasing behaviors near\ndifferent material interfaces based on evanescent interface modes. In addition, we derive the scaling laws of nano-EM\nsingle-qubit dephasing (SQD) and collective dephasing presented in the main text via analysis of evanescent interface\nmodes strongly confined to material interfaces.\nIn the nano-EM environment, SQD and collective dephasing are ubiquitously determined by contributions of evanes-\ncent interface modes. We explicate this by first considering the term eikz(zi+zj)in Eq. (S32), which can be written\nase−√\nq2−k2\n0(zi+zj)for integration over the q > k 0part. For low-frequency Im← →Gm(ri,rj, ω) important for nano-EM\nsuper-dephasing processes, this term leads to a cutoff momentum qcut∼1/(zi+zj)≫k0=ω/cfor the integral in\nEq. (S32). The integral in Eq. (S32) is thus dominated by q-integration around qcut. These EM waves of in-plane\nmomentum q∼qcut≫k0have large imaginary perpendicular momentum components on both sides of the interface,\ncorresponding to evanescent waves highly confined to the interface.\nWe illustrate the above discussions by analyzing the q-spectrum of [Im← →Gm(ri,ri, ω)]zznear superconducting nio-\nbium interfaces, which is the zzcomponent of the MDGF. In Fig. S1, we present the q-spectrum of [Im← →Gm(ri,ri, ω)]zz\natω= 2π×104,T= 8.6 K near superconducting niobium thin films ( Tc= 9.2 K) for zi= 2 nm (dashed curve) and\nzi= 50 nm (dotted curve). We demonstrate that [Im← →Gm(ri,ri, ω)]zzis dominated by contributions from the inte-\ngration over q≫k0, corresponding to the evanescent interface modes. For superconducting niobium, the q-spectrum\nof [Im← →Gm(ri,ri, ω)]zzhas distinct behaviors in high and low qregions, as shown by the solid curve corresponding\nto the q-spectrum in the absence of the cutoff function eikz(zi+zj)in Fig. S1. This is because high- qevanescent\ninterface modes depend on the real part of conductivity σ1dominated by the response of thermally broken Cooper\npairs (quasiparticles) at T= 0.8Tc. In contrast, low- qevanescent interface modes are influenced by the imaginary\npart of conductivity σ2, which is dominated by the response of the Cooper pair condensate.\nFIG. S1. The q-spectrum of [Im← →Gm(ri,ri, ω)]zznear superconducting niobium interfaces at ω= 2π×104,T= 8.6 K.\n[Im← →Gm(ri,ri, ω)]zzis dominated by contributions from the evanescent interface modes with q≫k0. High- qevanescent\ninterface modes are dominated by the real part of conductivity σ1dominated by the response of thermally broken Cooper pairs\n(Bogoliubov quasiparticles). Low- qevanescent interface modes are affected by the imaginary part of conductivity σ2dominated\nby the response of the Cooper pair condensate.\nTo this end, we derive the scaling of SQD in the nano-EM environment discussed in the main text based on the q-\nspectrum behaviors. Since qcut∼1/(zi+zj), Im← →Gm(ri,ri, ω) with small ziis dominated by high- qevanescent interface\nmodes contributions, as shown by the dashed curve in Fig. S1. Hence, spin qubits closer to the interface are more\nstrongly coupled to high- qevanescent interface modes. The q-spectrum behavior ∂[Im← →Gm(ri,ri, ω)]zz/∂ q∼q0e−2qzi11\natq∼qcutleads to ts\nϕ∼zifor spin qubits with quantization axes perpendicular to the interface, according to\nEqs. (S26, S32). In contrast, Im← →Gm(ri,ri, ω) with large ziis dominated by low- qevanescent interface modes’\ncontributions, as shown by the dotted curve in Fig. S1. Therefore, spin qubits at large distances from the interface\nare coupled to low- qevanescent interface modes. The q-spectrum follows ∂[Im← →Gm(ri,ri, ω)]zz/∂ q∼q3e−2qziat\nq∼qcut, resulting in ts\nϕ∼z4\nifor spin qubits with quantization axes perpendicular to the interface. The above\ndiscussions explain the scaling behaviors of ts\nϕnear the niobium thin film demonstrated in the main text.\nFor YIG and silver thin films, we find ∂[Im← →Gm(ri,ri, ω)]zz/∂ q∼q2e−2qziand∂[Im← →Gm(ri,ri, ω)]zz/∂ q∼q0e−2qzi\nforq≫k0, respectively. From Eqs. (S26, S32), it is easy to obtain the ts\nϕ∼z3\niandts\nϕ∼ziscaling laws demonstrated\nin the main text.\nS7. FILM THICKNESS EFFECTS\nIn this part, we provide supplemental discussions for the effects of film thickness ton nano-EM collective dephasing\nbehaviors. In the regime D≥t,ts\nϕ/tc\nϕdecays faster concerning Dcompared to the D < t regime discussed in the\nmain text. In Fig. S2, we demonstrate nano-EM collective dephasing behaviors for a pair of spin qubits at z= 15 nm\nfrom silver thin films of 600 nm thickness. We find ts\nϕ/tc\nϕ∼D−3forθ= 0 and ts\nϕ/tc\nϕ∼D−2forθ=π/2 in the D≥t\nregime, which decays faster compared to ts\nϕ/tc\nϕ∼D−1in the D < t regime.\nFIG. S2. Dependence of nano-EM collective dephasing ts\nϕ/tc\nϕonD/2znear silver films of 600 nm thickness. In the D≥t\nregime, ts\nϕ/tc\nϕdecays faster ( ∼D−2,D−3) than in the D < t regime ( ∼D−1) discussed in the main text.\nS8. DERIVATIONS OF NANO-EM SUPER-DEPHASING SCALING LAWS\nIn this section, we provide derivations for the nano-EM super-dephasing scaling laws concerning the number of spin\nqubits demonstrated in the main text. Furthermore, we extend the discussions to the state |ψ⟩DFSin the decoherence-\nfree subspace (DFS). We demonstrate nano-EM sub-dephasing for |ψ⟩DFSin the three regimes distinguished by the\nfigure of merit (FOM) introduced in the main text.\nWe consider Nspin qubits arranged in one-dimensional (1D) N= 1×nand two-dimensional (2D) N=n×narrays\nwith lattice constant bat distance zfrom lossy material interfaces. We focus on the decay of multiqubit coherence\ndefined through the l1norm measure Cl1(ρ) =/summationtext\nn̸=k|ρnk|[18], where ρis the density matrix of the multi-spin-qubit\nsystem and ρnkare off-diagonal elements of ρ. The multiqubit coherence decay time tmqdsatisfying Cl1(ρ(t=tmqd)) =\ne−1Cl1(ρ(t= 0)) can be obtained from Eqs. (S22, S25, S26). We find the decay power α=−∂lntmqd/∂lnNfrom\nNdependence of tmqd, which characterizes the scaling behaviors of nano-EM super-dephasing. In the following, we\ndenote the multiqubit coherence decay time for |ψ⟩GHZand|ψ⟩DFSastGHZandtDFS, respectively.12\nA. Nano-EM super-dephasing of the Greenberger–Horne–Zeilinger (GHZ) state\nTheN-qubit GHZ state is |ψ⟩GHZ=1√\n2(|0⟩⊗N+|1⟩⊗N). From Eq. (S22), we can obtain the decoherence function\nΦ2D\nGHZ(t;n) for|ψ⟩GHZin 2D n×nspin qubit arrays:\nΦ2D\nGHZ(t;n) = 2/integraldisplay\nn/summationdisplay\nxi, yi=1γϕ\ns(t;ri) +n/summationdisplay\nxi, xj, yi, yj=1\n(xi, yi)̸=(xj, yj)γϕ\nc(t;ri,rj)\ndt, (S49)\nwhere ri=bxiˆx+byiˆy,rj=bxjˆx+byjˆy.\nFor 1D 1 ×nspin qubit arrays, the decoherence function Φ1D\nGHZ(t;n) is:\nΦ1D\nGHZ(t;n) = 2/integraldisplay\nn/summationdisplay\nxi=1γϕ\ns(t;ri) +n/summationdisplay\nxi, xj=1\nxi̸=xjγϕ\nc(t;ri,rj)\ndt, (S50)\nwhere ri=bxiˆx,rj=bxjˆx.\nThe multiqubit coherence decay time tGHZcan thus be equivalently defined through Eqs. (S49, S50) as Φ GHZ(tGHZ;n) =\n1.\n1. Planar silver interfaces\nFrom Eqs. (S32, S25, S26), with mof all spin qubits perpendicular to the interface, we find that the ratio between\nnano-EM collective and single-qubit noise spectra Jc(ri,rj, ω)/Js(ri, ω) at low frequencies ω→0 near polycrystalline\nsilver films is:\nJc(ri,rj, ω→0)\nJs(ri, ω→0)≈2z/radicalbig\n(2z)2+ (ri−rj)2. (S51)\nSubstituting Eq. (S51) into Eqs. (S25, S26, S49, S50), we obtain:\nΦ2D\nGHZ(t;n)\nΦs(t)≈n/summationdisplay\nxi, xj, yi, yj=1r/radicalbig\nr2+ (xi−xj)2+ (yi−yj)2, (S52)\nΦ1D\nGHZ(t;n)\nΦs(t)≈n/summationdisplay\nxi, xj=1r/radicalbig\nr2+ (xi−xj)2, (S53)\nwhere r= 2z/bis the FOM introduced in the main text. In the following, we first derive the scaling behaviors of\nΦ2D\nGHZ(t;n)/Φs(t) and Φ1D\nGHZ(t;n)/Φs(t) concerning nin the three regimes corresponding to r≪1,r≳1, and the\nintermediate region.\n1. In the regime r≪1, summation terms corresponding to ( xi, yi) = (xj, yj) are much larger than other terms in\nEq. (S52). Therefore, we have:\nΦ2D\nGHZ(t;n)/Φs(t)∼n2. (S54)\n2. In the regime r≳n, we can convert the summation in Eq. (S52) into an integral through the Euler–Maclaurin\nformula:\nΦ2D\nGHZ(t;n)/Φs(t)∼n3/integraldisplay1\n0/integraldisplay1\n0/integraldisplay1\n0/integraldisplay1\n0dx1dx2dy1dy2/radicalbig\n(r\nn)2+ (x1−x2)2+ (y1−y2)2,\n= 4n3/integraldisplay1\n0/integraldisplay1\n0(1−x)(1−y)/radicalbig\n(r\nn)2+x2+y2dxdy,(S55)13\nwhere the last step is because the distribution of the absolute difference of two uniform random variables is\n(1−x). Converting the integral into the polar coordinates ( ϱ, θ), we have:\nΦ2D\nGHZ(t;n)/Φs(t)∼8n3/integraldisplayπ/4\n0dθ/integraldisplay1/cosθ\n0dϱϱ(1−ϱcosθ)(1−ϱsinθ)/radicalbig\nϱ2+ (r/n)2(S56)\n≈8n3/integraldisplayπ/4\n0dθ/integraldisplay1/cosθ\n0dϱ ϱ(1−ϱcosθ)(1−ϱsinθ)n\nr, (S57)\nwhere we employ r/n≳1 for the last equation. From Eq. (S57), we have:\nΦ2D\nGHZ(t;n)/Φs(t)∼n4, (S58)\n3. In the intermediate regime, from Eq. (S56), we can obtain:\nΦ2D\nGHZ(t;n)/Φs(t)∼8n3/integraldisplayπ/4\n0dθ/integraldisplay1/cosθ\n0dϱϱ(1−ϱcosθ)(1−ϱsinθ)/radicalbig\nϱ2+ (r/n)2\n≈8n3/integraldisplayπ/4\n0dθ/integraldisplay1/cosθ\n0dϱ(1−ϱcosθ)(1−ϱsinθ),(S59)\nforr/n≪1. Therefore, we have:\nΦ2D\nGHZ(t;n)/Φs(t)∼n3. (S60)\nFor 1D qubit arrays, from Eq. (S53), we can similarly find Φ1D\nGHZ(t;n)/Φs(t)∼nand Φ1D\nGHZ(t;n)/Φs(t)∼n2in the\nr≪1 and r≳nregimes, respectively. In the intermediate regime, through the Euler–Maclaurin formula, we obtain\nΦ1D\nGHZ(t;n)/Φs(t)∼nlogn.\nThe above analytical derivations match well with the nano-EM suoer-dephasing scaling behaviors demonstrated in\nthe main text. From Φ2D\nGHZ(t;n)/Φs(t), with Φ s(t)∝t, we can easily find that tGHZin the three regimes approximately\nfollows\ntGHZ∼\n\nN−1r≪1,\nN−1.5intermediate regime,\nN−2r≳1,(S61)\nfor 2D arrays ( N=n2), as presented in the main text. Meanwhile, differences between Φ2D\nGHZ(t;n)/Φs(t) and\nΦ1D\nGHZ(t;n)/Φs(t) in the intermediate regime shows the dimensionality effects on nano-EM super-dephasing phenom-\nena.\n2. Planar YIG interfaces\nFrom Eqs. (S32, S25, S26), with mof all spin qubits perpendicular to the interface, we find that the ratio be-\ntween nano-EM collective and single-qubit noise spectra Jc(ri,rj, ω)/Js(ri, ω) atω→0 near planar YIG interfaces\napproximately follows:\nJc(ri,rj, ω→0)\nJs(ri, ω→0)≈2(2z)5−(2z)3(ri−rj)2\n[(2z)2+ (ri−rj)2]5\n2. (S62)\nTherefore, we have:\nΦ2D\nGHZ(t;n)\nΦs(t)≈n/summationdisplay\nxi, xj, yi, yj=12r5−r3[(xi−xj)2+ (yi−yj)2]\n2[r2+ (xi−xj)2+ (yi−yj)2]5\n2, (S63)\nΦ1D\nGHZ(t;n)\nΦs(t)≈n/summationdisplay\nxi, xj=12r5−r3(xi−xj)2\n2[r2+ (xi−xj)2]5\n2, (S64)\nwhere r= 2z/bis the critical FOM. The scaling behaviors of Φ2D\nGHZ(t;n)/Φs(t) and Φ1D\nGHZ(t;n)/Φs(t) concerning nin\nthe three regimes are:14\n1. In the regime r≪1, Eq. (S63) is dominated by terms corresponding to ( xi, yi) = (xj, yj). Therefore, we have:\nΦ2D\nGHZ(t;n)/Φs(t)∼n2. (S65)\n2. In the regime r≳n, with the Euler–Maclaurin formula, we convert Eq. (S63) into the integral:\nΦ2D\nGHZ(t;n)/Φs(t)∼n/integraldisplay1\n0/integraldisplay1\n0/integraldisplay1\n0/integraldisplay1\n02(r\nn)2−(x1−x2)2−(y1−y2)2\n[(r\nn)2+ (x1−x2)2+ (y1−y2)2]5\n2dx1dx2dy1dy2,\n= 4n/integraldisplay1\n0/integraldisplay1\n02(r\nn)2−x2−y2\n[(r\nn)2+x2+y2]5\n2(1−x)(1−y)dxdy.(S66)\nIn polar coordinates ( ϱ, θ), we have:\nΦ2D\nGHZ(t;n)/Φs(t)∼8n/integraldisplayπ/4\n0dθ/integraldisplay1/cosθ\n0dϱ ϱ(1−ϱcosθ)(1−ϱsinθ)2(r/n)2−ϱ2\n(ϱ2+ (r/n)2)5/2(S67)\n≈8n/integraldisplayπ/4\n0dθ/integraldisplay1/cosθ\n0dϱ ϱ(1−ϱcosθ)(1−ϱsinθ)(n\nr)3, (S68)\nwhere we employ r/n≳1 for the last equation. From Eq. (S68), we have:\nΦ2D\nGHZ(t;n)/Φs(t)∼n4(S69)\n3. In the intermediate regime, taking r/n≪1 for the integral in Eq. (S67), we can obtain:\nΦ2D\nGHZ(t;n)/Φs(t)∼nlogn. (S70)\nFor 1D qubit arrays, from Eq. (S64), we can similarly find Φ1D\nGHZ(t;n)/Φs(t)∼nand Φ1D\nGHZ(t;n)/Φs(t)∼n2in the\nr≪1 and r≳nregimes, respectively. We obtain Φ1D\nGHZ(t;n)/Φs(t)∼nthrough the Euler–Maclaurin formula in the\nintermediate regime.\nThe above analytical analysis matches well with the nano-EM super-dephasing scaling behaviors near YIG interfaces\ndemonstrated in the main text. From Φ2D\nGHZ(t;n)/Φs(t), with Φ s(t)∝t, we can easily find that tGHZin the three\nregimes approximately follows\ntGHZ∼\n\nN−1r≪1,\n(N0.5logN)−1intermediate regime,\nN−2r≳1,(S71)\nfor 2D arrays ( N=n2), as presented in the main text. Here, differences between Φ2D\nGHZ(t;n)/Φs(t) near YIG and silver\nfilms manifest material influence on nano-EM super-dephasing scaling behaviors. Φ1D\nGHZ(t;n)/Φs(t)∼nin both the\nintermediate and r≪1 regimes reveals the absence of nano-EM sub-dephasing due to anti-correlations of collective\ndephasing and SQD near YIG films in 1D arrays.\nB. Nano-EM sub-dephasing of the decoherence-free subspace (DFS) state\nTo this end, we extend the above analysis to states in the decoherence-free subspace (DFS). We consider\n|ψ⟩DFS=1√\n2(|0101···⟩+|1010···⟩) with arrangements of |0101···⟩and|1010···⟩satisfying ˆ σz\nmˆσz\nk|0101···⟩<0\nand ˆσz\nmˆσz\nk|1010···⟩<0 for all pairs of nearest-neighbor qubits m, k in the 2D array.\nFrom Eq. (S22), the decoherence function Φ2D\nDFS(t;n) for|ψ⟩DFSin 2D n×nspin qubit arrays is:\nΦ2D\nDFS(t;n) = 2/integraldisplay\nn/summationdisplay\nxi, yi=1γϕ\ns(t;ri) +n/summationdisplay\nxi, xj, yi, yj=1\n(xi, yi)̸=(xj, yj)(−1)|xi−xj|+|yi−yj|γϕ\nc(t;ri,rj)\ndt, (S72)\nand the multiqubit coherence decay time tDFScan thus be equivalently defined through Eq. (S72) as Φ2D\nDFS(tDFS) = 1.15\nWe demonstrate nano-EM sub-dephasing behaviors of |ψ⟩DFSin 2D arrays at distance z= 20 nm from YIG and\nsilver films of thickness 2 µm, as shown in Fig. S3. We focus on collective effects on multiqubit decoherence behaviors\nin the three regimes distinguished by r. In the r≪1 regime near both material interfaces, tDFS∼N−1as shown\nby the green line in Fig. S3. This is because nano-EM collective dephasing is negligible, and multiqubit decoherence\nis dominated by the SQD processes. In the r≳nregime near both interfaces, nano-EM collective dephasing has\ncomparable rates as SQD near both materials. Therefore, collective dephasing strongly suppresses the decoherence of\n|ψ⟩DFSthrough destructive interference with SQD, leading to nano-EM sub-dephasing phenomena with longer tDFS.\nIn the intermediate regime, nano-EM collective dephasing can moderately suppress multiqubit decoherence of |ψ⟩DFS\nnear both materials. Meanwhile, tDFS∼N−1follows qualitatively similar behaviors as in the r≪1 regime.\nFIG. S3. Nano-EM sub-dephasing of |ψ⟩DFSnear YIG and silver interfaces. (a,b) Schematics of 2D n×nqubit arrays with\nlattice constant bat a distance zfrom (a) YIG and (b) silver films of thickness t. (c,d) Scaling of nano-EM sub-dephasing in\n|ψ⟩DFSin the three regimes corresponding to r≪1,r≳n, and the intermediate regime.\nS9. ENGINEERING NANO-EM SUPER-DEPHASING WITH META-STRUCTURES\nIn this section, we provide supplemental discussions for engineering nano-EM super-dephasing with hyperbolic meta-\nstructures. We derive that hyperbolic media can reduce the range of nano-EM super-dephasing, in stark contrast to\ntheir enhancement of the RDDI range [2]. As discussed in Ref. [19], long-range collective dephasing can be detrimental\nto quantum error correction.\nIn the main text, we consider a periodic silver grating structure with silver width p1and dielectric width p2. When\nthe grating periodicity p=p1+p2is smaller than the distance zbetween spin qubits and the grating structure ( p≪z),\nwe can obtain an approximate dielectric tensor← →εHMM for the grating structure using effective medium theory [20].\nFor the grating structure shown in the main text,← →εHMM is a diagonal tensor with diagonal elements [21]:\nεyy=εzz=p1εAg+p2εd\np1+p2,\n1\nεxx=p1/εAg+p2/εd\np1+p2,(S73)\nwhere εAgandεdare the silver and dielectric permittivity considered in the grating structure. In the low-frequency\nregime, |εAg| ≫ | εd|. Therefore, we have εyy=εzz≈p1\np1+p2εAgandεxx≈p1+p2\np2εd. This shows that the grating\nstructure exhibits properties of a type-II hyperbolic metamaterial (HMM) [21]. The Fresnel reflection coefficient rss16\nof this effective HMM is given by [22]:\nrss(q, ω) =A1B1+A2B2\nA1+A2,\nA1=1\ntanθkzok0\nq2+kzkzo,\nA2= tan θ√εyyk0/radicalig\nεxxεyyk2\n0−εxxq2x−εyyq2y+εyyk0kz\n√εyykz/radicalig\nεxxεyyk2\n0−εxxq2x−εyyq2y+k2zo,\nB1=kz−kzo\nkz+kzo,\nB2=kz−/radicalig\nεxxk2\n0−εxxq2x/εyy−q2y\nkz+/radicalig\nεxxk2\n0−εxxq2x/εyy−q2y,(S74)\nwhere k0=ω/c,q=qxˆ x+qyˆ y,q2=|q2|,kz=/radicalbig\nk2\n0−q2,kzo=/radicalbig\nk2\n0εyy−q2.θ= arccos ( q·ˆ x) is the angle between\nqandˆ x.\nSubstituting Eq. (S74) into Eq. (S32), we can obtain:\nJc(ri,ri+Dˆ x, ω→0)\nJs(ri, ω→0)∝Re/bracketleftigg\ni\nD/integraldisplay2π\n0dθ1\nsinθ+2iz\nD1− |sinθ|\n1 +|sinθ|/bracketrightigg\n, (S75)\nJc(ri,ri+Dˆ y, ω→0)\nJs(ri, ω→0)∝Re/bracketleftigg\ni\nD/integraldisplay2π\n0dθ1\ncosθ+2iz\nD1− |sinθ|\n1 +|sinθ|/bracketrightigg\n. (S76)\nfor two spin qubits with mperpendicular to the material interface and separated by interqubit distance D.\nIn the main text, we demonstrate that the range of nano-EM collective dephasing can be reduced when the qubits\nare aligned along the ˆ xdirection. This can also be derived by substituting Eqs. (S75, S76) into Eqs. (S25, S26). For\nlarge D, we have:\nts\nϕ(ri)\ntc\nϕ(ri,ri+Dˆ y)∼1\nD,ts\nϕ(ri)\ntc\nϕ(ri,ri+Dˆ x)∼1\nD2, (S77)\nwhich matches our numerical results demonstrated in the main text.\n[1] S. Fuchs, J. Crosse, and S. Y. Buhmann, Casimir-polder shift and decay rate in the presence of nonreciprocal media,\nPhysical Review A 95, 023805 (2017).\n[2] C. L. Cortes and Z. Jacob, Super-coulombic atom–atom interactions in hyperbolic media, Nature communications 8, 14144\n(2017).\n[3] L. S. Langsjoen, A. Poudel, M. G. Vavilov, and R. Joynt, Qubit relaxation from evanescent-wave johnson noise, Physical\nReview A 86, 010301 (2012).\n[4] W. Sun, S. Bharadwaj, L.-P. Yang, Y.-L. 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Belov, and Y. Kivshar, Hyperbolic metamaterials, Nature photonics 7, 948 (2013).\n[22] T. Sosnowski, Polarization mode filters for integrated optics, Optics Communications 4, 408 (1972)." }, { "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": "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": "2112.06523v1.Fluctuating_magnetic_droplets_immersed_in_a_sea_of_quantum_spin_liquid.pdf", "content": "1 Fluctuating magnetic droplets immersed in a sea of \nquantum spin liquid \nZ. H. Zhu1,∗, B. L. Pan1,∗, L. P. Nie2,∗, J. M. Ni1,∗, Y. X. Yang1, C. S. Chen1, Y. Y. Huang1, E. J. \nCheng1, Y. J. Yu1, A. D. Hillier3, X. H. Chen2,4,5, T. Wu2,4,5,†, Y. Zhou6,7,8,†, S. Y. Li1,4,9,†, & L. \nShu1,4,9,† \n \n1State Key Laboratory of Surface Physics , and Department of Physics, Fudan University, \nShanghai 200433, China \n2CAS Key Laboratory of Strongly -coupled Quantum Matter Physics, Department of Physics, \nUniversity of Science and Technology of China, Hefei, Anhui 230026, China \n3ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell Campus, \nDidcot, Oxfordshire OX11 0QX, United Kingdom \n4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China \n5CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai \n200050, China \n6Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese \nAcademy of Sciences, Beijing 100190, China \n7Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China. \n8Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological \nQuantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China. \n9Shanghai Research Center for Quantum Sciences, Shanghai 201315, China \n \n \n∗These authors contributed equally to this work. \n†Corresponding Authors. Email: wutao@ustc.edu.cn (T.W.); yizhou@iphy.ac.cn (Y.Z.); \nshiyan li@fudan.edu.cn (S.Y.L.); leishu@fudan.edu.cn (L.S.). \n \nThe search of quantum spin liquid (QSL), an exotic magnetic state with strongly -fluctuating \nand highly -entangled spins down to zero temperature, is a main theme in current condensed \nmatter physics. However, there is no smoking -gun evidence for deconfined spinons in any \nQSL candidate so far. The disorders and competing exchange interactions may prevent the \nformation of an ideal QSL state on frustrated spin lattices. Here we report comprehensive \nand systematic measurements of the magnetic susceptibility, ultra -low temp erature specific \nheat, muon spin relaxation ( µSR), nuclear magnetic resonance (NMR), and thermal \nconductivity for NaYbSe 2 single crystals, in which Yb3+ ions with effective spin -1/2 form a \nperfect triangular lattice. All these complementary techniques find no evidence of long -range 2 magnetic order down to their respective base temperatures . Instead, specific heat, µSR and \nNMR measurements suggest the coexistence of quasi -static and dynamic spins in NaYbSe 2. \nThe scattering from these quasi -static spins may ca use the absence of magnetic thermal \nconductivity. Thus, we propose a scenario of fluctuating ferrimagnetic droplets immersed in \na sea of QSL. This may be quite common on the way pursuing an ideal QSL, and provides a \nbrand- new platform to study how a QSL state survives impurities and coexists with other \nmagnetically ordered states . \n \nQuantum spin liquid (QSL) is a highly -entangled quantum state in which spins remain \ndisordered and dynamic even down to absolute zero temperature due to strong quantum \nfluctuat ions1–6. Such an exotic state was first proposed from the study of the triangular -lattice \nHeisenberg antiferromagnets in 1973 by Anderson1. Since QSL has potentially tight relationship \nwith high -temperature superconductivity7 and quantum -information applic ations8, it has gained \ncontinuous attention in condensed matter physics. The QSL s tates are characterized by fractional \nspin excitations, such as spinons, and the detection of these excitation s is a crucial issue for \nidentifying QSL in real materials2–6. Several QSL candidates have been suggested by experiments , \ntypical examples include triangular -lattice organic compounds κ- (BEDT -TTF) 2Cu2(CN) 39–11 and \nEtMe 3Sb[Pd(dmit) 2]212–14, kagome -lattice ZnCu 3(OH) 6Cl215–19, honeycomb lattice α -RuCl 320–23 \nand H 3LiIr 2O6 (ref. 24). Despite numerous efforts made in both theoretical and experimental sides, \nfinding realistic “smoking -gun” evidence for QSLs still remains the most challenging task in this \nfield. One of the obstacle s comes from the ambiguous role played by the impurities and competing \nexchange interactions: are they fatal or vital to the survival of a QSL? \nIn recent few years , the inorganic compound YbMgGaO 4, in which Yb3+ ions with effective \nspin- 1/2 form a perfect triangular lattice, was argued to have a QSL ground state25–27. Although \nmuon spin relaxation ( µSR) experiments are consistent with persistent spin dynamics and no static \nmagnetism ≳ 0.003 µB per Yb ion28,29, the absence of magnetic thermal conductivity at extremely \nlow temperature casts doubts30, and the observation of frequency -dependent peak of ac magnetic \nsusceptibility suggests a spin -glass ground state in YbMgGaO 431. It was argued that the random \noccupation between Mg2+ and Ga3+ can mimic a spin -liquid -like state32. Thus , a random spin \nsinglet s tate, or valence bond glass, w as proposed to account for the observations33–35. \nCompared with YbMgGaO 4, the family of Yb dichalcogenide delafossites NaYb(O, S, Se) 2 \nwith effective spin -1/2 is free from the Mg -Ga disorders in non- magnetic layers , thus is of crucial \nimportance to clarify whether there is a QSL ground state in clean triangular lattice of Yb3+ ions36,37. \nAll three compounds are free from long -range order (LRO) down to 50 mK determined from zero \nfield (ZF) specific heat and µSR measurements37–40. Including CsYbSe 2 with the same structure, 3 all of them have field -induced magnetic orders38,41–44. Very recently, the ground state of NaYbSe 2 \nis claimed to be a QSL with spinon Fermi surface45. Pressure -induced superconductivity is also \nobserved in Na YbSe 246,47, opening up a promising way to study the mechanism of \nsuperconductivity in QSL candidates. \nHere w e report the magnetic susceptibility, specific heat, µSR , nuclear magnetic resonance \n(NMR), and ultra -low-temperature thermal conductivity measurements on NaYbSe 2 single crystals. \nThe absence of magnetic order and spin glass is confirmed by different techniques down to 50 mK. \nWith decreasing temperature in zero field , a hump followed by a linear temperature dependent \nspecific heat is observed. In µSR and NMR measurements, both quasi -static and dynamic spins are \nfound clearly in NaYbSe 2. Furthermore , the residual linear term of thermal conductivity at all fields \nare negligible, pointing to the absence of itinerant ferm ionic magnetic excitations in NaYbSe 2. Our \ndata reveal that NaYbSe 2 hosts a ground state of fluctuating ferr imagnetic droplets immersed in a \nsea of quantum spin liquid on Yb3+ triangular lattice. \nNaYbSe 2 crystallizes in the space group R 3�m36. As shown in F ig. 1a, in the structure of \nNaYbSe 2, magnetic Yb3+ ions form flat triangular layers and each Yb3+ ion has 6- fold coordination \nwith O atoms to form a n YbO 6 octahedron which is edge -sharing with neighboring octahedrons. \nInterlinked between these flat triangular layers are sheets of pure Na atoms, which structurally \nremoves the widely discussed issue on the site mixing in YbMgGaO 4. \nThe temperature dependences of magnetic susceptibility χ of NaYbSe 2 in external magnetic \nfield µ0H = 1 T in two different directions are plotted in Fig. 1b. The absence of magnetic phase \ntransition is confirmed down to 2 K. There is no splitting between zero- field cooling (ZFC) and \nfield cooling (FC) curves of magnetic susceptibility (Fig. S2a), suggesting no spin glass in the \nsystem down to 2 K . The inset of Fig . 1b presents a Curie -Weiss (CW) fit with field perpendicular \nto the c axis. The data above 100 K can be well fitted by CW law, giving effective moment µ eff = \n4.54 µB and the CW temperature Θ CW = −49.0 K. The value of µ eff agrees with the theoretical \nprediction 4.54 µB for trivalent Yb3+ ion with J = 7/2. Compared with the CW temperature of \nYbMgGaO 4 (−4 K)25, ΘCW is much larger in NaYbSe 2. Similar to YbMgGaO 425, magnetization M \nremains unsaturated but is smaller up to 7 T at 2 K in NaYbSe 2 (Fig. S2b). The larger absolute \nvalue of the CW temperature and smaller M indicate stronger AFM interactions in NaYbSe 2. \nThe temperature dependence of specific heat of NaYbSe 2 in various fields (H || c) from 0.05 \nto 20 K are shown in F ig. 1c. Consistent with magnetic susceptibility and former reports43,45, no \nsharp anomaly of LRO is observed in NaYbSe 2. With decreasing temperature, a broad hump of \nspecific heat shows up, whose position shifts to higher temperature in magnetic field. H owever, \nwe do not observe the field-induced transition peak reported previously, due to the lack of the 4 sufficiently strong field43. For the ZF data, after subtracting the contributions from phonon and \nnuclear Schottky anomaly (Fig. S3), we obtain the magnetic contribution of specific heat C Mag/T \nas shown in Fig. 2d. As guided by the red dashed line, the temperature independent behavi or of \nCMag/T below 0.25 K is consistent with a spinon Fermi surface45. By integrating C Mag/T, we obtain \nthe magnetic entropy S Mag, as shown in Fig. 2d. For an effective spin- 1/2 system, the theoretical \nmagnetic entropy is Rln2, where R is the gas constant. The residual entropy of NaYbSe 2 at 50 mK \nis only 5.2% of total entropy. Such little entropy remaining suggests low temperature physics is \ndominated by quantum fluctuations rather than thermal fluctuations, indicating the existence of \nQSL. \nBoth µSR and NMR , which measure spin dynamics at different frequency ranges, are \npowerful tools in clarifying the static and /or dynamic nature of the magneti c ground state . µSR, \nwhich uses muon as a probe , is more sensitive to local magnetic field28,48–51. As shown in Fig. 2a, \nthe time spectra of muon polarization P (t) in ZF clearly indicates that LRO is absent in ZF -µSR \ndown to 88 mK. The relaxation process of ZF -µSR can be well described by the sum of a Kubo-\nToyabe (KT) term and an exponential term: \nP (t) = f G KT(σ, t) + (1 − f ) exp( −λt), (1) \nwhere f is the fraction of the KT term. The fitting function is exactly the same as in the NaYb S2 \ncase40. The KT term originates from an isotropic Gaussian distribution of randomly oriented static \nor quasi - static local fields, whose relaxation rate σ is proportional to the root -mean -square (rms) \nwidth of the distribution48. The exponential term with relaxatio n rate λ, originates from dynamic \nspins. The successful fitting with the above function strongly suggests the coexistence of \ndistinguishable quasi -static spins and dynamic spins. \nThe temperature dependence of f , σ, and λ are plotted in Fig s. 2b, and 2c. At high temperature, \nthe value of f is equal to 1, indicating a trivial paramagnetic state. With decreasing temperature \nbelow 20 K , f decreases continuously due to the role of magnetic exchange interaction and \nsignificant spin dynamic appears which comes t o the second term in Eq. (1). Above 10 K, the \nsystem remains in a paramagnetic state without quasi -static spins, which is also supported by a \ntemperature- independent NMR intensity (see Fig. S6b and related discussion in Supplementary \nMaterials ). Below 6 K , the temperature- dependent f deviates from the descending behavior and \nshows an upturn behavior , while the temperature -dependent σ also shows a clear increasing \nbehavior below 6 K . These results strongly suggest the formation of quasi -static spins at low \ntemperatures . However, both σ and f saturates to a finite value at low temperature s, and t he \nsaturation value of f indicates that only 23% of the sample becomes quasi -static at base temperature. \nOn the other hand, the temperature -dependent λ also exhibits an increasing behavior below 4 K, 5 supporting the enhancement of spin dynamics at low temperatures . The temperature independent \nbehavior of λ below 0.2 K suggests the existence of persistent spin dynamics. Additional evidence \nof the coexistence of quasi -static and dynamic spins in NaYbSe 2 comes from longitudinal field \n(LF) µSR, which yields that the fluctuation rate ν c at 0.1 K is 2.8 MHz (Fig. S4), larger than 1.7 \nMHz in NaYbS 2 (ref. 40) . \nSimilar evidence for the coexistence of quasi -static an d dynamic spins is also found in 23Na \nNMR experiments. As shown in Fig. 3a, the three- peak structure of 23Na NMR spectra at high \ntemperature comes from quadrupole splitting of nuclei with spin number I = 3/2 (Fig. 3a). With \ndecreasing temperature, a remarkable broadening of line shape occurs below 10 K, suggesting the \nformation of short -range spin correlations52. Meanwhile, t he NMR spectra, besides a group of three \nrelatively sharp peaks , start to develop a broad Gaussian background as a new component (Fig. \nS5a), which should be ascribed to the quasi -static spins as suggested by ZF -µSR. As shown i n Fig. \n3b, t he temperature dependence of full width at half maximum (FWHM) shows a similar \nincreasing behavior for these two components below 10 K, which indicates a close correlation \nbetween these two components beyond simple competition. In addition, the quasi -static moment \ncan be estimated from the broad part of NMR spectra, yielding a small value of 0.15 µ B (Fig. S6). \nThis result indicates that the quas i-static spins in NaYbSe 2 are still fluctuating which is sharp \ncontrast to traditional spin glass53,54. \nBesides NMR spectrum, the nuclear spin -lattice relaxation also supports the coexistence of \nquasi -static and dynamic spins. An inhomogeneous spin dynamic s is indeed observed below 2 K \naccompanied by the above two -component behavior in spectrum. As shown in Fig. S5 c, the \nstretching exponent β, which usually depicts the inhomogeneity of spin dynamics, shows a clear \ndecreasing below 2 K with the value well below one. Especially, at the lowest temperature of 0.25 \nK, there is a clear two -component behavior appearing in the recovery curve of T 1 process (Fig. \nS5b), which is beyond a single T 1 fitting with stretching exponent . This is in line with the scenario \nproposed above with the coexistence of quasi -static and dynamic spins . Finally, the temperature \ndependence of the spin- lattice relaxation rate 1 /T1 extracted from the stretched exponential fitting \nis plotted in Fig. 3c. The broad hump feature around 50 K is usually ascribed to the development \nof strong spin correlation at low temperature or crystal electric field (CEF) effect41. The absence \nof magnetic order is confirmed again by the absence of any significant critical fluctuation at low \ntemperatures . Below 2 K, 1/ T1 saturates to a constant, coinciding with the persistent spin dynamics \nobserved in µSR experiments. This result also excludes the possibility of a trivial spin glass phase, \nsuggesting a novel magnetic ground state in NaYbSe 2. \nTo further check the existence of gapless magnetic excitations, we perform the thermal 6 conductivity measurements to probe the possible itinerant e xcitations. As for a QSL candidate, \nthermal conductivity measurement is highly advantageous in probing such elementary excitations, \nsince it is only sensitive to itinerant excitations. In a solid, the contributions to thermal conductivity \ncome from various quasi -particles, such as phonons, electrons, magnons, and spinons. Since \nNaYbSe 2 is an insulator, electrons do not contribute to the thermal conductivity at ultra -low \ntemperatures. Additionally, the contribution of magnons can be ruled out due to the abse nce of \nmagnetic order down to 50 mK. Therefore, thermal conductivity κ at ultra -low temperatures can \nbe described by the formula \n \nκ = aT + bTα, (2) \n \nwhere aT and bTα represent the contribution of possible itinerant gapless fermionic magnetic \nexcitations and phonons, respectively55,56. Due to the specular reflections of phonons at the sample \nsurfaces, the power α in the second term is typically between 2 and 355,56. The experiment results \nin ZF are shown in Fig. 4. The fitting to the data below 0.4 K gives the residual linear term κ0/T ≡ \na = −0.038 ± 0.007 mW K−2 cm−1, and α = 1.66 ± 0.04. This behavior is very similar to \nYbMgGaO 430, suggesting the absence of itinerant gapless magnetic excitations (Fig. S7). \nWe now turn to discuss the ground state of NaYbSe 2. The absence of LRO and spin glass in \nNaYbSe 2 is confirmed down to 50 mK. Both µSR and NMR experiments point out that a minority \nof quasi -static spins and a majority of dynamic spins coexist in NaYbSe 2 down to the base \ntemperature. In fact, our spe cific heat measurements also hint at this picture. The broad hump of \nCMag/T around 0.8 K comes from the correlations of quasi -static spins, while the temperature \nindependent behavior below 0.25 K suggests the existence of well -defined magnetic excitations, \nwhich is an essential feature of gapless QSLs24,37,44. Comparing our results in ZF, these two \ncharacteristic temperatures coincides with our µSR data astonishingly. As for the thermal \nconductivity measurements, the dynamic spins should result in a finite residual linear term κ 0/T in \nNaYbSe 257–59. However, the gapless spinons may be strongly scattered by the quasi -static spins, \nleading to a negligible κ 0/T. \nWe propose here a possible picture of mixed state of fluctuating short -range ferrimagnetic \ndroplets and QSL. As for the minority quasi -static spins, there are no long- range but at least short -\nrange correlations between them. They are not static like spin glass, and our NMR result suggests \nthat they are still fluctuating. They only take 23% of t he total spins, suggesting they distribute in \nthe system like droplets. When it comes to the dynamic spins, they remain disordered and \nfluctuating down to our base temperature, exactly matching the definition of QSL. Additionally, \nthere is only less than 5 .2% residual entropy at zero temperature, also indicating the presence of 7 QSL. \nNow it brings us to why other methods like magnetic susceptibility and neutron scattering \ndo not observe such ferrimagnetic droplets36,45. For magnetic susceptibility technique, it is more \nsensitive to slower fluctuations, whose limit is about 104 Hz, while NMR and µSR are more \nsensitive to faster fluctuations. Hence a state could be dynamic in magnetic susceptibility \nmeasurements, but quasi -static in NMR and µSR . In inelastic ne utron scattering experiments, such \nfluctuating droplet s could also mimic spinon continuum due to its randomness, making it difficult \nto differentiate. \nThe ferrimagnetic droplet s immersed in a sea of QSL is illustrated in Fig. 5 , on which an up -\nup-down magnetic structure forms within each droplet in accordance with field-induced magnetic \norder s in Yb3+ compounds on triangular lattice38,41 –44. It is natural to assume that such a \nferrimagnetic ordering state has slightly higher energy than the QSL state, w hich allows the \nnucleation of ferrimagnetic droplet around a defect. Meanwhile, the thermal fluctuation of these \nmagnetic droplets will give rise to the residual entropy . The ratio between the residual entropy and \nthe total magnetic entropy is estimated to be r ln(1+m/3)\n𝑚𝑚 ln 2, where r is the volume fraction of droplets \nand each droplet carries m Yb3+ ions. It is expected that the size of the fluctuating droplets will be \nenhanced by an external magnetic field, resulting in a long ranged up- up-down magnetic order in \nbulk when the applied magnetic field exceeds some threshold38,41 –44. \nIn summary, we present specific heat, µSR , NMR, and thermal conductivity measurements \non triangula r-lattice compound NaYbSe 2 single crystals to figure out its ground state. The absence \nof long -range magnetic order and spin glass is confirmed down to 50 mK. Specific heat, µSR and \nNMR measurements all find a majority of dynamic spins and a minority of quasi -static spins mixed \nin NaYbSe 2, which is further supported by thermal conduc tivity measurements. The ground state \nof NaYbSe 2 can be regarded as a mixed state with both QSL and fluctuating short -range \nferrimagnetic droplets, providing a platform to study how disorder influence the QSL state. \n \nReferences \n1. Anderson, P. W. Resonating valence bonds: A new kind of insulator? 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Lett. 95, 036403 (2005). \n59. Werman, Y., Chatterjee, S., Morampudi, S. C. & Berg, E. Signatures of Fractionalization in \nSpin Liquids from Interlayer Thermal Transport. Phys. Rev. X 8, 031064 (2018). \n60. Schleid, T. & Lissner, F. Single crystals of NaMS 2 (M: Ho -Lu) from reactions of the \nLanthanoids with Sulfur in the presence of NaCl. Eur. J. Solid State Inorg. Chem. 30, 829 \n(1993). 12 61. Suter, A. & Wojek, B. Musrfit: A Free Platform -Independent Framework for µSR Data \nAnalysis. Phys. Procedia 30, 69–73 (2012). \n \n \nMethods \n \nSample preparation High -quality NaYbSe 2 single crystals were grown by a modified flux method \nfollowing Schleid and Lissner59. Analytically pure Yb powder, Se powder and NaCl as flux in a \nmolar ratio of 2:3:90 were sealed in a quartz tube and heated to 950 ◦C for 7 days, followed by a \nmaintainin g at 950 ◦C for 7 days. The mixture was slowly cooled down to 600 ◦C at a rate of 50 ◦C \nper day. In the end, reddish black platelets with largest size of 7- 8 mm, as shown in the inset of \nFig. S1, were separated by dipping in water. The large natural surfac e was determined to be the \n(001) plane by X -ray diffraction (XRD), as illustrated in Fig. S1 and no impurity phases were \nobserved, indicating a relatively high crystallization quality. \nMagnetic measurements The magnetic susceptibility measurements were per formed in \ncommercial SQUID and the specific heat was measured in the physical property measurement \nsystem (PPMS) (Quantum Design) by the relaxation method. \nµSR measurements In a µSR experiment, a beam of nearly 100% spin polarized muon is \nimplanted into th e sample. Muon spin precesses and relaxes due to inhomogeneous local magnetic \nfield. One can measure the time spectra of muon spin polarization, and the relaxation process can \nreveal the distribution of local field27,48,49. Besides, muon is extremely sensi tive to small field, \nwhich is a powerful technique to check the essence of magnetic order50. ZF and LF µSR \nmeasurements were performed down to 88 mK on MuSR spectrometer at ISIS, Rutherford \nAppleton Laboratory, Chilton, UK. Single crystals of NaYbSe 2 were aligned so that the c axis was \nnormal to the sample’s planar surface and parallel to the initial muon spin polarization, and \nmounted onto a silver holder covering a circle area of 1 inch in diameter, and 3 mm in thickness. \nµSR data were analyzed using the MANTID PROJECT and MUSRFIT software package60. Subtracting \nthe constant background signal due to silver sample holder, ZF muon spin polarization spectra P (t) \ncan be described by the formula P (t) = f GKT(σ, t) + (1 − f ) exp(−λt), where \n𝐺𝐺KT=1\n3+2\n3(1−𝜎𝜎2𝑡𝑡2)exp �−1\n2𝜎𝜎2𝑡𝑡2� is the Kubo- Toyabe (KT) function47. \nNMR measurements The 23Na NMR measurements are taken on one piece of NaYbSe 2 single \ncrystal with the mass of 2.6 mg. Because the nuclear gyromagnetic ratios for 23Na \n(γNa = 11.2625 MHz /T) is very close to 63Cu (γCu = 11.285 MHz /T), we chose the Ag wire to wind \nNMR coil. In order to define the exact external magnetic field, we fill a small piece of Al foil into 13 the coil. The NMR spectra are obtained by the fast Fourier transformation (FFT) sum of the \nstandard spin- echo signals. The linewidth is extracted from Gauss fitting. Especially, when fitting \nthe spectra of dynamic part between 0.25 K and 1.5 K, we fixed the linewidth of the central line \nand the satellite line to be the same. The nuclear spin -lattice rel axation rate 1 /T1 is measured by \nsaturation method below 1.5 K and inverse method for high er temperature. The recovery curve of \nthe nuclear magnetization M (t) is fitted with the function \n1−𝑀𝑀(𝑡𝑡)\n𝑀𝑀(∞)=𝐼𝐼0�0.1exp �−�𝑡𝑡\n𝑇𝑇1�𝛽𝛽\n�+0.9exp[−�6𝑡𝑡\n𝑇𝑇1�𝛽𝛽\n]�. The error bars are determined by the least \nsquare method. \nThermal conductivity measurements The single crystal selected for the thermal conductivity \nmeasurements was a rectangular shape of dimensions 5.38 × 1.3 mm2 in the ab plane, with a \nthickness of 0.04 mm along the c axis. The thermal conductivity was measured in a dilution \nrefrigerator, using a standard four wire steady -state method with two RuO 2 chip thermometers, \ncalibrated in situ against a reference RuO 2 thermometer. Magnetic fields were applied along the c \naxis for specific heat and thermal conductivity measurements and perpendicular to the heat current \nin the thermal conductivity measurements. \nResidual entropy Assuming each ferrimagnetic droplet carries m spin- 1/2 ( 𝐽𝐽𝑒𝑒𝑒𝑒𝑒𝑒=1/2 local \nmoment) and the fractional volume of the ferrimagnetic droplets is r , then the effective spin of \neach droplet is 𝑆𝑆 = 𝑚𝑚\n3×1\n2, which gives rise to the upper bound of the residual entropy \nR ln�1+2𝑆𝑆�=R ln (1+m/3). Thus, the ratio between the residual entropy and the total \nmagnetic entropy at high temperatures has an upper bound of r ln(1+m/3)\n𝑚𝑚 ln 2. \n \nData availability \n \nAll data needed to evaluate the conclusions in the paper are present in the main text or the \nsupplementary information. \n \nAcknowledgments \n \nWe thank Y. Xu for helpful discussions . This research was funded by the National Natural Science \nFoundations of China (Grant No. 12034004, No. 11774061, and No. 11774306) , the Shanghai \nMunicipal Science and Technology (Major Project Grant No. 2019SHZDZX01 and No. \n20ZR1405300), the National Research and Development Program of China, No. \n2016YFA0300503, and the Strategic Priority Research Program of Chinese Academy of Sciences \n(No. XDB28000000) . 14 \nAuthor Contributions \n \nL.S. and S.Y.L. planned the project. B.L.P. synthesized and characterized the sample. Z.H.Z., Y.X.Y. \nand C.S.C. carried out the µSR experiments with experimental assistance from A.D.H. L.P.N. carried \nout the NMR experiments. B.L.P., J.M.N., Y.Y.H., E.J.C. and Y.J.Y. performed the thermal \nconductivit y measurements. L.S., S.Y.L., T.W., Z.H.Z., B.L.P., L.P.N. and J.M.N. analyzed the data. \nY.Z. provided the theoretical explanation. L.S., S.Y.L., T.W., Y.Z., X.H.C., Z.H.Z., B.L.P. and \nL.P.N. wrote the paper. 15 \n \nFigure 1: Basic properties of NaYbSe 2. (a) The u nit cell of NaYbSe 2. Green spheres: Na. Red \nspheres: Yb. Blue spheres: Se. (b) The t emperature dependence of magnetic susceptibility at \nµ0H = 1 T of NaYbSe 2. The inset shows the fitting result of Curie -Weiss law at T > 100 K. (c) The \ntemperature dependence of specific heat C at µ0H = 0, 4, 8 T. (d) The m agnetic specific heat C Mag/T \nand the calculated magnetic entropy S Mag at zero field. The red dashed line is a guide to eyes to \nshow that C Mag/T is temperature independent at low temperature. 16 \n \nFigure 2: Ze ro-field µSR experiment. (a) Time spectra of ZF -µSR at representative temperatures. \nThe curves are the fittings using Eq. 1. (b) The t emperature dependence of fraction of quasi -static \nspins. (c) The t emperature dependence of relaxation rate due to quasi -static and dynamic spins σ \n(blue spheres), and λ (red spheres), respectively. \n 17 \n \nFigure 3: 23Na NMR experiments at 4.5 T. (a) NMR spectra of 23Na nuclei with external field \nµ0H = 4.5 T parallel to c-axis. The sharp peak guided by the black dashed line is the 27Al peak \nwhich is used to calibrate the magnetic field. (b) The t emperature dependence of NMR linewidth \nderived from the spectra as described in SM. The quasi -static (blue spheres) and dynamic (red \nspheres) components can be easily separated. (c) The t emperature dependence of the spin- lattice \nrelaxation rate 1 /T1. 18 \n \nFigure 4: In -plane thermal conductivity at zero field. The red solid line is the fit to the data \nbelow 0.4 K using Eq. 2. \n \nFigure 5 : Magnetic droplets immersed in a sea of quantum spin liquid. Each droplet has an up-\nup-down ferrimagnetic structure . \n1 Supplementary Information for \nFluctuating magnetic droplets immersed in a sea of \nquantum spin liquid \nZ. H. Zhu1,∗, B. L. Pan1,∗, L. P. Nie2,∗, J. M. Ni1,∗, Y. X. Yang1, C. S. Chen1, Y. Y. Huang1, E. \nJ. Cheng1, Y. J. Yu1, A. D. Hillier3, X. H. Chen2,4,5, T. Wu2,4,5,†, Y. Zhou6,7,8, †, S. Y. Li1,4,9,†, & \nL. Shu1,4,9,† \n \n1State Key Laboratory of Surface Physics , and Department of Physics, Fudan University, \nShanghai 200433, China \n2CAS Key Laboratory of Strongly -coupled Quantum Matter Physics, Department of Physic s, \nUniversity of Science and Technology of China, Hefei, Anhui 230026, China \n3ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell Campus, \nDidcot, Oxfordshire OX11 0QX, United Kingdom \n4Collaborative Innovation Center of Advance d Microstructures, Nanjing 210093, China \n5CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai \n200050, China \n6Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese \nAcademy of Sciences, Beijing 100190, China \n7Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China. \n8Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological \nQuantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China. \n9Shanghai Research Center for Quantum Sciences, Shanghai 201315, China \n \n∗These authors contributed equally to this work. \n†Corresponding Authors. Email: wutao@ustc.edu.cn (T.W.); yizhou@iphy.ac.cn (Y.Z.); \nshiyan li@fudan.edu.cn (S.Y.L.); leishu@fudan.edu.cn (L.S.). 2 1 X-ray diffraction \nRoom -temperature X -ray diffraction pattern from the largest natural surface of sing le \ncrystalline NaYbSe 2 is plotted in Fig. S1. Only (00 l) peaks are observed, demonstrating that the \nlargest natural surface is (001) plane. A photo of NaYbSe 2 single crystal is also shown in the inset \nof Fig. S1. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S1: X -ray diffract ion pattern of single crystalline NaYbSe 2. Inset: photograph of a NaYbSe 2 \nsingle crystal. \n \n \n2 Magnetic properties \nTo check if there is any evidence of spin glass, we performed zero -field cooling (ZFC) and \nfield cooling (FC) measurements of magnetic susceptib ility. As plotted in Fig. S2a, in 1 T external \nfield, no splitting between ZFC and FC curve s of magnetic susceptibility is observed with field \nparallel and perpendicular to the ab- plane, indicating the absence of spin glass. \n3 We also measured the field dependence of magnetization at 2 K with field parallel and \nperpendicular to the ab- plane. Both curves show no sign of saturation up to 7 T. Although it was \nreported that a 4 T field in the ab-plane can induce magnetic order1, the transition temperature is \nabout 1 K which is below the temperature range of our experiment. \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S2: Magnetic properties of NaYbSe 2. (a) The temperature dependence of magnetic \nsusceptibility at µ 0H = 1 T (|| ab and || c) with FC and ZFC. (b) The field dependence of the \nmagnetization at 2 K with µ 0H || ab and || c . \n4 3 Specific heat \nThe analysis process of the specific heat C in zero field is shown in Fig. S3. First, we subtract \nthe lattice contribution from the total specific heat as shown in Fig. S3a. The lattice contribution \nis obtained from the specific heat result from the work of Dai et al.2. \n \n \n \n \n \n \n \n \n \n \nFigure S3: Specific heat results o f NaYbSe 2. (a) The temperature dependence of the total and lattice \nspecific heat in zero field. (b) The fitting of the nuclear Schottky anomaly. Blue spheres: total \nspecific heat subtracting the lattice contribution. Red curve: the fitting curve using Eq. S1. Red \ndashed line: the T -linear term in the fitting function. Grey spheres: the magnetic contribution as \nshown in Fig. 1d in the main text. \n5 We now consider the ultra low-temperature specific heat. Based on the work of Dai et al.2, a \nT-linear specific heat is expected after subtracting nuclear Schottky anomaly contribution. We fit \nthe data below 0.25 K using a T -linear term plus a two -state Schottky anomaly term: \n 𝐶𝐶 = 𝛾𝛾𝛾𝛾 + 𝑛𝑛𝐶𝐶Schottky , (S1) \nwhere \n 𝐶𝐶Schottky =𝑅𝑅𝑔𝑔0\n𝑔𝑔1�Δ\n𝑘𝑘B𝑇𝑇�2exp (Δ/𝑘𝑘B𝑇𝑇)\n�1+𝑔𝑔0\n𝑔𝑔1exp�Δ\n𝑘𝑘B𝑇𝑇� �2 . (S2) \nHere, R is the gas constant, and k B is the Boltzmann constant. The fitting result is shown as the red \nsolid curve in Fig. S3b. The fitting yields γ = 1.62 J mol−1 K−2 shown as the red horizontal dashed \nline, the value of which is close to previous report2. The area between the red solid curve and \ndashed line represents the Schottky anomaly term. In the Schottky anomaly term, g 0/g1 is the ratio \nbetween degeneracies of the ground and excited state, which is set to 1 to simplify the fitting, n = \n0.026 is the fraction of atoms that induce nuclear Schottky anomaly, and ∆ = 9.03 µeV is the energy \ndifference between the two states, respectively. Subtracting the obtained nuclear Schottky anomaly \nterm, we get the magnetic contribution C Mag/T as shown in Fig. 1d in the main text or the grey \nspheres shown in Fig. S3b. The calculated magnetic entropy S Mag shown in Fig. 1d is derived by \nintegrating CMag/T. We assume that the spins are totally free above 20 K, meaning that S Mag is Rln2 \nat 20 K, and then derive the residual entropy at base temperature is 5.2% of Rln2. \n4 Longitudinal field muon spin relaxation \nRepresentative longitudinal field µSR (LF -µSR) at 0.1 K in various fields µ 0H along c-axis \nare shown in Fig. S4a. In LF -µSR experiments, with field increasing, the muon spin is gradually \ndecoupled from the static and quasi -static local field, and affected by the external field3. In the time \nspectra, this effect displays as a suppression of the relaxation . But the relaxation due to dynamic \nfield is much more robust. Thus we can differentiate the quasi -static and dynamic field. The \nconstant background derived from the fitting of ZF -µSR is also subtracted. The LF polarization \nspectra P (t) can be well described by the formula \n 𝑃𝑃(𝑡𝑡)=𝑓𝑓𝐺𝐺LFKT(𝜇𝜇0𝐻𝐻,𝜎𝜎,𝑡𝑡)+(1−𝑓𝑓)𝑒𝑒−𝜆𝜆𝜆𝜆, (S3) 6 where f and σ are fixed at the value derived from the fitting of ZF -µSR at 0.1 K, and \n 𝐺𝐺LFKT(𝜇𝜇0𝐻𝐻,𝜎𝜎,𝑡𝑡)=1−2𝜎𝜎2\n�𝛾𝛾𝜇𝜇𝜇𝜇0𝐻𝐻�2�1−𝑒𝑒−1\n2𝜎𝜎2𝜆𝜆2cos�𝛾𝛾𝜇𝜇𝜇𝜇0𝐻𝐻𝑡𝑡�� \n +2𝜎𝜎4\n�𝛾𝛾𝜇𝜇𝜇𝜇0𝐻𝐻�3∫𝑒𝑒−1\n2𝜎𝜎2𝜏𝜏2\nsin�𝛾𝛾𝜇𝜇𝜇𝜇0𝐻𝐻𝜏𝜏�d𝜏𝜏𝜆𝜆\n0, (S4) \nis the static LF -KT function, where γ µ/2π = 135.54 MHz/T is the gyromagnetic ratio of muon3. \nSince all other parameters are fixed, the only derived parameter is the dynamic relaxation rate λ . \nThe field dependence of λ at 0.1 K is shown in Fig. S4b. Normally, the field dependence of λ can \nbe described by the conventional Redfield formula which supposes the time correlation function \nof dynamic spin takes a simple exponential Markovian form4 \n 𝜆𝜆=�𝛾𝛾𝜇𝜇𝐵𝐵locrms�2𝜏𝜏𝑐𝑐\n1+�𝛾𝛾𝜇𝜇𝜇𝜇0𝐻𝐻𝜏𝜏𝑐𝑐�2, (S5) \nwhere 𝐵𝐵locrms is the root -mean -square (r ms) of local fields, and τ c is the correlation time. However, \nsimilar to YbMgGaO 4 and NaYbS 2, the field dependence of λ cannot be described by the Redifield \nformula5,6. Instead, the correlation function takes a more general form S( t) ∼ (τ/t)x exp(−νt), whe re \nτ and 1/ν are the early and late time cutoffs, respectively4. When x = 0, the correlation function \ndegenerates back into the Markovian case. Such a more general correlation function yields4 \n 𝜆𝜆(𝐻𝐻)=2Δ2𝜏𝜏𝑥𝑥∫𝑡𝑡−𝑥𝑥exp(−𝜈𝜈𝑡𝑡)cos(𝛾𝛾𝜇𝜇𝜇𝜇0𝐻𝐻𝑡𝑡) d𝑡𝑡∞\n0 (S6) \nAs shown in Fig. S4b, the fitting of the field dependence of λ to Eq. S6 gives x = 0.39 and ν = 2.8 \nMHz, which is close to the NaYbS 2 case ( x = 0.44 and ν = 1.7 MHz)6, and comparable to \nYbMgGaO 4 (x = 0 .66 and ν = 9.4 MHz)5. \n \n \n \n \n 7 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S4: LF -µSR results at 0.1 K with µ 0H parallel to c -axis. (a) Time spectra of polarization of \nLF-µSR at 0.1 K. The curves are the fits to the data with Eq. S4. (b) Field dependence of the \ndynamic relaxation rate λ at 0.1 K. The red curve is the fit with Eq. S6. \n5 Nuclear magnetic resonance \nThe fitting details to the 23Na NMR spectra below 10 K are shown in Fig. S5a. The spectra at low \ntemperatures can be decomposed into two components. One is a broad Gaussian background \nowing to the quasi -static spins, and the other is three narrow peaks which stand for the dynamic \nspins. The fittings to the recovery curve are plotted in Fig. S5b. The most of recovery curves at \n8 different temperatures can be fitt ed by a standard T1 formula with a stretch ing exponent β which \nusually depicts the inhomogeneity of spin dynamics8. The temperature dependence of the \nstretching exponent β is shown in Fig. S5c. The decrease of β below 2 K suggests a strong \ninhomogeneity in NaYbSe 2. It should be noted that, with the continuous decrease of β , two \nrelaxation process es with different relaxation rates can be observed at the lowest temperature of \n0.25 K, supporting a coexistence of quasi -static and dynamic components . The blue and green \ndashed curves are the quasi -static an d dynamic components, respectively. In addition, the broad \nhump around 50 K in 1/T 1 could be the indicator for the development of strong spin correlation at \nlow temperatures or the crystal electric field (CEF) effect in NaYbSe 27. \n \n \n \n \n \n \n \n \n \n \n \nFigure S5: The details for the fitting of NMR spectra and recovery curves . (a) Fitting of the 23Na \nspectra with two components at low temperature s. Three green narrow peaks stand for the dynamic \ncomponent, and the blue broad peak represents the quasi -static component. The red curve is the \ntotal fitting to the raw data. (b) The recovery curve of nuclear magnetization . At 0.25 K, a \nremarkable two -component behavior appears, which is also ascribed to the coexistence of quasi -\n9 static and dynamic spins. The blue and green das hed curves represent the quasi -static and dynamic \ncomponents in the total fitting . (c) The temperature dependence of the stretching exponent β. \nTo further figure out the nature of the observed quasi -static spins , we have analyzed the \nKnight shift in more detail s. In single spin component system, the Knight shift is proportional to \nthe bulk magnetic susceptibility χ: \n 𝐾𝐾(𝛾𝛾) ~ 𝐴𝐴𝑠𝑠𝜒𝜒(𝛾𝛾) (S7) \nIn this case, K − χ plot can be used to estimate the magnetic hyperfine coupling tensor A s. As shown \nin Fig. S6a , by analyzing the K − χ plot, we find that the value of A s is estimated to be −0.095 T/µB \nin the temperature range from 268 and 98 K. Then the val ue of A s changes to − 0.4115 T/µB in the \ntemperature range from 98 and 5 K. The change of As around 98 K can be ascribed to the CEF \neffect7. After know ing the value of A s, we try to estimate the magnitude of the quasi -static spins. \nThe line width of the quas i-static component at 0.25 K is about 0.7 MHz. Using the gyromagnetic \nratio γ Na = 11.2625 MHz/T, we can get the distribution of internal field at Na site is about 0.062 T. \nSuch a distribution of internal field should come from the quasi -static component of Yb3+ ion . \nHence we can estimate that the moment of the quasi -static component is about 0.15 µ B, which is \nmuch smaller than µ eff = 0.5 µB with H || c estimated from low -temperature magnetization curves . \nThis result indicates the fluctuating nature of the quasi -static spins. \nIt should be noted that the wipe -out effect in NMR spectrum might lead to a n underestimation \nof the line width. To verify the wipe -out effect on line width , the temperature -dependent integral \narea of 23Na NMR full spectra multiplied by temperatu re is plotted in Fig. S6b. Usually , if there is \nno wipe -out effect or intensity loss, it means that the magnetic system should be still in a \nparamagnetic state. As shown in Fig.S6b, t here is no detectable wipe -out effect above 2 K, \nconfirming our conclusi on on a trivial paramagnetic state above 6 K from our µSR results (Fig. 2). \nBelow 2 K, a clear wipe- out effect appears before taking consideration of the T2 effect . In order to \nfurther check the T2 effect, w e have measured T2 at the frequency corresponding to the central line \nof the dynamic component at 0.25 K. As shown by the red point in Fig. S6b, although the precise \nmeasurement on T 2 is not trivial for an inhomogeneous system, it is quite clear that the intensity \nloss is mainly caused by the T2 effect instead of st atic spin order . Anyway, the limited wipe -out \neffect rules out a conventional spin glass in NaYbSe 2. 10 \n \n \n \n \n \n \n \n \n \n \nFigure S6: Analysis on h yperfine coupling tensor and wipe out effect . (a) The K − χ plot. The red \ndash line is the fitting for hyperfine coupling tensor by using the data between 268 and 98 K . The \ngreen dash line is the fitting for hyperfine coupling tensor by using the data between 98 and 5 K. \n(b) The temperature dependence of the integral area of 23Na NMR full spectra multiplied by \ntemperature. The value is normalized to that at 30 K. The red point the revised data point at 0.25 \nK by considering the correction of T 2 effect . \n6 Thermal conductivity \nThe temperature dependence of in -plane thermal conductivity κ/T in various fields (µ0H || c) \nare shown in Fig. S7a. The residual linear term κ 0/T is virtually zero in all fields, indicating the \n11 absence of itinerant gapless magnetic excitations. Fig. S7b plots the field dependence of the κ/ T at \n0.2, 0.3, and 0.4 K. For µ0H < 3 T, κ/T is independent of fields. With increasing fields, the spins \nare increasingly polarized, thus reducing the scattering of phonon, leading to the rapid \nenhancement of thermal conductivity from 3 to 5 T. \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S7: In-plane t hermal conductivity of NaYbSe 2. (a) In-plane t hermal conductivity of \nNaYbSe 2 at various fields (µ0H || c). (b) The field dependence of κ/ T at 0.2, 0.3, and 0.4 K. \nReferences \n1. Ranjith, K. M. et al. Anisotropic field- induced ordering in the triangular -lattice q uantum spin \nliquid NaYbSe 2. Phys. Rev. B 100, 224417 (2019). \n12 2. Dai, P. -L. et al. Spinon Fermi Surface Spin Liquid in a Triangular Lattice Antiferromagnet \nNaYbSe 2. Phys. Rev. X 11, 021044 (2021). \n3. Hayano, R. S. et al. Zero - and low -field spin relaxation studie d by positive muons. Phys. Rev. \nB 20, 850–859 (1979). \n4. Keren, A., Bazalitsky, G., Campbell, I. & Lord, J. S. Probing exotic spin correlations by muon \nspin depolarization measurements with applications to spin glass dynamics. Phys. Rev. B 64, \n054403 (2001). \n5. Li, Y . et al. Muon Spin Relaxation Evidence for the U(1) Quantum Spin- Liquid Ground State \nin the Triangular Antiferromagnet YbMgGaO 4. Phys. Rev. Lett. 117, 097201 (2016). \n6. Sarkar, R. et al. Quantum spin liquid ground state in the disorder free triangular la ttice NaYbS 2. \nPhys. Rev. B 100, 241116 (2019). \n7. Ranjith, K. M. et al. Field -induced instability of the quantum spin liquid ground state in the \n triangular -lattice compound NaYbO 2. Phys. Rev. B 99, 180401 (2019). \n8. Mitrovic, V . F. et al. Similar glassy featur es in the 139La NMR response of pure and disordered \nLa1.88Sr0.12CuO 4. Phys. Rev. B 78, 014504 (2008). \n \n" }, { "title": "2308.11128v2.Structural__morphological__and_magnetic_characterizations_of__Fe0_25Mn0_75_2O3_nanocrystals__a_comprehensive_stoichiometric_determination.pdf", "content": "1Structural, morphological, and magnetic characterizations of (Fe 0.25Mn 0.75)2O3\nnanocrystals: a comprehensive stoichiometric determination\nJohn C. Mantillaa,b*, Luiz C. C. M. Nagaminea*, Daniel R. Cornejoa, Renato Cohena, Wesley de Oliveirac, \nPaulo E. N. Souzab, Sebastião W. da Silvab, Fermin F. H. Aragónb, Pedro L. Gasteloisd; Paulo C. Moraisb,e, \nJosé A. H. Coaquirab\naUniversidade de São Paulo, Instituto de Física, São Paulo SP 05508-090, Brazil \nbUniversidade de Brasília, Instituto de Física, Núcleo de Física Aplicada, Brasília DF 70910-900, Brazil \ncUniversidade Federal de Mato Grosso do Sul, Instituto de Física, Campo Grande MS 79070-900, Brazil \ndCentro de Desenvolvimento da Tecnologia Nuclear – CDTN, Av. Antônio Carlos, 6627, Pampulha, Belo \nHorizonte MG 31270-90, Brazil \neUniversidade Católica de Brasília, Programa de Pós-Graduação em Ciências Genômicas e Biotecnologia, \nBrasília DF 70790-160, Brazil \nABSTRACT\nThis study focuses on the structure and magnetism of a nanostructured compound, initially labeled as \nFeMnO 3, prepared by the sol-gel method. Through Mössbauer spectroscopy analysis and Rietveld \nrefinement of X-ray diffraction data, the composition of the compound was well determined: a majority \nbixbyite phase (86 mol%, 94 wt%) with (Fe 0.25Mn 0.75)2O3 stoichiometry and average crystallite size of ~48 \nnm, plus a minority hematite phase (14 mol%, i.e., 6 wt%) with an average crystallite size of ~8 nm. \nThe Raman spectrum exhibits characteristic vibrational bands at 659 cm−1, 519 cm-1 and 416 cm-1, \nconfirming the majority phase, with no features associated with the minority phase. X-ray photoelectron \nspectroscopy analysis confirmed the presence of oxygen vacancy onto the (Fe 0.25Mn 0.75)2O3 particle surface, \nwith varying oxidation states (Fe3+, Fe2+, Mn3+, and Mn4+). X-band magnetic resonance data revealed a \nstrong and broad resonance line in the whole temperature range (4.3 K≤T≤300 K), dominated by the\nmajority phase, with g-value decreasing monotonically from (2.93 ± 0.01) at 50 K down to (2.18 ± 0.01) at \n300 K. The temperature dependence of both resonance field and resonance linewidth shows a remarkable \nchange in the range of 40-50 K, herein credited to surface spin-glass behavior. The model picture used to \nexplain the MR data in the lower temperature range (below about 50 K) assumes (Fe 0.25Mn 0.75)2O3 \nnanoparticles with a core-shell structure. Results indicate that below about 50 K the shell’s spin system\nreveals a paramagnetic to spin-glass-like transition upon cooling, with a critical temperature estimated at \n(43 ± 1) K. In the higher temperature range (above ~50 K), the superparamagnetic minority phase \ncontributes remarkably to the temperature dependence of the resonance linewidth. Zero-field-cooled and \nfield-cooled data show strong irreversibility and a peak in the ZFC curve at ~33 K, attributed to a \nparamagnetic-ferrimagnetic transition of the majority phase. \nKEYWORDS: FeMnO 3, (Fe xMn 1-x)2O3, Sol-gel method, Magnetic nanoparticles, Magnetic phase \ntransition, Mössbauer spectroscopy, Magnetic resonance. \n* Corresponding Author: mantilla52@gmail.com; nagamine@if.usp.br 21. INTRODUCTION\nNanotechnology has engendered a paradigm shift across various domains of science and \ntechnology, empowering manipulation and governance of matter at the nanoscale. The \nextraordinary and distinct properties of nanomaterials have ushered in novel prospects for \napplications in medicine, electronics, energy, and a plethora of other fields. Moreover, \nnanotechnology has yielded substantial progress in the fabrication of more efficient and \nminiaturized devices, fostering a world that is increasingly interconnected and \ntechnologically sophisticated. Given its capacity to tackle intricate challenges and \nenhance the quality of life, nanotechnology continues to stand as a captivating and \nauspicious arena for research and development endeavors. \nNanostructured magnetic semiconductors, including mixed metal oxides, hold particular \ninterest for technological applications due to the versatile nature of their physical and \nchemical properties, derived from both stoichiometry and particle size. As widely \nacknowledged, these properties often differ from those observed in the same materials \nbut with micrometric and larger particle sizes. Among this extensive array of \nnanomaterials, manganites, which are perovskite oxides with the general formula R 1–\nxMxMnO 3, where R is normally a trivalent rare earth ion and M is usually a transition \nmetal or a divalent alkaline ion, are widely recognized for exhibiting intriguing properties. \nMaterials from the R 1–xMxMnO 3 family, especially when the particle size is in the \nnanometer range, can exhibit remarkable densities of ionic and electronic defects, making \nthem important candidates for the development of solid-state fuel cells, chemical gas \nsensors, magnetic refrigeration, and eventually as memory storage devices [ 1-3]. Just as \nin all perovskites, the composition and structural parameters strongly influence the \nphysico-chemical properties of manganites [ 4]. On the other hand, it is well established \nthat the fundamental state of a given manganite can be modified by variations in basic \nthermodynamic variables such as pressure and strain, as well as by the application of \nelectric and magnetic fields [ 5-8]. The basic building block of manganites with a \nperovskite structure is the MnO 6 octahedron. According to Travis et al. [ 9], a compound \nfrom the manganite family will exhibit a stable perovskite structure when the \nGoldschmidt tolerance factor t is within the range of 0.89 < t < 1.02. Considering a \nparticular magnetite with composition AMnO 3 (where A represents a rare earth ion or a \ntransition metal ion), the Goldschmidt tolerance factor is defined as follows: 3=\n√2+\n(+) \nbeingr A(rMn) the ionic radius of the A+(Mn+) cation andr Othe ionic radius of the O-\nanion. Note thatt= 1 corresponds to the ideal perovskite structure [ 9]. The perovskite\nlattice undergoes structural deformation primarily due to two types of distortions: one\nresults from the tilting of the MnO 6octahedron, whereas the other arises from the\nasymmetry in the six Mn-O bond lengths surrounding the Mn atom within the MnO 6\noctahedron [ 10] (the Jahn-Teller distortion [ 11]). Simultaneously, there has been a\nremarkable surge in interest towards investigating non-perovskite structures of the ABO 3\ntype (where A and B represent cations such as Co, Ni, Fe, Mn, etc.). This heightened\ninterest can be attributed to their multifunctional properties and promising potential for\ndiverse device applications [ 12,13 ]. Notably, manganese oxide has garnered extensive\nattention due to its intriguing magnetic and multiferroic characteristics [ 14]. Manganese\nexists in various valence states, thereby endowing the material with a diversity of\nmagnetic properties contingent on its composition. Furthermore, altering the composition\nresults in concurrent changes in structure and valence states. For instance, in the case of\nMnO, which adopts a halite-type cubic structure, the Mn ion exhibits a valence state of\n2+. In contrast, for Mn 2O3, with a bixbyite structure (either cubic or orthorhombic), the\nvalence state is 3+. Similarly, in Mn 3O4, possessing a spinel cubic structure, the Mn ion\ndisplays a mixed valence of 2+ and 3+ [ 15]. Remarkably, AMnO 3-type compounds,\nincluding the bixbyite compound FeMnO 3, have found versatile applications, such as in\nnegative temperature coefficient (NTC) thermistors, oxidation catalysis, and\nsuperparamagnetic materials. Caoet al. [ 16] extensively investigated FeMnO 3as an\nanode material for lithium-ion batteries. This compound has gained prominence in\nelectrochemistry owing to its intriguing redox behavior, superior theoretical capacity for\nlithium-ion batteries (500–1000 mAh/g), and lower operating potential. Furthermore,\nFeMnO 3has been effectively employed in microwave devices and catalysts. The\nbixbyite-based (Mn3+, Fe3+)2O3compound adopts theβ-Mn 2O3crystal structure (cubic,\nIa-3,a= 9.41 Å) [ 17]. It comprises two metal sites situated at the 24d and 8b\ncrystallographic positions, whereas the oxygens occupy the 48e positions. Both cations\nexhibit octahedral coordination with oxygen, with the 8b site displaying a slight trigonal\ndistortion, whereas the 24d site exhibits a more substantial distortion. Rayaprolet al.\n[18,19] conducted a thorough investigation of the magnetic and magnetocaloric4properties of FeMnO 3synthesized through mechano-synthesis. Their findings reveal that\nFeMnO 3exhibits ferrimagnetic order at room temperature but undergoes a phase\ntransition to the antiferromagnetic state at 36 K. Below approximately 150 K, the\nmagnetization exhibits a pronounced deviation from Curie-Weiss behavior. It would be\nanticipated that Fe3+and Mn3+ions, octahedrally coordinated to oxygen, would possess\nhigh-spin configurations with quenched orbital moments, leading to magnetic moments\nof 5.9μ B(Fe3+) and 4.9μ B(Mn3+), resulting in an expected average of 5.5μ Bper metal\nsite. Through fitting the magnetization data in the 200–380 K range using the Curie-Weiss\nlaw, the authors previously derived an average magnetic moment per site of 4.1μ B,\nsuggesting the presence of some degree of antiferromagnetic correlation even at higher\ntemperatures [ 19]. Various preparation techniques have been successfully employed to\nsynthesize nanosized FeMnO 3, including physical methods such as high-energy ball\nmilling, as well as chemical techniques like chemical co-precipitation, sol-gel, and\ncombustion [ 20-21 ]. The physical and chemical attributes of the materials produced\nthrough the sol-gel approach, such as particle size, surface area, and mechanical\nproperties, can exhibit significant variations depending on factors such as working\ntemperature, operating conditions, and chemical precursors utilized. However, it has been\ndocumented that the sol-gel method offers a highly reproducible means of fabricating\nnanomaterials with elevated surface area and improved mechanical properties when\ncompared to alternative synthetic routes [ 22]. Nonetheless, it is noteworthy that a\nlimitation of the sol-gel method lies in the sensitivity of the precursor's hydrolysis to water\naddition. Even under vigorous stirring, the rate of hydrolysis is so rapid that particles tend\nto precipitate immediately upon water introduction into the reaction medium. A reduced\nrate of hydrolysis, however, can result in particle size reduction and an increase in surface\narea, aspects of great interest in catalysis [ 23]. The stability of various compounds formed\nwithin the FeMnO 3system is contingent not only on the Fe/Mn/O ratios but also on the\npreparation method and calcination temperature [ 24]. Among the identified compounds\nare mixed oxides (Fe 1-xMn x)1-yO, which have been utilized in water-splitting reactions\n[25], rock-salt oxides Fe xMn 1-xO [26], defect spinelγ-FeMnO 3analogous toγ-Fe 2O3[27],\nand manganese-substituted magnetite Mn xFe3-xO4[28]. In a study conducted by Ponceet\nal. [29], it was revealed that the stability of Mn4+ions play a critical role in determining\nthe catalytic activity of manganites for methane oxidation in the temperature range of\n200-800 °C. Generally, high surface area is associated with small particle sizes, leading5to a larger surface area exposed to gases for a given mass of nanoparticles [ 30].\nMorphology control stands as a paramount requisite for enhancing catalyst activity. As\nanticipated, nanosized FeMnO 3exhibits behavior characteristic of a single magnetic\ndomain, and its magnetic properties are significantly influenced by factors such as size,\nshape, stoichiometry, inversion parameter, crystallinity, and surface termination. These\naspects can be effectively manipulated through preparation and post-treatment synthetic\nmethodologies [ 31]. \nConsidering the FeMnO 3compound as paradigmatic for this family, there are few studies\nthat have explored the physical and morphological characteristics of compounds with\nexcess or deficiency of Fe, with the exception of the first-principles study by Bazhenova\nand Honkala [ 32]. \nIn this investigation, we present a comprehensive exploration of the physical properties\nexhibited by Fe 0,5Mn 1,5O3nanoparticles synthesized using the sol-gel method. A\nmeticulous evaluation of the structural, morphological, optical, and magnetic\ncharacteristics was undertaken through a combination of cutting-edge techniques,\nincluding x-ray diffraction (XRD), x-ray photoelectron spectroscopy (XPS), transmission\nelectron microscopy (TEM), Raman spectroscopy, Mössbauer spectroscopy (MS),\nmagnetometry, and electron magnetic resonance (EMR). \nThe proficient integration of MS, XPS and XRD techniques, the latter skillfully resolved\nthrough Rietveld analysis, enabled a meticulous elucidation of the system’s precise\nstoichiometry. Additionally, this approach facilitated the unequivocal identification of the\ndistinct phases present in the sample, their corresponding fractions, and the mean sizes of\ntheir crystallites. The magnetic properties of the compound were exhaustively\ncharacterized using the aforementioned techniques, and the results were effectively\ncorrelated with the morphology and composition of the studied material. \n \n2. EXPERIMENTAL DETAILS \n \n2.1 S ample preparation \nThe (Fe 0.25Mn 0.75)2O3 nanoparticles (NPs) reported here were synthesized using sol-gel \npolymerization method, which utilized transition metal nitrates as precursors. The process \ninvolved dissolve manganese nitrate tetrahydrate (Mn(NO 3)2·4H 2O), ferric nitrate \nnonahydrate (Fe(NO 3)3·9H 2O) and citric acid ( C6H8O7) in a small qu antity of double 6distilled water. Aqueous solutions of manganese (0.4 mol/L) and iron (0.8 mol/L) nitrates \nwere mixed in stoichiometric proportion (Fe3+:Mn2+::1:1). The mixture was then diluted \nin 50 mL of ethylene glycol (99% purity) while keeping the 1:1 volume ratio. The \nhomogeneous reaction medium was stirred at 80 °C to ensure uniformity. Once the gel \nwas formed the temperature was raised to 250 °C for self-ignition reaction. The resulting \nproduct was subsequently calcined under ambient atmosphere (900 °C, 72 hours) to \nobtain the nanosized compound. \n \n2.2 Characterization Details \nStructural analysis of the as-synthesized powder at room temperature were performed \nusing a commercial XRD diffractometer (Rigaku, model D/max) equipped with copper \nradiation source Cu-Kα ( =1.5418 Å) and operating in steps (2 ) of 0.05°. Crystal \nstructures were refined using the Rietveld method [ 33] with GSASII suite program. \nRepresentative TEM micrographs of the sample, including energy dispersive x-ray \nspectroscopy (EDX), were obtained using a commercial microscope (JEOL, model JEM-\n2100). Raman spectra were recorded using a triple spectrometer (Jobin-Yvon, model \nT64000) equipped with a 2048512 pixels nitrogen-cooled CCD (Charge-Coupled \nDevice) camera. Compositional XPS analysis was performed on a Specs surface analysis \nsystem equipped with a Phoibos 150 electron analyzer using a monochromatized Al K \nradiation (1486.6 eV) at a power of 350 W. Casa XPS software was used to process the \nrecorded data and to estimate the sample surface atomic concentration. C -1s signal (284.6 \neV) was used as reference for calibration of the binding energies (BE) of different \nelements. Electron magnetic resonance (EMR) spectra (from 3.8 to 300 K) were collected \nusing a commercial X-band spectrometer (Bruker, model EMX) equipped with a \nrectangular cavity. 57Fe Mössbauer spectra in transmission mode were recorded at 80 K \nand 300 K using a conventional spectrometer . The sample was mixed with boron nitrate \n(reaching 0.1 mg 57Fe per cm2) and homogeneously dispersed within a nylon-based \nsample holder. The spectrometer was equipped with a 25 mCi 57Co source, immersed in \nRh matrix. The Mössbauer source was coupled to the driver (room temperature) while a \nsinusoidal velocity driver was used. A cryostat (Oxford Cryosystems) equipped with a \ntemperature controller was able to keep the sample in the desirable temperature. The dc \nmagnetic measurements were performed using a commercial SQUID (MPMS, Quantum 7Design) magnetometer varying the temperature in the range of 5 to 300 K while applying \nmagnetic fields up to ± 70 kOe.\n \n3. RESULTS AND DISCUSSION \n \n3.1 Mössbauer Spectroscopy \nThe Mössbauer spectra recorded at 295 K and 80 K and their fits, including the subspectra \n(colored solid lines), are shown in Figs. 1a and 1b, respectively. These fits were \nperformed using two doublets corresponding to 24d and 8b crystallographic positions of \nthe (Fe,Mn) 2O3 phase, plus two sextets identified as hematite (α-Fe 2O3), and amorphous \nhematite. The obtained hyperfine parameters are collected in Table 1. \n \n0.880.920.961.00\n-10 -5 0 5 100.880.920.961.00 EXP\n FIT\n 24d\n 8b\n hematite\n amorphous Fe2O3295 K(a)\n(b)Transmission\nVelocity (mm/s)80 K\n \nFigure 1. Fittings of 57Fe Mössbauer spectra of the as-synthesized sample measured at 295 K and 80 K. \nThe open circles represent experimental data whereas the solid lines represent curve fittings. One doublet \nrepresents Fe3+ occupying 24d sites (solid red line) and the other doublet represents Fe3+ occupying 8b sites \n(solid blue line). The sextets are assigned to crystalline (solid magenta line) and amorphous (solid green \nline) hematite. \n 8The Mössbauer sextet (1) is assigned to crystalline α-Fe 2O3 in a weakly ferromagnetic-\nlike spin state for both temperatures, as suggested by the quadrupole splitting values of \n= -0.21 mm/s at T = 295 K and = -0.17 mm/s at T = 80 K, typical of this magnetic \nordering [ 34]. Therefore, no Morin transition (T M) is expected in the range of 295 - 80 K. \nMoreover, according to Amin and Arajs [ 35] T M = 264.2 - 2194/d, where d is the \nnanoparticle diameter (in nm) and T M is the Morin temperature (in K units). Then, using \nthe lower temperature value (T M = 80 K), one can estimate the mean size of the as-\nsynthesized α-Fe 2O3 nanoparticles below 12 nm. This result agrees with the mean size \nestimated by XRD analysis (8 nm). According to the literature, the Morin temperature \ndecreases as the mean particle size decreases, tending to be quenched for particles smaller \nthan about 8 nm in mean size [ 35,36 ]. Negative values of quadrupole splitting are also \nfound for sextet (2), indicating weakly ferromagnetic coupling for this component, at both \ntemperatures (80 and 295 K). Bulk hematite is a weak ferromagnet below the Néel \ntemperature (948 K T N 963 K) which undergoes a magnetic phase transition (to \nantiferromagnet and presenting spin reorientation) at the Morin temperature (T M 263 \nK). For the doublets, as can be seen in Table 1, the low temperature (80 K) isomer shift \n() and quadrupole splitting ( ) of 24d site, are 0.61 mm/s and 0.96 mm/s, respectively, \nwhich are higher than the 0.31 mm/s and 0.90 mm/s found for the 8b site. The larger \nvalues of both and , obtained for 24d site, agree well with the values reported by Nell \net al. [ 37] in natural and synthetic samples of FeMnO 3. However, the values for both \nsites are significantly higher than the values reported at room temperature in other studies \n[27,37 ]. This is likely related to the preparation method of the samples used here, which \nresults in more distorted nanocrystals than those reported in the literature. The area ratio \nof the doublets, Area 24d/Area 8b= 1.79 was extracted at 80 K. This value will be used as a \nconstraint for the Rietveld analysis. For the Mössbauer sextet (2), the unusual large \nlinewidth value ( = 0.87 mm s-1, T = 80 K) obtained for the amorphous hematite confirms \nthe degree of amorphization, with the decrease of the hyperfine field, as compared to the \ncrystalline hematite (see Table 1). Moreover, Kolket al. [ 38] studying hematite \nnanoparticles have also reported the presence of an additional component with broader \nlinewidth and hyperfine field of about 480 kG, which was attributed to amorphous -\nFe2O3, whereas the crystalline hematite showed hyperfine field of 525 kG at 15 K.\n 9Table 1. List of hyperfine parameters obtained from the fits of the Mössbauer spectra, where is the isomer\nshift, is the quadrupole splitting, is the linewidth, Area is the percentage area of the corresponding \nsubspectrum, and B hf is the hyperfine field. is given relative to Fe.\nT(K) Subspectrum \n(mm s-1) \n(mm s-1) \n(mm s-1)Area \n(%) Bhf (kG)\n Doublet (1), 24d 0.61(1) 0.96(1) 0.42(1) 41(2) - \n Doublet (2), 8b 0.31(1) 0.91(1) 0.33(1) 23(2) - \n80 Sextet (1) 0.47(2) -0.17(2) 0.48(2) 25(2) 528(2) \n Sextet (2) 0.55(2) -0.09(2) 0.87(2) 11(2) 482(2) \n Doublet (1), 24d 0.49(1) 0.95(1) 0.43(1) 45(2) - \n Doublet (2), 8b 0.18(1) 0.90(1) 0.33(1) 25(2) - \n295 Sextet (1) 0.39(2) -0.21(2) 0.33(2) 16(2) 508(2) \n Sextet (2) 0.38(2) -0.18(2) 1.27(2) 14(2) 473(2) \n \n \nThe spectral area ratio of the doublets (0.64 0.02) at 80 K and the total area of all \nsubspectra will also be used as constraint for the XRD data Rietveld refinement, as \ndiscussed later on in this report. It is worth noting that the subspectra area ratio value \nestimated at 295 K are slightly higher than the value estimated at 80 K. The difference in \nthis regard is likely related to the non-negligible superparamagnetic relaxation of hematite \nnanoparticles (~ 8 nm) at 295 K. As expected, at room temperature, a given fraction of \nthe nanoparticles in the sample becomes superparamagnetic (featured as doublet in the \nMössbauer spectra). Therefore, it is assumed that the superparamagnetic-related \nsubspectra area has been incorporated into the two paramagnetic doublets area (8b and \n24d sites), justifying the increase of the relative area attributed to the (Fe 0.25Mn 0.75)2O3 \nphase at room temperature. Therefore, as the relaxation is an unwanted effect, the area \nratio determined at 80 K was used as the best constraint for Rietveld refinements . \nMoreover, this effect can also justify the strong decreasing of values found for the \ndoublets at 295 K when compared with the corresponding values observed at 80 K. \n \n3.2 X-ray diffraction \nFigure 2 shows the room-temperature XRD pattern of the as-prepared powder sample. \nThe XRD data was refined using the Rietveld refinement method (GSASII software ). The \nrefinement analysis indicated formation of a cubic structure, space group Ia-3, bixbyite \n(Fe3+Mn3+)O3 type structure as the majority phase (~86 mol%), and another minority 10phase identified as hematite (~14 mol%) as shown in Fig. 2. The parameters obtained \nfrom the refinement are listed in Table 2. \n \n20 40 60 8054.0 54.4Intensity (u.a.)\n2 (degree)22 24Intensity (u.a.)\n2 (degree) Y exp\n Y calc\n Yexp - Ycalc \n Fe(0.25Mn0.75)2O3 (94 wt%)\n Fe2O3 (6 wt%)\nIntensity (arb. units)\n2 (degree)Fe2O3Fe2O3\n \nFigure 2. Room temperature XRD pattern of the as-synthesized sample, with the calculated data \nrepresented by the solid red line and black symbols indicating experimental data. The solid green line at \nthe bottom shows the difference between the experimental (Yexp) and calculated (Ycalc) data. Bragg’s\nreflections of the standard (Fe 0.25Mn 0.75)2O3 and -Fe 2O3 phases are indicated by vertical black and blue \nticks, respectively.\n \nThe fractional positions for the bixbyite phase as well as the fitting parameters of the α-\nFe2O3 phase are included in Table 2. A lattice constant of a = 9.412(1) Å corresponding \nto the bixbyite phase was found, which is consistent with the value reported in the \nliterature (9.41 ± 0.03) Å for the FeMnO 3 formed in the bixbyite phase and prepared via \nmechano-synthesis [ 18]. \n 11Table 2. Structural and statistical parameters obtained from the Rietveld refinement of x-ray diffraction. V \nis the cell volume, GOF is the goodness of fit andχ2 is the chi-square quality parameter. χ2 = 1.69; GOF = \n1.30. \n \n a (Å)b (Å)c (Å)V(Å)3Space group Phase \npercentage \n(mol%) \n(Fe 0.25Mn 0.75)2O39.412(1) 9.412(1) 9.412(1) 833.818(1) Ia-3 86.2 \nα-Fe 2O35.036(1) 5.036(1) 13.730(2) 301.514(1) R-3c 13.8 Fractional positions of (Fe 0.25Mn 0.75)2O3 \nAtom Wyckoff \nPosition x y z Occupancy \nFe1 8b 0.2500(2) 0.2500(2) 0.2500(2) 0.356 \nMn 1 8b 0.2500(2) 0.2500(2) 0.2500(2) 0.644 \nFe2 24d 0.0308(2) 0.0000 0.2500(2) 0.212 \nMn 2 24d 0.0353(2) 0.0000 0.2500(2) 0.788 \nO 48e 0.3342(2) 0.1048(2) 0.1196(2) 1.0000 \nFractional positions of Fe 2O3 \nFe 12 0.0000 0.0000 0.3544(3) 1.0000 \nO 18 0.3723(2) 0.0000 0.2500(2) 1.0000 \n \nThe stoichiometry (Fe 0.25Mn 0.75)2O3 was deduced from the Fe and Mn occupancies (see \nTable 2). In agreement with the results obtained from MS, it was verified that the Fe \natoms site occupancy ratio (24d/8b) is 1.79. It was also possible to calculate the ratio of \nFe atoms in the (Fe 0.25Mn 0.75)2O3 phase with respect the total amount of Fe atoms (all \nphases), resulting in (0.61 0.02). This finding is also in agreement with the result \nobtained from MS, considering the uncertainties, and allows one to conclude that this is \nthe most reliable stoichiometry for the as-synthesized sample.\nThe mean crystallite size () of the (Fe 0.25Mn 0.75)2O3 phase was estimated using \nthe modified Scherrer’s equation. This was accomplished while plotting lnδ versus \nln(1/cos ) to obtain the y-axis intercept ln(kλ/), using least square linear regression, \nwhere k is a constant (0.89 for spherical nanoparticle), λ is the wavelength of the x-ray 12(Cu-Kα), is the full width at half maximum (FWHM) of the x-ray diffraction line, and\nθ is the corresponding diffraction angle [ 39]. The mean crystallite size ~ 48 nm \nwas estimated for the FeMnO 3 phase using the (211), (400), (332) and (440) Bragg planes \n(see solid black circles in Fig. 2). To estimate the mean crystallite size of the secondary \nα-Fe 2O3 phase, the Scherrer’s equation was used only for the (116) plane (see thesquare \ngray in Fig. 2), once it represents the highest XRD intensity peak observed for this phase. \nThen, ~ 8.0 nm was estimated for the α-Fe 2O3 phase.\n \n3.3 Morphology \nFigure 3a shows a representative TEM micrograph of the FeMnO 3 like sample produced \nat 900 °C. As see in Figs. 3a and 3b, irregularly shaped particles with a wide range of \nsizes extending from 20 to 820 nm are observed. The crystal structure of the \n(Fe 0.25Mn 0.75)2O3 was also examined by selected-area electron diffraction (SAED).\nThe selected area electron diffraction (SAED) pattern, obtained from the \n(Fe 0.25Mn 0.75)2O3 sample (highlighted by the white circle in Fig. 3b), reveals a distinctive \narrangement of broad concentric diffraction rings. Notably, the rings correspond to \ncrystallographic planes such as (200), (211), (321), (222), (411), and (422), attributing \nthem to the cubic phase of the (Fe 0.25Mn 0.75)2O3 sample (depicted in Fig. 3d). Remarkably, \nthe absence of the hematite phase in this SAED pattern suggests that it does not coat the \nparticle surface. Instead, a secondary phase is formed. This intriguing observation is \nunderscored by the clear absence of the aforementioned electron diffraction associated \nwith hematite in this particular region. The crystallographic planes of each diffraction \nring are explicitly indicated in Fig. 3d. \nThe EDX spectrum recorded from the as-synthesized sample is shown in Fig. 3e. As \nobserved, the peaks corresponding to Mn, Fe and O confirm the presence of these \nelements in the major phase (Fe 0.25Mn 0.75)2O3. The C and Cu signal appearing in the EDX \nspectrum is due to the tape and grid used for sample preparation, respectively. \n \n \n 13 \n \n \nFigure 3. (a) and (b) TEM images of the as-synthesized sample (scale bar in nm). (c) HRTEM image of \nthe as-synthesized sample with Zone 1 and Zone 3 referring to interplanar distance of the FeMnO 3 phase, \nwhereas Zone 2 and Zone 4 refer to the α-Fe 2O3 phase (scale bar in nm). (d) SAED pattern for the as-\nsynthesized sample (scale bar in nm). (e) Shows the EDX spectrum of the as-synthesized sample recorded \nin the indicated position (white circle) of panel (b).\n \n \n3.5 Raman Spectroscopy \nThe crystallographic symmetry is the bixbyite structural type with the Ia-3 space group \nhaving 22 Raman active modes described by: 4A g + 4E g +14F g [40], 10 inactive modes \n5Au + 5E u, and 16 T u IR modes [41]. Despite the large number of Raman active modes \n(22), the number of modes actually observed in the Raman spectrum is reduced [40]. \nFigure 4 shows the room-temperature Raman spectrum of the as-synthesized powder \nsample in the 200-800 cm-1 range. According to the above-mentioned crystallographic \nsymmetry, the bixbyite structural type is the source of the Raman active modes in the \nspectrum, which is consistent with the XRD result. Furthermore, the bands exhibit \nbroadness, which is unexpected given the higher average particle size reported by XRD \nand TEM. Concurrently, this implies a significant degree of disorder, particularly at the \nparticle surface, as the Raman shows a greater surface contribution than the XRD.\n0 1 2 3 4 5 6 7 8 9 10O\nMn\nCMn\nFeCu\nCuIntensity (a.u.)Energy (eV)\nMn\nFe(a) (b) \n(c) (d) \nZone1\nZone4\nZone2\nZone3\n(220) (MnFeO 3)\n(217) (Fe 2O3)\n(217) (Fe 2O3)\n(411) (FeMnO 3)\n500nm\n500nm(e) 14 \nFigure.4. Raman spectrum of the as-synthesized sample. The experimental data are represented by open \nblack symbols whereas the solid red line portrays the best fitting obtained by incorporating blue colored \ncomponents. \n \nThe characteristic vibrational bands observed at 659 cm−1, 519 cm-1 and 416 cm-1 \ncorrespond to three of 14 F g modes which are labeled as F g(1), Fg(2), and, F g(3), respectively. \nThe band located at 335 cm−1 is assigned to the E g + F g mode, meanwhile, the 274 cm-1 \nband is linked to the E g mode. To display the effect of the presence of iron in the bixbyite \nholding matrix we can highlight the spectrum reported by Chen et al. which displays the \nFg(1) located at 652 cm-1 linked to the Mn3+-O mode vibration. When iron is present in the \nMn/FeO 6 octahedral, as in the case of the FeMnO 3, vibrations can cause the Raman \nfrequencies to shift to lower values; these downshift being also observed in the F g(2) and \nFg(3) vibrational modes [42].\n \n3.5 Surface composition analysis \nXPS measurements were performed to examine the oxidation state of Fe and Mn and the \npresence of oxygen specimens on the surface of the nanoparticles. The background was \nmodeled using a Shirley-type function implemented in Casa XPS software, and all \npositions were corrected using the expected 284.6 eV position of the C 1s binding energy. \nThe high-resolution spectrum of Fe 2p, displayed in Fig. 5a, clearly shows Fe 2p 1/2 and \nFe 2p 3/2 enlarged photoelectron lines (PL), suggesting that there are two sets of peaks 200 300 400 500 600 700 800 \nIntensity (a.u.)\nRaman Shift(cm-1) 659 \n 519 335 274 416 15corresponding to two different oxidation states. In this regard, a good fit was obtained \nusing two PL for each Fe 2p peak, deconvoluted into two mixed Gaussian-Lorentzian-\nshaped peaks. The doublets binding energies (BEs) obtained have been located at 710.2 \neV/723.6 eV, and 712.5 eV/725.9 eV, which were assigned to Fe2+ and Fe3+ oxidation \nstates, respectively. In addition, satellite peaks were observed at 715.1 eV/719.7 eV and \n728.3 eV/733.3 eV which are also characteristic of Fe2+ and Fe3+, respectively. The \npercentage of each species was estimated from the integrated areas of each peak to be \naround ~43% of Fe3+ and ~57 % of Fe2+. These values are consistent with the crystalline \nstate identified with different occupational sites for Fe atoms using XRD and MS.\nAs shown in Fig 5b, the fitting of the Mn 2p core level high-resolution XPS spectra \nreveals two component features as well, which are centered at 641.0eV/643.1eV and \n652.4 eV/654.2 eV. The Mn 2p 3/2 and Mn 2p 1/2 features centered respectively at binding \nenergies ~641.0eV and ~ 652.4 eV, are indication of the presence of Mn3+ ions with a \nspin-orbit split equals to ~11.4 eV. Moreover, the binding energies ~643.1 eV and 654.2 \neV are indication of presence of presence of Mn4+ ions with a spin-orbit split equals to \n~11.1 eV. The satellites of the Mn 2p 1/2 peak are located at 663.3 eV and 667.4 eV, with \na separation between them of about 9 eV (see Fig. 5b), which is a fingerprint of the Mn3+ \nand Mn4+ oxidized state, respectively [ 43]. \nThe XPS spectrum of O 1s, revealing three peaks, with binding energies of approximately \n529.9, 531.3 and 533.0 eV, present integrated areas of 34%, 45% and 20%, respectively. \nSimilar observation is reported in BaTiO 3 [44], which has been assigned to O2− ions, O1− \nions, and OChem species. Regarding the high percentage of OChem species observed (20%) \nit could be associated with oxygen vacancies present on the surface of the FeMnO 3 \nnanocrystals. It is worth noting that this technique can detect only oxides at the surface \n(up to 2 nm deep into the nanoparticle). The core of the nanoparticles may have different \ncomposition, very likely prevailing the oxide with Fe3+ and Mn3+.\n 16740 735 730 725 720 715 710 705Intensity (a. u.)\nB.E. (eV)Fe 2p3/2Fe 2p1/2\nFe2+Fe3+(a)\nsat.sat.\n670 660 650 640Intensity (a. u.)\nB.E. (eV)Mn 2p3/2\nMn 2p1/2\nMn4+\nMn3+(b)\nsat.\n536 534 532 530 528Intensity (a. u.)\nB.E. (eV)O 1S\nOchem.O-1(c)O-2\n \nFigure 5. XPS spectra of the as-synthesized sample, with panels (a), (b), and (c) representing the high-\nresolution XPS spectra of Fe 2p, Mn 2p, and O 1s, respectively.\n3.6 Magnetic measurements \nFigures 6a and 6b show the magnetization hysteresis loop for the as-synthesized sample \nat 300 K and 5 K, respectively. The magnetization at 300 K is consistent with the presence \nof mainly a paramagnetic behavior, superimposed to superparamagnetic response, the \nlatter saturating at relatively small magnetic field. The dashed line in Fig. 6a indicates the \nsaturation magnetization of the superparamagnetic contribution credited to the α-Fe 2O3 \nsecondary phase [45,46 ] according to the percentage of phases listed in Table 2. The inset \nin Fig. 6a shows the central part of the 300 K M vs H curve (magnetic field up to ± 10 \nkOe). Therefore, the magnetization data confirm the presence of α-Fe 2O3 secondary phase \nas determined from the Rietveld refinement data analysis. Moreover, the negligible value \nfor the coercive field at 300 K confirms the superparamagnetic regime for these phases. \nIt is worth noting that the particles of the hematite phase have been determined as blocked \nat 300 K, once the Mössbauer spectra for them are two magnetic sextets. 17-80 -60 -40 -20 0 20 40 60 80-6-4-20246\n-10 -5 0 5 10-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0(Fe0.25Mn0.75)2O3 \nm= 12.65 mg\nT= 300 K \n \nM(emu/g)\nH(kOe)(a)\nFe2O3 \n \nM(emu/g)\nH(kOe)\n \n \n \nFigure 6. Magnetization hysteresis loops in the ± 70 kOe range obtained (a) at 300 K (the inset shows a \nzooming of the 300 K plot in the ± 10 kOe range) and (b) at 5 K (note the coercivity and magnetic hysteresis \nin the low field range). Mass of the measured powder is quoted in the legend and the dashed line in panel \n(a) represents mainly the contribution of the α-Fe 2O3 (secondary) considering the phase percentage obtained\nby the Rietveld refining (see Table 2) and the saturation magnetization of bulk α-Fe 2O3 at 300 K (Ms = 84 \nemu/cm3). \n \nThis disagreement is due to different time of measurements involved for the \nmagnetization (102 s) and Mössbauer spectroscopy (10-8 s) techniques, resulting in higher \nTB for the latter (MS). While comparing Figs. 6a and 6b, one finds remarkable differences -80 -60 -40 -20 0 20 40 60 80-20-1001020\n(Fe0.25Mn0.75)2O3 \nm= 12.65 mg\nT= 5 K \n \nM(emu/g)\nH(kOe)(b)\nFe2O318between the M vs H curves collected at 300 K and 5 K, reflecting the magnetic evolution \nof the dominant phase. For instance, the coercive field increase, which is discussed in \ndetail in the next paragraphs.\n \nThe temperature dependence of zero-field-cooled (ZFC) and field-cooled (FC) \nmagnetization curves of the as-synthesized sample obtained with a magnetic field of 20 \nOe is shown in Fig. 7. The ZFC curve presents a sharp peak at T C ~ 33 K (assigned to the \nferrimagnetic transition), following a sharp drop with decreasing T. Simultaneously, a \nrising feature is observed in the FC curve while decreasing T, showing a strong \nirreversibility between ZFC and FC curves. This strong irreversibility can be assigned to \nthe presence of the secondary α-Fe 2O3 phase, besides the majority phase (Fe 0.25Mn 0.75)2O3. \nMoreover, the temperature related to the peak observed in the ZFC curve (T C ~ 33 K) is \nlower than the value reported for the FeMnO 3 system prepared by mechano-synthesis (T C \n~ 40 K) [ 15,26 ]. This is mainly due to samples with different stoichiometry.\n0 50 100 150 200 250 3000.030.040.050.060.070.08\n0 50 100 150 200 250−0.20.00.20.40.60.8\ndMZFC/dT (a.u.)\nTemperature (K)TBTC\nTBZFC \nM(emu/g)\nTemperature (K)(Fe0.25Mn0.75)2O3 \nH = 20 OeFC\nTC = 33 K \n \nFigure 7. Temperature dependence of zero-field-cooled (ZFC) and field-cooled (FC) traces obtained while \napplying a magnetic field of 20 Oe. The inset shows the derivate (dM ZFC/dT) used to identify the blocked \ntemperature (T B) of the -Fe 2O3 and the ferrimagnetic transition temperature (T C). \n 19Importantly, in Fig. 7 one can observe a valley around 150 K. This feature is attributed to \nthe blocking temperature (T B) of the hematite phase. To determine the exact blocked \ntemperature d(ZFC)/dT is plotted as a function of temperature (see inset in Fig. 7). The \nd(ZFC)/dT versus T plot shows a visible downturn around 140 K, which is attributed to \nthe blocked temperature of the -Fe 2O3 phase. To determine the type of magnetic \ntransition involved in the as-synthesized sample, the temperature dependence of the \ninverse magnetic susceptibility ( -1) measured at 20 Oe is plotted in Fig. 8.\n-200 -150 -100 -50 0 50 100 150 200 250 3000.100.150.20\n−5−4−3−2−103.203.213.223.233.243.253.263.273.28 Experimental Data FitLog1\nLogtmTK= 3.5 K=0.511( 104 Oe.g/emu)\nT(K) (Fe0.25Mn0.75)2O3\n C-W Fitmeff = 5.2 mBH = 20 Oe\nTG= 101 K\n \nFigure 8. Temperature dependence of the inverse susceptibility ( -1), at 20 Oe (black symbols), along with \nthe Curie-Weiss fit (solid red line) in the 150-300 K range using Eq. 1. The inset shows the temperature \ndependence of the dc susceptibility (following Eq. 2) plotted in a double logarithmic scale. \n \nData in Fig. 8, regarding the high-temperature range (150-300 K), were fitted (solid red \nline) to the Curie-Weiss law [ 47]: \n=\n (1) \nwhere C=Nµ eff2/3kB is the Curie constant (N is the number of magnetic ions, µ eff is the \neffective magnetic moment per ion, is the characteristic Curie-Weiss temperature, and \nkB is the Boltzmann constant. 20It is worth mentioning that the Curie-Weiss data analysis (see Fig. 8) must reflect mainly \nthe (Fe 0.25Mn 0.75)2O3 phase. The best fit provides = (-177 ± 2) K. The negative and \nrelatively high θ value suggests strong AFM interactions between Fe3+ and Mn3+ magnetic \nmoments in the bixbyite crystallites formed by several unit cells [ 48]. Moreover, they also \nindicate the presence of magnetic frustration in the nanocrystals [ 18]. The literature on \nFeMnO 3 [49] has established the frustration ratio parameter (defined as the ratio between \nthe absolute value of the Curie-Weiss temperature, ||, and T C, i. e. ||/) as an \nindicator of the degree of magnetic frustration in the sample. For our (Fe 0.25Mn 0.75)2O3 \nsample, this ratio is ||/ = 177/33 = 5.4, a value smaller than the one ( ||/ = 336/32 \n= 10.5) obtained by Roth et al. for (Fe 0.56Mn 0.44)2O3 [49,50], where a long-range magnetic \norder was not observed at low temperatures, indicating a prevailing spin glass state. \nHowever, our frustration ratio value is higher than the one ( ||/ = 69/36 = 1.9) obtained \nby S. Rayaprol et al. [18,19] for (Fe 0.5Mn 0.5)2O3, in which an antiferromagnetic order is \nestablished at low temperature. \nA Curie constant of C = 0.36 emu/mol K was obtained from the fit. From this value and \nassuming the structure has approximately 16 pairs of Fe 0.5Mn 1.5 at sites 8b and 24d per \nunit cell, the experimental value of the effective magnetic moment per pair of Fe 0.5Mn 1.5 \nis µ effexp = 5.2 µ B. This value is higher than the value of 2.8 µ B determined by Seifu et al. \nin FeMnO 3 compounds produced by the mechanical alloying technique [ 26]. This is \nattributed to differences in stoichiometry between the investigated samples Assuming that \nthe electronic configuration of the system is (Fe3+0.5) (Mn3+1.5)(O 32-) and taking into \naccount that the theoretical spin-only values are: µ Fe3+ =5.90 µ B and µ Mn3+ = 4.90 µ B (Fe3+ \nand Mn3+ ions in high spin state), the theoretical value of the effective magnetic moment \nis estimated as [ 51]: µ efftheo = [(1-f) (µMn3+)2 + f (µFe3+)2 ]1/2 = 5.2 µ B, where f = 0.25 is \nthe percentage of Fe over the total magnetic ions of the system. This result is in excellent \nagreement with the experimental one, confirming that Fe3+ and Mn3+ ions are in the high \nspin state configuration.\nThe downward deviation of the temperature-dependent inverse magnetic susceptibility \n(1) from the ideal Curie-Weiss law (see Fig. 8) signals the presence of the Griffiths \nphase (GP) in the system. The temperature at which -1(T) deviates from the ideal Curie-\nWeiss law is known as the Griffiths temperature (T G) herein ascribed to 101 K, as shown \nin Fig. 8. This feature is a unique characteristic of the GP. 21A similar downturn was also observed in the antiferromagnetic TbFe 0.5Cr0.5O3 and \nFeMnO 3 phases fabricated by the mechano-synthesis method [ 52,53 ], in which the FM \ncorrelation among neighboring short-range FM cluster accounts for the observed \ndeviation. Herein, T G is defined as the onset temperature, where -1(T) data deviates from \nthe Curie-Weiss law, and a local AFM ordered area begins to develop [ 54]. The GP \nregime is usually characterized by the temperature dependence of the inverse \nsusceptibility, which follows a power law: \n 1()(− )1 (2) \nwith 0≤ < 1. The power law in the previous equation is a generalized Curie-Weiss law. \nHere, T K is the critical temperature of the ferromagnetic (or ferrimagnetic) clusters [ 18] \nthat can be estimated from = 0 in the Curie-Weiss regime [ 55,56 ], which is equivalent \nto the Curie-Weiss temperature ( ). The parameter appearing in the exponent shows the \nstrength of the GP. The double logarithmic plot of the dc susceptibility against reduced \ntemperature ( =\n−), reproduced in the inset of Fig. 8, shows a linear behavior at \nlow t m values, and confirm the proposed GP. The fitted value of is 0.51 for the as-\nsynthesized (Fe 0.25Mn 0.75)2O3 sample which is in the range (0≤ < 1) expected from a \nsystem exhibiting the Griffiths phase. The solid red line in the inset of Fig. 8 represents \nthe best fit of the -1 versus T data using Eq. 2, in the range of 5K < T < T G. Actually, Eq. \n2 represents the temperature dependence of the order parameter ( -1) in the context of the \nclassical Landau second-order phase transition, with the critical exponent (δ = 1- ) very \nmuch close to ½ [ 57]. \n \n3.7 Magnetic resonance \nThe MR spectra obtained from the as-synthesized sample at a fixed frequency of 9.50 \nGHz, in the temperature range of 4.3 - 293 K are presented in Fig. 9. Throughout this \ntemperature range, a strong and broad resonance signal, with a nearly symmetrical shape, \ncan be observed. This shape is strictly Lorentzian, suggesting a strong interaction between \nmetal ions through the exchange interaction [ 58]. Moreover, it suggests that the majority \n(Fe 0.25Mn 0.75)2O3 phase dominate the spectra. By utilizing g = h /µBHR, where denotes \nthe spectrometer’s operating frequency (9.50 GHz),µ B represents the Bohr magneton and \nHR represents the extracted MR field, the temperature dependence of the extracted g-\nvalues can be plotted, as shown in the inset of Fig. 9. The g-values for the as-synthesized 22(Fe 0.25Mn 0.75)2O3 nanocrystals have been found to decrease systematically with increasing \ntemperature, in agreement with the behavior found to perovskites compounds [ 59,60 ]. \nThis behavior is largely attributed to a decreasing of the magnetic moments occurring as \nthe temperature rises [ 18,61 ]. Specifically, these values decreased from (2.93 ± 0.01) at \n50 K to (2.18 ± 0.01) at 300 K. \n0 2 4 6-60-50-40-30-20-10010203040506070\n50 100 150 200 250 3002,12,22,32,42,52,62,72,82,93,0\n \ndA/dH(a.u.)\nMagnetic Field (kOe)Temp.(K)\n 4.3\n 6.9\n 9.0\n 15.1\n 19.9\n 25\n 30\n 40\n 50\n 60\n 80\n 120\n 160\n 200\n 240\n 293\n \ng- value\nT(K)\n \nFigure 9. X-band MR spectra, representing the first derivate of the absorption, recorded from the as-\nsynthesized sample. The spectra were recorded at various temperatures, ranging from 4.3 K to 293 K. The \ntemperature dependence of the g- factor is also plotted in the inset (the solid line is just to guide the eyes).\n \nThe inset of Fig. 10 shows H R vs T, where the vertical dotted black lines indicate T C, 2T C, \nand T G temperatures. For T G< T < 300K, the majority (Fe 0.25Mn 0.75)2O3 phase is \nparamagnetic and H R reduces smoothly with decreasing T, due to the enhancement of the \nmagnetic moment [ 18,61 ]. For 2T C < T < T G, the (Fe 0.25Mn 0.75)2O3 phase is still in the \nparamagnetic state but with the presence of clusters formed at T G, and H R decreases \nsharply due to the enhancement of the magnetic moment and the presence of these clusters. \nImportantly, the Griffiths-like phase is an intermediate state between the disordered \nparamagnetic and the ordered ferromagnetic (or ferrimagnetic) state, where it begins to \nappear magnetic interaction between the FM (or ferrimagnetic) clusters. In the study of 23FeMnO 3 nanoparticles prepared by mechano-synthesis method, Rayaprol et al. [ 18] \nshowed that at T2T C a change in the unit volume cell occurred, and they argued that \nthere is a strong coupling between the structure and the magnetic ordering occurring at \nTC, provoking large variations of magnetic entropy in this temperature range. Although \nanother method of synthesis has been herein employed, H R for T < 2T C also reduces \nsharply while decreasing the temperature, confirming that the as-synthesized sample can \nalso show similar behavior on H R from 66 K (2T C) down to 4.2 K. Therefore, this sharp \ndrop is attributed to the increasing magnetic interaction influenced by structural changes \nand long-range ferrimagnetic coupling as the system approaches T C, as well as the spin-\nglass-like behavior occurring at the surface of the nanoparticles. Figure 10 shows the \nvariation of peak-to-peak line width (Δ ) as function of temperature for the \n(Fe 0.25Mn 0.75)2O3 nanocrystals, showing a sharp peak at T max = 42 K (~1.27 T C), where \nTmax is assigned to the temperature where the Δ is maximum. It is known that for \nsystems showing spin-glass-like behavior, Δ presents noticeable broadening below \nthe spin-glass temperature [ 62]. As shown in Fig. 10, higher value for T max (43 K) while \ncompared to T C (33 K) is attributed to a surface spin-glass-like behavior of \n(Fe 0.25Mn 0.75)2O3 nanocrystals. \nActually, the study of spin-glass-like systems has relied on the temperature dependence \nof ΔHpp to obtain information about the spin freezing phenomenon and the corresponding \nfreezing temperature (T f). Recently, an exponential relationship between Δ and T for \ndiluted magnetic semiconductors exhibiting spin-glass behavior has been proposed [ 63]: \nΔ=Δ∞+Δ0(−/)(/) (3) \nThe above-presented relationship incorporates parameters such as∆H 0 and∆H , which \ndescribes Δ respectively at low and high temperatures, and correlates with the \nconcentration of magnetic ions and the dominant magnetic interaction above the \ntransition temperature. The Curie-Weiss temperature ( ) is included in the pre-\nexponential term, whereas the exponential term allows for determination of the freezing \ntemperature (T f). Black symbols in Fig. 10 represent the temperature dependence of Δ , \nwith the experimental data fitted to Eq. 3 in the low temperature region, up to 50 K. While \nEq. 3 successfully fits the Δ versus T data below 50 K, it fails to account for the \nexperimental data above this temperature. Notably, the Δ values systematically \ndecrease from about 4200 Oe to about 3600 Oe as the temperature is increased from about 2450 K to 300 K, representing the opposite trend observed below about 50 K. Importantly, \nbelow and above this temperature (50 K), the concavity is upward and downward \nrespectively, signaling a remarkable change in the dominant magnetic behavior.\nA key issue in the present study was to analyze the Δ versus T data in the whole \ntemperature range (4.3 K to 293 K). In order to accomplish this goal, an additional term \nΔtanh( /) was included into Eq. 3 [ 64]: \nΔ=Δ∞+Δ0(−/)/+Δtanh( /) (4)\nThis extra term has proven to be successful in describing theΔHpp of superparamagnetic \nparticles in a wide range of temperature. Actually, it is related to thermally-induced jumps \nbetween two energy minima, which correspond to two distinct orientations of \nnanoparticle’s magnetic moment with respect to the easy axis of magnetization[ 65,66 ]. \nThe third term (extra term) on right-hand side of Eq. 4 includes the pre-factor ∆H spm = \n5gSn/R3, with S, n and R representing the effective spin of the magnetic center, number \nof magnetic centers inside the superparamagnetic particle and average particle-particle \ndistance, respectively. Also, included into the extra term is a characteristic temperature \nTspm = ΔE/2k B, where ΔE is the energy barrier between the two orientations of the \nnanoparticle’s magnetic moment . The solid red line in Fig. 10 represents the best fit of \nthe experimental data while using Eq. 4, with the following fitted values:∆H = (358 ± \n13) Oe,∆H 0 = (84 ± 2) Oe, = (-150 ± 5) K, T f = (5.1 ± 0.1) K,∆H spm = (3849 ± 10) Oe, \nand T spm = (1194 ± 5) K. Some of these values can be compared with the values obtained \nexperimentally, as instance = -177 K (see legend of Fig. 8) and 948 K≤T N ≤963 K\nobtained for bulk -Fe 2O3 [67]. Higher value obtained for Tspm should be related to the \nnanosized characteristic of -Fe 2O3 particles.\nImportantly, it is herein claimed that the extra term included into Eq. 4 describes mainly \nthe contribution of the secondary phase, namely the superparamagnetic hematite \n(estimated XRD mean size of 8 nm), being dominant at high temperatures (above 50 K), \nalthough contributing to a relatively small change in the Δ values, roughly from 3600 \nOe to 4200 Oe. Broadening of the MR line while lowering the temperature of \nsuperparamagnetic hematite has been widely reported in the literature [ 68-70 ]. The \nsuccessful fitting of the data in the full temperature range of investigation has allowed for \na better understanding of the model picture. Overall, the findings have provided valuable \ninsights into the behavior of magnetic nanoparticles at varying temperatures. 250.00 0.05 0.10 0.15 0.20 0.25600120018002400300036004200\n0 50 100 150 200 250 3000500100015002000250030003500\nHR(Oe)\nT(K) data\n Linear FitFiM GP PM\nTG\n2Tc\nTcTSG\nTmax= 43 K \n \nHpp(Oe)\n1/T(K) data\n Fit\n \nFigure 10. Experimental values of the magnetic resonance linewidth (ΔH pp) tracked as a function of \ntemperature (T), as indicated by the solid black symbols. The solid red line represents the best fit using Eq. \n5. The inset shows the temperature dependence of the resonance field (solid black-white symbols). \n \nIndeed, the present report emphasizes the applicability of the MR technique, while using \nthe Δ versus T data, to unveil the magnetic contributions of a multi-phase \nnanomaterial presenting distinct magnetic ordering. As indicated by both Mössbauer and \nXRD data evaluation, the as-synthesized sample comprises mainly two distinct \nnanophases; the majority (~86 mol%) iron manganese trioxide (Fe 0.25Mn 0.75)2O3 phase, \nwith mean size around 48 nm, and the minority (~14 mol%) hematite ( -Fe 2O3) phase \nwith mean size around 8 nm. Typically, different iron oxide nanophases (e.g. iron sulfate, \niron oxide, Cd/Zn/Cu/Ni/Mn-ferrite), pristine or surface-dressed, synthetic or extracted \nfrom living organisms, present broad MR lines (up to about 4 kOe) and resonance field \nbelow about 3.5 kOe [71,72,64,65,68,73,74]. Therefore, it is not surprising that the \nmajority (Fe 0.25Mn 0.75)2O3 phase dominates the MR spectra shape in the wide temperature \nrange of our investigation (4.3 - 293 K). Nevertheless, the signature of the minority \nhematite phase is clearly identified in the temperature dependence of Δ above about 2650 K, as described by Eq. 4. Whereas the larger (48 nm) magnetic phase ( iron manganese \ntrioxide ) dominates the Δ trend in the lower temperature range (below about 50 K) \nthe smaller (8 nm) magnetic phase (hematite) contributes remarkably to the Δ trend \nin the higher temperature range (above about 50 K) . It is worth noting that the spin-glass-\nlike behavior should be concentrated at the surface of the nanocrystals (0.6 nm tick), \nwhereas the core can be magnetic, as reported in other magnetic nanoparticles [ 60,75 ]. \nMoreover, it is not completely ruled out that the surface spin-glass-like behavior can also \noccur for -Fe 2O3 nanoparticles. The third term on the right hand-side of Eq. 4 accounts \nfor the magnetic behavior of the hematite phase, with negligible increment below 60 K, \nbut with non-negligible contribution above 60 K . Therefore, it is not surprising that Eq. 3 \ncan fit nicely the Δ versus T data below about 60 K and fails to perform the fitting in \nthe whole temperature range (4.3-293 K). Likewise, it is not surprising the need of an \nextra term to account for the superparamagnetic contribution of the hematite phase above \n60 K, as included into Eq. 4. Moreover, as the temperature drops below 60 K, it is claimed \nthat a phase transition occurs from a continuous paramagnetic phase to a spin-glass-like \nphase in the shell layer of the (Fe 0.25Mn 0.75)2O3 phase . Regarding a rough estimation of \nthe start of this transition, two linear fits have been carried out for H R vs T, one of them \nfrom 20 up to 50 K and the other from 120 to 295 K, where the intersection of these lines \nwas determined as spin-glass temperature T SG = 60 K (see inset of Fig. 10). This transition \nextends down to a value of T f = 5.1 ± 0.1 K, below which the spins in the shell layer \nbecome completely frozen in a specific configuration across the surface of the sample. \nThe linear behavior emphasized in the inset of Fig. 10, namely H R vs T, has been reported \nin the literature while associating the slope of straight line with the nanoparticle size \n[73,74 ].\nExtra evidence of the spin-glass-like characteristic of the (Fe 0.25Mn 0.75)2O3 nanoparticles’\nshell layer is the upshift of the g-value while lowering the temperature of the sample, \nparticularly below about 60 K (see inset of Fig. 9). It is worth mentioning that high g-\nvalues are characteristic of magnetically isolated transition metal ions in a low-symmetry \nenvironment, attributed to the presence of relatively strong crystalline fields due to the \nlack of symmetry translation, as reported in the literature [ 75]. \n 274. CONCLUSIONS \nThe present study reports on the structural and magnetic characterizations of a \nnanostructured powder compound, which was initially labeled as FeMnO 3, successfully \nsynthesized through the sol-gel method. Mössbauer spectroscopy data analysis provided \nevidence supporting the predominant formation of a (Fe,Mn) 2O3 phase and a minority \nhematite (α-Fe 2O3) phase. The data extracted from the Mössbauer spectrum at 80 K were \nused as constraints in the Rietveld analysis of the x-ray diffraction data. The Rietveld \nrefinement method confirmed the formation of the majority bixbyite phase (86 mol%, 94 \nwt%), with (Fe 0.25Mn 0.75)2O3 stoichiometry and mean crystallite size of ~ 48 nm, plus the \nminority hematite phase (14 mol%, i.e. 6 wt%, and mean crystallite size of ~8 nm). \nTEM images depict agglomerates of nanoparticles exhibiting a wide range of sizes and \nshapes. HRTEM images distinctly reveal crystalline planes identified as belonging to the \n(Fe 0.25Mn 0.75)2O3 and α-Fe 2O3 phases. This observation suggests that the smaller α-Fe 2O3 \nnanoparticles may be decorating the larger (Fe 0.25Mn 0.75)2O3 nanoparticles. \nThe Raman spectrum of the compound displays five Raman active modes (three Fg modes \nat 659 cm-1, 519 cm-1, and 416 cm-1; Eg + Fg mode located at 335 cm−1; and Eg mode at \n274 cm-1). These modes are characteristic of an (Fe x,Mn 1-x)2O3 phase. The spectrum did \nnot show features that could be associated with the minority phase. On the other hand, X-\nray photoelectron spectroscopy analysis confirmed the presence of oxygen vacancy onto \nthe (Fe 0.25Mn 0.75)2O3 particle surface, with varying oxidation states (Fe3+, Fe2+, Mn3+, and \nMn4+).\nThe small size of the hematite crystallites explains why the Morin transition is not \nobserved. At 300 K, the magnetization data indicated a dominant paramagnetic behavior \ncredited to the (Fe 0.25Mn 0.75)2O3 phase plus a weak superparamagnetic contribution \ncoming mainly from the α-Fe 2O3 phase. Moreover, the hysteresis cycle recorded at 5 K \nis characteristic of ferrimagnetic ordering, revealing a phase transition associated to the \n(Fe 0.25Mn 0.75)2O3 phase while decreasing the temperature. The ZFC peak at 33 K was \nattributed to the paramagnetic-ferrimagnetic transition (T C), in addition to a valley \nobserved at T B = 140 K, herein interpreted as a signature of the blocking temperature \nassociated to the α-Fe 2O3 phase. Above Griffiths temperature (T G), the Curie-Weiss law \nrules the temperature dependence of the susceptibility, indicating a paramagnetic phase \nwith strong antiferromagnetic short-range correlation, with = -177 K and meff = 5.2 µ B 28per magnetic ion pair in the (Fe 0.25Mn 0.75)2O3 phase. Importantly, the downward deviation \nof the inverse magnetic susceptibility ( -1) versus temperature data from the ideal Curie-\nWeiss law, observed below T G, strongly suggest the onset of the Griffiths phase (GP) in \nthe system. Consequently, a power law using a generalized Curie-Weiss expression was \nused, and the value of = 0.51 was obtained, which is within the limit (0≤ < 1) expected \nfrom a system exhibiting a GP regime. Hysteresis curve at 5 K shows a low coercive field \nof 4 kOe, with the magnetization not reaching saturation at 70 kOe, suggesting the \noccurrence of a ferrimagnetic core with a magnetic disorder at surface, characteristic of \ncore-shell spin-glass-like behavior.\nX-band magnetic resonance (MR) data revealed a strong and broad resonance line in the \nwhole temperature range (4.3 K≤T≤300 K), dominated by the majority phase, withg-\nvalue decreasing monotonically from (2.93 ± 0.01) at 50 K down to (2.18 ± 0.01) at 300 \nK. The temperature dependence of both resonance field and resonance linewidth shows a \nremarkable change in the range of 40-50 K, herein credited to surface spin glass behavior. \nThe model picture used to explain the MR data in the lower temperature range (below \nabout 50 K) assumes (Fe 0.25Mn 0.75)2O3 nanoparticles with a core-shell structure. Results \nindicate that below about 50 K the shell’s spin system reveals a paramagnetic to spin-\nglass-like transition upon cooling, with a critical temperature estimated at (43 ± 1) K. In \nthe higher temperature range (above about 50 K), the superparamagnetic minority phase \ncontributes remarkably to the temperature dependence of the resonance linewidth.\n 29Author Contributions \nJohn C. Mantilla conduct all experiments and characterizations performing the data \nanalysis. Luiz C. C. Nagamine performed and assisted interpretation of MS. Daniel R. \nCornejo and Renato Cohen performed the magnetic measurement. Wesley de Oliveira \nhelped sintering the nanocrystals and the experimentation. Paulo Souza performed MR \nmeasurement and helped analyzed the data. Sebastião W. da Silva conducted Raman \nspectroscopy experiment and analyzed the results. Fermin F. H. Aragon and Pedro L. \nGastelois performed XPS measurements and analyzed the results. Luiz. C. C. M. \nNagamine, Paulo C. Morais, Daniel R. Cornejo and Jose A. H. Coaquira wrote the \nmanuscript and contributed to the structural and magnetic analysis. All authors \ncontributed to the analysis and discussion of the results and have approved the final \nversion of the manuscript. \n \nConflicts of interest \nThe authors declare they have no competing financial interest or personal \nrelationships that could have appeared to influence the work reported in this study. \n \nAcknowledgements \nAuthors acknowledge the Laboratório Multiusuário de Microscopia de Alta Resolução \n(Labmic/UFG), IF/UnB and Laboratory of Crystallography, IF/USP for XRD \nmeasurements. John C. Mantilla acknowledges partial financial support from Brazilian \nNational Research Council (CNPq Grant # 101441/2022-3). \n \nReferences \n \n1. C. R. H. Bahl, D. Velázquez, K. K. Nielsen, K. Engelbrecht, K. B. Andersen, R.\nBulatova, N. Pryds, High performance magnetocaloric perovskites for magnetic\nrefrigeration. Appl. Phys. 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Matter. 6 (1994) 9415-9428. https://doi.org/10.1088/0953-\n8984/6/44/020 " }, { "title": "0810.4128v1.Study_of_the_mixed_Ising_spins__1_2_3_2__in_a_random_crystal_field.pdf", "content": "arXiv:0810.4128v1 [cond-mat.stat-mech] 22 Oct 2008Study of the mixed Ising spins (1\n2,3\n2)in a random\ncrystal field\nL. Bahmad , A. Benyoussef∗, and A. El Kenz†\nFacult´ e des Sciences, D´ epartement de Physique,\nLaboratoire de Magn´ etisme et Physique des Hautes Energies.\nB.P. 1014, Rabat, Morocco\nAbstract\nWe study the magnetic properties of a mixed Ising ferrimagnetic\nsystem, in which the two interacting sublattices have spins σ, (±1/2)\nand spins S, (±3/2,±1/2) in the presence of a random crystal field,\nwith the mean field approach. The obtained results show the exis-\ntence of some interesting phenomena, such as the appearance of a\nnew ferrimagnetic phase namely the partly ferrimagnetic phase ( mσ=\n−1\n2,mS= +1)andconsequentlytheexistenceofthreetopologicallydif-\nferent types of phase diagrams. The effect of increasing the exch ange\ninteraction parameter J, at very low temperature is investigated. The\ntransitions shown in these phase diagrams are in good agreement wit h\nthose obtained in the ground state case.\nPACS: 05.50.+q; 75.10Hk;75.50.Gg;\nKeywords: Mixed Ising, Ferrimagnetism, Random, Crystal-fi eld.\n1 Introduction\nThe magnetic properties of two sublattices mixed spins −1/2 and spin\nS >1/2 Ising system, with a crystal field interaction, have been re cently\nstudied both experimentally and theoretically. In fact, a c rystal-field in-\nteraction effects on the transition temperature are investig ated by several\nmethodssuch as effective fieldtheory [1, 2, 3, 4], finitecluste r approximation\n[5], mean field theory [6], Migdal-Kadanof renormalisation group method [7]\nand cluster variational method [8]. However, there are some disagreements\namong those theoretical studies such as the existence of tri critical points\n∗benyous@fsr.ac.ma\n†e-mail: elkenz@fsr.ac.ma\n1and other features. Experimentally, the MnNi(EDTA)−6H2Ocomplex\nhas been shown to be an example of a mixed spin system [9]. Anot her inter-\nest towards the mixed spin Ising models can be related to the m odelling of\nmagnetic structures suitable for describing a ferrimagnet ism of certain class\nof insulating materials. Indeed, these systems provide sim ple but interest-\ning models to study molecular magnetic materials that are co nsidered to\nbe possibly useful materials for magneto-optical recordin gs. Furthermore,\nimportant advances have been made in the synthesis of two and three di-\nmensional ferrimagnets, such as 2 dorgano-metallic ferrimagnets [10, 11],\n2dnetworks of the mixed-metal materials [ P(Phenyl)4][MnCr(oxalate)3]n\n[12]. However, the possibility of many compensation points in a variety of\nferrimagnetic systems has been clarified theoretically [13 , 14].\nSince the ferrimagnetic order plays an important role in the se materials, the\ninvestigation of ferrimagnetism in mixed spin systems has r apidly become\na very rich field of research. As it is well known, these system s have less\ntranslational symmetrythantheirsingle-spincounterpar tssincetheyconsist\nof two interpenetrating and non equivalent sublattices. Fr om a theoretical\npoint of view, many different methods have been employed in the se studies.\nIn particular, the mixed spin −1/2 and spin −1 Ising model has been solved\nexactly in special cases [15, 16]. On the other hand, the use o f approximate\nmethods such as mean field theory, free-fermion approximati on, effective\nfield theory, high-temperature series expansions, renorma lisation group and\nMonte Carlo simulation, have revealed interesting results .\nMore recently, new magnetic properties and compensation be haviour are\nfound for mixed spins in the presence of a crystal-field [17]. Such magnetic\ncompensation behaviours were predicted by Nel theory of fer rimagnetism\n[18], but not revealed in previous works.\nOn the other hand, the effective-field theory study, and multic ritical be-\nhaviour, of the mixed spin Ising model with different anisotro pies have re-\nvealed contradictions withearlierworksobtainedwithint hesametheoretical\nframework. But the mean-field theory study based on the Bogol iubov in-\nequality fortheGibbsfreeenergy[19], it hasbeenshowntha t inthepresence\nof a single-ion anisotropy, the phase diagrams obtained exh ibit a variety of\nmulticritical points such as tricritical points and isolat ed critical points.\nOn the other hand, some recent experimental studies perform ed on various\nsingle-crystal samples RVO3(R=La,Nd,Sm,Gd,Er, andY) [20], showed\na temperature-induced and magnetization reversal.\nThe purpose of this work is to study, via mean field approximat ion (MFA),\nthe influence of crystal-field disorder on the phase diagrams and magneti-\nzations of a mixed-spin ferrimagnetic Ising system, in whic h the two inter-\npenetrating sublattices have spins σ=±1/2 andS=±3/2,±1/2. The\nmost interesting result emerging from this study is the appe arance of new\ntypes of phase diagrams, in the particular case of two-value d distribution\nof crystal-field. Consequently, three topologically differe nt types of phase\n2diagrams occur. This paper is organized as follows: in the se ction 2, we in-\ntroduce the model and give the details of the MFA. The ground- state phase\ndiagram is discussed in section 3. In section 4 we present and discuss our\nresults. Finally, section 5 is devoted to summarizes and con clusions.\n2 Model and method.\nSince the MFA method neglects correlations between different spins, it is\ninteresting to study the behaviour of complex spin systems, such as the\nferrimagnetic mixed Ising models. The model we are studying consists of\ntwo interpenetrating sublattices. Onesublattice has spin sσassumed to take\nthe values ±1/2, the other sublattice has spins Sthat can take four values:\n±3/2, and±1/2. The spins Shave only the spins σas nearest neighbours\nand vice versa. The interaction between the spins σandSis assumed to be\nan antiferromagnetic exchange. The Hamiltonian of this mod el is written\nas:\nH=J/summationdisplay\nσiSj+N/2/summationdisplay\ni=1∆iS2\ni (1)\nwhereNis the total number of lattice sites. The exchange interacti on\nparameter Jis assumed to be positive. The first summation is carried out\nonly over nearest pairs of spins and ∆ iis a quenched random crystal field\ndistributed according to the probability distribution [21 , 22, 23, 24]:\nP(∆i) =1\n2[δ(∆i−∆(1+α))+δ(∆i−∆(1−α))] (2)\nwhereαis a positive constant.\nAn analogous probability distribution has been used to inve stigate the crit-\nical behaviour of3He−4Hemixtures in random media (aerogel) modelled\nby the spin −1 Blume-Capel model. In this model, the negative crystal-fie ld\nvalue corresponds to the field at the pore-grain interface an d the positive\none is a bulk field that controls the concentration of3Heatoms [25, 26, 27].\nThe variational principle based on the Gibbs-Bogoliubov in equality for\nthe free energy per site is described by [28, 29, 30]:\nF ≤Φ =−Tln(Z0)+0 (3)\nLet us denote by hσandhSthe molecular fields associated with the order\nparameters mσ=< σ >0andmS=< S > 0, respectively, expressed as:\nhσ=Jz/summationdisplay\nj=1< Sj>0=zJmS (4)\n3hS=Jz/summationdisplay\nj=1< σ >0=zJmσ (5)\nwherezis the number of nearest neighbours and < ... > 0=Tr...exp (−βH0)\nexp(−βH0)\ndenotes the average value performed over the Hamiltonian H0.\nThe effective Hamiltonian of the system is given by:\nH0=hσN/2/summationdisplay\ni=1σi+hSN/2/summationdisplay\ni=1Si+N/2/summationdisplay\ni=1∆iS2\ni. (6)\nThe partition function generated by the above Hamiltonian i s :\nZ0=Tr(exp(−H0\nT))\n= (2cosh(βhσ\n2))N/2(2exp(−9β∆i\n4)cosh(3βhS\n2)+2exp(−β∆i\n4)cosh(βhS\n2))N/2(7)\nwhereTis absolute temperature and the Boltzmann’s constant has be en set\nto unity. The total free energy is given by:\nΦ =NJzm σmS\n2−Nhσmσ\n2−NhSmS\n2\n−NT/integraldisplay\nLog(Z0)P(∆i)d∆i. (8)\nAfter the integration over the probability distribution, t he free energy per\nspin is given by:\nΦ\nN=NJzm σmS\n2−Nhσmσ\n2−NhSmS\n2−t\n2[Log(2coshhσ\n2t)\n+1\n2(Log(2exp(−9d(1+α)\n4t)cosh(3hS\n2t)+2exp(−d(1+α)\n4t)cosh(hS\n2t))\nLog(2exp(−9d(1−α)\n4t)cosh(3hS\n2t)+2exp(−d(1−α)\n4t)cosh(hS\n2t)))](9)\nIn order to investigate the magnetizations of the system, th e order pa-\nrameters mσandmSare defined by minimizing the free energy. Then, the\nmean-field equations of state are expressed as follows:\nmσ=−1\n2tanh(z\n2tJmS) (10)\nmS=−1\n2[A\nB+C\nD] (11)\nwhere,\nA= 3exp(−9d\n4t(1+α))sinh(3zmσ\n2t)+exp(−d\n4t(1+α)))sinh(zmσ\n2t)\n4B= 2exp(−9d\n4t(1+α))cosh(3zmσ\n2t)+2exp(−d\n4t(1+α)))cosh(zmσ\n2t)\nC= 3exp(−9d\n4t(1−α))sinh(3zmσ\n2t)+exp(−d\n4t(1−α)))sinh(zmσ\n2t)\nD= 2exp(−9d\n4t(1−α))cosh(3zmσ\n2t)+2exp(−d\n4t(1−α)))cosh(zmσ\n2t)\nIn the above equations, and in all the following, tandddenote the reduced\ntemperature T/Jand the reduced crystal field ∆ /J, respectively. The co-\nordination number zis set to be 4 (square lattice).\nThe solutions of the Eqs. 10–11 are not unique, the stable one s are those\nminimizing the free energy Eq. 8, while the others are the uns table ones. If\nthe order parameters are continuous (discontinuous), the t ransitions are of\nsecond (first) order.\n3 Ground state\nThegroundstatephasediagramofthesystemunderinvestiga tion, according\nto the distribution of the crystal field law (Eq. 2), is illust rated in Fig. 1.\nIndeed, for very low temperatures and depending on the value s ofα≥0\nand the reduced crystal field d= ∆/J. Eqs. 10–11 lead to three solutions:\n(mσ=−1/2,mS= 3/2), (mσ=−1\n2,mS=1\n2), and (mσ=−1\n2,mS= 1).\nBy comparing the energies for all possible configurations, w e established the\nground state phase diagram. This phase diagram is drawn in th e reduced\n(d,α) plane. One can distinguish four cases:\n1- Forα= 0, a first order transition between the phase ( mσ=−1/2,mS=\n3/2) and the phase ( mσ=−1\n2,mS=1\n2) occurs at d= +1.\n2- For 0< α <1, two first-order transition lines occur: in one hand betwee n\nthe phase ( mσ=−1/2,mS= 3/2) and the phase ( mσ=−1\n2,mS= 1)\naccording to the equation line d=z/(4(1 +α)) and in the other hand\nbetween the phase ( mσ=−1\n2,mS= 1)and the phase ( mσ=−1\n2,mS=1\n2)\naccording to the equation line d=z/(4(1−α)).\n3- Forα= 1, a first order transition between the phase ( mσ=−1\n2,mS=3\n2)\nand the ferrimagnetic phase ( mσ=−1\n2,mS= 1) appears at d=1\n2.\n4- Forα >1, two first-order transition lines between the ( mσ=−1\n2,mS= 1)\nphase and the ( mσ=−1\n2,mS=3\n2) phase and the ( mσ=−1\n2,mS=3\n2)\nphase and the ( mσ=−1\n2,mS= 1) phase occur at d=z/(4(1−α)) and\nd=z/(4(1+α)), respectively.\nIt is worth to note that for d= 0, only the phase ( mσ=−1\n2,mS=3\n2)\nis stable at T= 0K. For a higher temperature, the phase diagrams for\ndifferent values of αanddwill be discussed in all the following.\n50,0 0,5 1,0 1,5 2,0 2,5 3,0 -3 -2 -1 01234Fig. 1 \n(-1/2,1/2) \n(-1/2,3/2) (-1/2,1) \n(-1/2,1) d\na\nFigure 1: The ground state phase diagram established in the ( d= ∆/J,α)\nplane. (−1/2,1/2), (−1/2,3/2) and (−1/2,1) are the only stable phases for\nvery low temperatures.\n4 Phase diagrams and discussions\n4.1 Phase diagrams\nA detailed discussion dealing with finite temperature phase diagrams is dis-\ncussed in this section. For this purpose, we solve numerical ly the Eqs. 10, 11\nand 8. A rich variety of phase transitions is observed both wh en varying\nαin the (tc=Tc/J,d= ∆/J) plane, and din the (tc,α) plane. Indeed,\nthe critical temperature is plotted as a function of dforα= 0 (Fig. 2a),\nα= 0.5 (Fig. 2b), α= 1 ( Fig. 2c) and α= 2 (Fig. 2d). In Fig.\n2a ,α= 0, the paramagnetic (0 ,0) and ferrimagnetic phases are separated\nby a second-order transition line (solid line). For very low temperatures,\nwe found a first-order transition line (dashed line) separat ing the phases\n(mσ=−1\n2,mS=3\n2) and (mσ=−1\n2,mS=1\n2). This first-order line is termi-\nnated by anisolated critical point located at ( dtr= 1,ttr= 0.099). Whereas,\nforα= 0.5 see Fig. 2b, we found two first-order transition lines. In on e\nhand, a first-order transition line separating the phases ( mσ=−1\n2,mS=3\n2)\nand(mσ=−1\n2,mS= 1), andterminated by an isolated critical point located\nat(dtr= 2/3,ttr= 0.15). Ontheotherhand,afirst-ordertransitionlinesep-\narating the phases ( mσ=−1\n2,mS= 1) and ( mσ=−1\n2,mS=1\n2), terminated\nby an isolated critical point located at ( dtr= 2,ttr= 0.045). Above these\npoints, a continuous passage to the paramagnetic phase occu rs. In Fig. 2c,\nplotted for α= 1, the phase ( mσ=−1\n2,mS=1\n2) disappears at low tempera-\nture and the only first-order transition line, present in thi s region, separates\n6the phases ( mσ=−1\n2,mS=3\n2) and (mσ=−1\n2,mS= 1), and terminated by\nan isolated critical point located at ( dtr= 1/2,ttr= 0.034). Therefore, the\nsecond order transition-line temperatures are greater tha n those found for\nα= 0 and α= 0.5. This phenomenon is still present for a higher value of α\n(see Fig. 2d for α= 2). Indeed, the ferrimagnetic phase ( mσ=−1\n2,mS= 1)\nemerges for large and positive values of the crystal-field. T his is due to a\ncompetition between positive and negative values of the cry stal-field which\nfavours the ferrimagnetic phase. Consequently, the system exhibits two\nfirst-order transition lines, for very low temperatures, te rminated by two\nisolated critical points: ( d=−1,t= 0.11) and ( d= 1/3,t= 0.19) separat-\ning the phases ( mσ=−1\n2,mS= 1) and ( mσ=−1\n2,mS=3\n2), and the phases\n(mσ=−1\n2,mS=3\n2) and (mσ=−1\n2,mS= 1) respectively.\nTo give more details concerning the first-order transition l ines, we have\ndeveloped, at low temperature, the free energy and entropy o f the system.\nIndeed, the free energy and entropy are plotted as function o fdfort= 0.035\nand two αvalues: 0 .25 and 1, in figures 3 aand 3b, respectively. In accor-\ndance with Figs. 2 band 2c, we observe a discontinuous change of the free\nenergy slope at first-order transition temperatures. Conse quently the en-\ntropy is discontinuous at these temperatures (see inset of F igs. 3aand 3b).\nFor the second-order transition lines, we have plotted the t otal free energy\nand entropy, versus temperature, for α= 0.25 andd=−3 (see Fig. 3 c).\nWe can remark that the entropy slope varies discontinuously at a second-\norder transition temperature. As consequence, the specific heat will exhibit\na discontinuity at this transition temperature.\nIn order to outline the effect of system behaviour as a function of the\nparameter αwe plot, in Figs. 4, the transition temperatures tcfor several\nvalues of the crystal field d. Indeed, for d= 2, Fig. 4a shows the existence\nof a second-order transition line between the paramagnetic and the partly\nferrimagnetic phases and a first-order transition between t he phases ( mσ=\n−1\n2,mS=1\n2) and (mσ=−1\n2,mS= 1) terminated by an isolated critical\npoint (α= 3/2,t= 0.055). Whereas for d= 0 , see Fig. 4b, the second\norder transition line from the ferrimagnetic phase to the pa ramagnetic one\nis independent on αvalues becauseof the absence of thecrystal field. InFig.\n4c, plotted for d=−1, we found a first order transition line separating the\nphases (mσ=−1\n2,mS=3\n2) and (mσ=−1\n2,mS= 1), which terminates at an\nisolated critical point ( α= 2,t= 0.09) between the ferrimagnetic and partly\nferrimagnetic phases. At high temperatures, a second order transition line\nbetween these phases and the paramagnetic one is found.\n4.2 Magnetic properties\nIn this part, we focus our interest on the magnetizations cor responding to\nthe previous phase diagrams as a function of the crystal field dand the\nparameter α, for fixed temperature values. Indeed, Figs. 5a and 5b show\n7the temperature dependence of magnetizations as a function ofdforα= 0\nandd= 0.5respectively. Theformerfigurepresentsafirstordertrans itionof\nthe magnetization mS, for a low temperature t= 0.05: from 3 /2 to 1/2 and\na second order transition of the magnetization mS, for a higher temperature\nt= 1.5: from 3 /2 to the paramagnetic phase. The second figure, Fig. 5b,\nshows a double first order transition of the magnetization mS, for a very low\ntemperature t= 0.04: from 3 /2 to 1 and from 1 to 1 /2, and a second order\ntransition for a higher temperature t= 1.1: from 3 /2 to the paramagnetic\nphase for increasing values of the crystal field d. In Fig. 5c we present\nthe behaviour of the magnetization mSas a function of the crystal field for\nα= 1 and two temperature values: t= 0.03 andt= 0.6. In agreement with\nFig. 2c, we found a discontinuity of this magnetization at a fi rst order point\nbetween the phases ( mσ=−1\n2,mS=3\n2) and (mσ=−1\n2,mS= 1), for a low\ntemperature ( t= 0.03). On the other hand, a continuous passage occurs\nfrom the same phases for a higher temperature ( t= 0.6).\nThe re-entrant behaviour found in Fig. 2d, is well illustrat ed in Figs.\n6a and 6b plotted, for α= 2, and two temperatures t= 2.2 andt= 2.3,\nrespectively. Indeed, Fig. 6a shows that for t= 2.2, the magnetizations\nmσ,mSand consequently M= (mσ+mS)/2 drop to zero in the region\n0< d <3.5. This is in good agreement with Fig. 2d. On the other hand,\nFig. 6b shows that for a higher temperature ( t= 2.3) this phenomenon is\ninverted in the region −3< d <0, so that the magnetizations mσ,mS, and\nconsequently M= (mσ+mS)/2, are nulls out of this region. This is once\nagain in agreement with Fig. 2d.\nLet us now show the behaviour of the magnetizations mσ,mS, andM=\n(mσ+mS)/2 as a function of the parameter αfor fixed values of crystal\nfield and temperature. For this purpose we plot in Figs. 7a and 7b these\nmagnetizations as a function of αfor (d= 2,t= 2) and ( d=−1,t= 2.5)\nrespectively. The first figure shows that for positive values of the crystal\nfield, the increasing αvalues effect is to force the ordered phase, in a region\nof temperatures lower than tc. This is the case in Fig. 7a, for t= 2. For a\nnegative value of the crystal field, this phenomenon is inver ted as it is shown\nin Fig. 7b, plotted for d=−1. One can note that the results found in Figs.\n7a and 7b, are in good agreement with those illustrated in Fig s. 4a and 4c,\nrespectively. On the other hand, it is obvious that the magne tizations are\nindependent of the parameter αin absence of the crystal field, as it is well\nillustrated in Fig. 4b.\nTo complete this study, we have investigated the effect of incr easing the\nexchange interaction parameter J, at very low temperature when keeping α\nconstant. Indeed, the results we found for a low temperature t= 0.025 and\nselected values of α, showed that thetransitions obtained inthegroundstate\nphase diagram (Fig. 1) are still present. For α= 0.5, Fig. 8a showed that\nthe system can exhibit the phases ( mσ=−1\n2,mS=1\n2), (mσ=−1\n2,mS= 1)\n8and (mσ=−1\n2,mS=3\n2) when increasing the parameter Jat a positive and\nconstant crystalfield( d >0). Forα= 1.0, thesystemundergoesatransition\nfrom the phase ( mσ=−1\n2,mS= 1) to the phase ( mσ=−1\n2,mS=3\n2) for\npositive and constant crystal field, when increasing the exc hange interaction\nJ, see Fig. 8b. For a large value of α(α= 2), Fig. 8c shows that the effect\nof increasing the parameter Jon the phase transitions ( mσ=−1\n2,mS=1\n2),\n(mσ=−1\n2,mS= 1) and ( mσ=−1\n2,mS=3\n2) is to displace these transitions\ntowards large and positive values of the crystal field.\n95 Conclusions\nWe have investigated a mixed spin σ= 1/2 and spin S= 3/2 Ising model on\na square lattice, using the mean field approximation ( MFA). The effect of a\nrandomcrystal fieldonthemagnetic propertiesofthesystem is investigated.\nIndeed, our results revealed many interesting phenomena, n amely, several\ntopologically different types of phase diagrams. Furthermor e, these phase\ndiagrams present rich varieties of phase transitions with fi rst and second\norder phase transition lines. These lines are linked by tric ritical points and\nterminated at isolated critical points. Finally, the effect o f increasing the\nexchange interaction parameter J, at very low temperature when keeping α\nconstant, is investigated. The results we found for a very lo w temperature\nand sleeted values of α, showed that the transitions obtained in the ground\nstate phase diagram are still present, but displaced toward s large and posi-\ntive values of the crystal field.\nReferences\n[1] T. Kaneyoshi, Physica A 153, 556 (1988).\n[2] T. Kaneyoshi, J. Magn. Magn. Mat. 92, 59 (1990).\n[3] A. Benyoussef, A. El Kenz and T. Kaneyoshi, J. Magn. Magn. Mat.\n131, 173 (1994).\n[4] A. Benyoussef, A. El Kenz and T. Kaneyoshi, J. Magn. Magn. Mat.\n131, 179 (1994).\n[5] N. Benayad, A. Klumper, J. Zittartz and A. Benyoussef, Z. Phys. B\n77, 333 (1989)\n[6] O. F. Abubrig, D. Horvak and M. Jascur, Physica A 296, 437 (2001).\n[7] N. Benayad, and J. Zittartz, Z. Phys. B 81, 107 (1990).\n[8] J. W. Tucker, J. Magn. Magn. Mat. 237, 437 (2001).\n[9] M. Drillon, E. Coronado, D. Beltran and R. Georges, J. Che m. Phys.\n79, 449 (1983).\n[10] H. Tamaki, Z. J. Zhong, N. Matsumoto, S. Kida, M. Korkawa , N.\nArchiwa, Y. Yashimoto, and H. 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X. de Carvalho and S. R. Salinas, Phys. Re v. B63,\n184415 (2001).\n[25] A. Maritan, M. Cieplak, M. R. Swift, F. Toigo and J. R. Ban avar, Phys.\nRev. Lett. 69, 221 (1992).\n[26] C. Buzano, A. Maritan and A. Pelizzola, J. Phys.: Conden s. matter 6,\n327 (1994).\n[27] N. S. Branco and B. M. Boechat, Phys. Rev. B 56, 11673 (1997).\n[28] N. N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947).\n[29] R. P. Feynmann, Phys. Rev. 97, 660 (1955).\n[30] J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newm an ”The\ntheory of critical phenomena”, Clarendon Press (1992).\n11-8 -6 -4 -2 0 2 4 6 80,0 0,5 1,0 1,5 2,0 2,5 3,0 \n(0,0) Fig. 2a \n(-1/2,1/2) (-1/2,3/2) \notc\nda=0 \n-8 -6 -4 -2 0 2 4 6 80,0 0,5 1,0 1,5 2,0 2,5 3,0 Fig. 2b \n(-1/2,1/2) (-1/2,1) \n(-1/2,3/2) \noo(0,0) tc\nda=0.5 \n-8 -6 -4 -2 0 2 4 6 80,0 0,5 1,0 1,5 2,0 2,5 3,0 \n(0,0) \n(-1/2,1) (-1/2,3/2) Fig. 2c \notc\nda=1.0 \n-8 -6 -4 -2 0 2 4 6 80,0 0,5 1,0 1,5 2,0 2,5 3,0 Fig. 2d \n(-1/2,3/2) \n(-1/2,1) (-1/2,1) \noo(0,0) tc\nda=2.0 \nFigure 2: The critical temperature as a function of dplotted for: α= 0\n(a),α= 0.5 (b),α= 1 (c) and α= 2 (d). The full lines correspond to the\nsecond-order transitions, whereas the dashed lines repres ent the first-order\ntransitions. The tiny circles denote the isolated critical points. 120.0 0.5 1.0 1.5 2.0 2.5 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 \n0.0 0.5 1.0 1.5 2.0 20 30 40 50 60 70 80 90 Entropy \ndα =0.25 \nt=0.035 Free energy \ndα =0,25 \nt=0,035 \n0.0 0.5 1.0 1.5 2.0 2.5 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 \n-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 55 60 65 70 75 80 85 90 Entropy \ndα =1,0 \nt=0,035 Free energy \ndα =1,0 \nt=0,035 \n0 1 2 3 4 5-7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 \n0 1 2 3 4 50.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Entropy \ntα =0.25 \nd=-3.0 Free energy \ntα =0.25 \nd=-3.0 \nFigure3: In aandb,thefreeenergyandentropy(fromexpressionsdeveloped\nat low temperature) are plotted versus the reduced crystal- fielddfort=\n0.035 and two αvalues: 0 .25 and 1, respectively. In c, the same physical\nquantities are plotted as a function of temperature for α= 0.25 andd=−3.130 1 2 3 40,0 0,5 1,0 1,5 2,0 2,5 \nCFig. 3a \n(0,0) \n(-1/2,1) (-1/2,1/2) \notc\nad=2.0 \n0 1 2 3 40,0 0,5 1,0 1,5 2,0 2,5 Fig. 3b \n(-1/2,3/2) (0,0) tc\nad=0 \n0 1 2 3 40,0 0,5 1,0 1,5 2,0 2,5 3,0 \n(0,0) \n(-1/2,1) (-1/2,3/2) \noFig. 3c tc \nad=-1.0 \nFigure 4: The transition temperatures tcas a function of the parameter α\nfor selected values of the crystal field: d= 2 (a) , d= 0 (b), and d=−1 (c).\n14-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 0,00 0,25 0,50 0,75 1,00 1,25 1,50 a=0 Fig. 4a magnetization: m s\nd t=0.05 \n t=1.5 \n-1 0 1 2 3 40,00 0,25 0,50 0,75 1,00 1,25 1,50 a=0.5 Fig. 4b Magnetization: m s\nd t=0.04 \n t=1.1 \n-1,0 -0,5 0,0 0,5 1,0 1,5 0,9 1,0 1,1 1,2 1,3 1,4 1,5 a=1 Fig. 4c Magnetization: m s\nd t=0.03 \n t=0.6 \nFigure 5: The dependency of magnetization mSas a function of the reduced\ncrystal-file dfor a fixed value of αand two reduced temperature values:\nα= 0 and ( t= 0.05 andt= 1.5) (a),α= 0.5 and (t= 0.04 andt= 1.1) (b)\nandα= 1.and (t= 0.03 andt= 0.6) (c), respectively..15-10 -5 0 5 10 -0,25 0,00 0,25 0,50 Fig. 5a \nα=2 \nt=2.2 Magnetizations \nd m s\n m σ\n m s+m σ\n-5 -4 -3 -2 -1 0 1 2-0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 α=2.0, t=2.3 Fig. 5b Magnetizations \nd m s\n m s\n m s+m s\nFigure6: Thedependencyof magnetizations mS,mσandM= (mσ+mS)/2\nas a function of the reduced crystal-file dforα= 2, and two temperature\nvalues:t= 2.2 (a) and t= 2.3 (b), respectively.\n160,0 0,5 1,0 1,5 2,0 2,5 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 Fig. 6a \nd=2 \nt=2 Magnetizations \nα m s\n m σ\n m s+m σ\n0,0 0,5 1,0 1,5 2,0 2,5 -0,1 0,0 0,1 0,2 0,3 Fig. 6b \nd=-1 \nt=2.5 Magnetizations \nα m s\n m σ\n m s+m σ\nFigure 7: The behaviour of the magnetizations mσ,mS, andM= (mσ+\nmS)/2 as a function of the parameter αfor fixed values crystal field and\ntemperature for ( d= 2,t= 2) in (a) and ( d=−1,t= 2.5) in (b).\n170 1 2 3 4 5 60246810 12 \n(−1/2,1/2) \n(−1/2,1) \n(−1/2,3/2) Fig. 7a Δ\nJα=0.5 \n0 1 2 3 4 5 60,0 0,5 1,0 1,5 2,0 2,5 3,0 \n(−1/2,1) \n(−1/2,3/2) Fig. 7b Δ\nJα=1 \n0 1 2 3 4 5-3 -2 -1 012\n(−1/2,1) \n(−1/2,3/2) \n(−1/2,1) Fig. 7c Δ\nJα=2 \nFigure 8: Phase diagrams in the plane (∆ ,J), at very low temperature t=\n0.025 forα= 0.5 (a),α= 1 (b) and α= 2 (c). In (a) the system exhibits the\nphases (mσ=−1\n2,mS=1\n2), (mσ=−1\n2,mS= 1) and ( mσ=−1\n2,mS=3\n2), in\n(b) the system undergoes a transition from the phase ( mσ=−1\n2,mS= 1) to\nthe phase ( mσ=−1\n2,mS=3\n2), while (c) shows the phases ( mσ=−1\n2,mS=\n1\n2), (mσ=−1\n2,mS= 1) and ( mσ=−1\n2,mS=3\n2) when increasing the\nexchange interaction parameter Jfor fixed values of the crystal field ∆.18" }, { "title": "1612.06300v1.Completely_compensated_ferrimagnetism_and_sublattice_spin_crossing_in_the_half_metallic_Heusler_compound_Mn1_5FeV0_5Al.pdf", "content": "arXiv:1612.06300v1 [cond-mat.str-el] 19 Dec 2016Completely compensated ferrimagnetism and sublattice spi n crossing in the\nhalf-metallic Heusler compound Mn 1.5FeV0.5Al.\nRolf Stinshoff,1Ajaya K. Nayak,1,2Gerhard H. Fecher,1Benjamin Balke,3\nSiham Ouardi,1Yurii Skourski,4Tetsuya Nakamura,5and Claudia Felser1\n1Max Planck Institute for Chemical Physics of Solids, 01187 D resden, Germany\n2Max Planck Institute of Microstructure Physics, 06120 Hall e, Germany\n3Institut f¨ ur Anorganische und Analytische Chemie,\nJohannes Gutenberg - Universit¨ at, 55099 Mainz, Germany\n4Dresden High Magnetic Field Laboratory (HLD), 01328 Dresde n, Germany\n5Japan Synchrotron Radiation Research Institute, SPring-8 , Hyogo 679-5198, Japan\n(Dated: March 22, 2018)\nThe Slater–Pauling rule states that L21Heusler compounds with 24 valence electrons do never\nexhibit a total spin magnetic moment. In case of strongly loc alized magnetic moments at one of the\natoms (here Mn) they will exhibit a fully compensated half-m etallic ferrimagnetic state instead, in\nparticular, when symmetry does not allow for antiferromagn etic order. With aid of magnetic and\nanomalous Hall effect measurements it is experimentally dem onstrated that Mn 1.5V0.5FeAl follows\nsuch a scenario. The ferrimagnetic state is tuned by the comp osition. A small residual magneti-\nzation, that arises due to a slight mismatch of the magnetic m oments in the different sublattices\nresults in a pronounced change of the temperature dependenc e of the ferrimagnet. A compensation\npoint is confirmed by observation of magnetic reversal and si gn change of the anomalous Hall ef-\nfect. Theoretical models are presented that correlate the e lectronic structure and the compensation\nmechanisms of the different half-metallic ferrimagnetic st ates in the Mn-V-Fe-Al Heusler system.\nPACS numbers: 75.50.Gg, 75.50.Cc, 75.30.Gw, 72.15.Jf\nKeywords: Compensated ferrimagnets, half-metallic ferri magnets, Heusler compounds\nHalf-metallic ferromagnets are promising candidates\nfor application in spintronics because they exhibit 100%\nspin polarization. They are metallic in one spin direc-\ntion and semiconducting in the other [1, 2]. However,\nferromagnets produce a large dipole field that hinders\nthe device performance. For example, the dipolar mag-\nneticanisotropybecomesverylargeforin-planemagnetic\nsystems leading to large switching fields. For this reason,\nthere is a great interest on zero magnetic moment spin-\ntronics, as such systems do not produce dipole fields and\nare extremely stable against external magnetic fields [3–\n6]. The concept of half-metallic antiferromagnetism was\nintroduced by van Leuken and de Groot [7]. It turns out,\nhowever, that symmetry does not allow half-metallic an-\ntiferromagnets and the materials are half-metallic com-\npensated ferrimagnets [8]. Recently, Hu [9] presented a\ntheoretical work on possible half-metallic antiferromag-\nnets for spintronic applications. However, the identical\nelectronic structure of both spin directions makes most\nof the conventional antiferromagnets unable to carry a\nspin polarized current.\nHeusler materials are well known for their tunable\nmagnetic structure due to the presence of one or more\nmagnetic sublattices. Depending on the constituting ele-\nments or crystal structure, ferromagnetic, ferrimagnetic,\nantiferromagnetic, or canted spin structures may be re-\nalised [10–14]. In particular, the Heusler compounds\nwithL21orC1bstructure are well known for their half-\nmetallic behaviour[1]. These materialsfollowthe Slater–\nPauling rule [15, 16] related to the half-metallicity [17–19]. According to this rule the spin magnetic moment\n(m) in cubic Heusler compounds with L21structure is\ndefined by m=Nv−24, where Nvis the accumulated\nnumber of valence electrons. As a direct consequence,\nHeusler compounds with Nv= 24 never exhibit a macro-\nscopic magnetic moment.\nIncertaincases,however,the DO3orL21Heuslercom-\npounds with 24 valence electrons are able to exhibit a\nfully compensated half-metallic behaviour [8]. In that\nconcept, the Slater–Pauling rule is combined with the\nK¨ ubler rule [20]. The latter states that Mn on the octa-\nhedrally coordinated position (4 b) in Heusler compounds\ntends to a high, localised magnetic moment. This mo-\nment has to be completely compensated by the magnetic\nmoments of the remaining atoms to satisfy the Slater–\nPauling rule.\nAlthough there are several theoretical predictions,\nmost of the suggested materials either do not exist or\nappear only in a different crystal structure [7]. Recently\nit was demonstrated that a compensated ferrimagnetic\nstate may be realized in the tetragonal Mn-Pt-Ga sys-\ntem [21]. However, it is known that Heusler materials\nwith tetragonal distortion do not show half-metallicity.\nKurtet al[22, 23] have shown that a compensated mag-\nnetic state with considerable spin polarization may be\nachieved in a cubic thin film of Mn 2RuxGa with compo-\nsition falling between C1bandL21Heusler compounds.\nDespite several attempts by different research groups\nthere is no experimental evidence of a compensated mag-\nnetic structure in the classical 24 valence electron based2\ncubicHeuslercompounds. Inthepresentworkitisshown\nby experiments and calculations that the Heusler com-\npound Mn 1.5V0.5FeAl with L21structure exhibits a com-\npletely compensated magnetic state. Further, the pres-\nence of a temperature and composition dependent sub-\nlattice spin compensation is demonstrated in the investi-\ngated system, while keeping the half-metallicity.\nPolycrystalline ingots of Mn 1.5V0.5FeAl were prepared\nby arc melting. The composition and structure of the\nsamples was determined by energy dispersive X-ray spec-\ntroscopy (EDX) and X-ray powder diffraction (XRD).\nLow field magnetic measurements were carried out by\nmeans of a vibrating sample magnetometer (MPMS 3,\nQuantum Design). Pulsed, high magnetic field exper-\niments were performed at the Dresden High Magnetic\nField Laboratory. The transport measurements were\ncarried out utilising a physical property measurement\nsystem (PPMS, Quantum Design). X-ray magnetic cir-\ncular dichroism (XMCD) investigations were performed\nat beamline BL25SU of SPring-8. The electronic struc-\nture was calculated in the local spin density approxima-\ntion. The selfconsistent electronic structure calculations\nwere carried out using the spin polarized fully relativistic\nKorringa–Kohn–Rostockermethod (SPRKKR) provided\nby Ebert et al[24, 25].\n/s53/s48/s53\n/s69/s110/s101/s114/s103/s121 /s32/s32/s32 /s69\n/s70/s32/s91/s101/s86/s93\n/s40/s98/s41\n/s32\n/s70 /s32/s45/s52/s51 /s109/s40/s97/s41\n/s40/s99/s41\n/s40/s100/s41/s65/s108\n/s77 /s110/s73\n/s77 /s110/s73/s73\n/s70/s101\n/s86\n/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115/s32/s32/s32 /s110 /s40/s69 /s41/s32/s91/s101/s86/s45/s49 \n/s93\n/s32/s32\n/s70/s32/s109/s45/s51/s109\nFIG. 1. (Color online) Crystalline and electronic structur e of\nMn1.5V0.5FeAl\n(a) The L21type cubic Heusler structure with space group\nF m3m(225). (b) The Xtype inverse cubic Heusler struc-\nture with space group F-43m(216). Different atoms are rep-\nresented by balls with different colours, shown in between th e\ntwo structures. The spin resolved density of states is shown\nforF m3min (c) and for F43min (d).\nThe XRD analysis indicates that Mn 1.5V0.5FeAl crys-tallizes in a cubic Heusler structure with a lattice param-\neter ofa= 5.83˚A. The two most energetically favoured\ncrystal structures are shown in Figure 1(a) and (b) to-\ngether with their magnetic order. In the regular L21\ncubic Heusler structure with space group F m3m(225),\nthe Al atoms occupy the 4 aposition, the 4 bposition\nis equally occupied by V and Mn atoms and a statis-\ntical distribution of the Mn and Fe atoms at 8 cis ex-\npected. In a less probable situation, an ordering of the\nMn and Fe atoms can split the 8 cposition to 4 cand\n4d, as shown in Figure 1(b). In order to determine the\ndensity of states (DOS) of Mn 1.5V0.5FeAl, the calcula-\ntions were performed using SPRKKR with coherent po-\ntential approximation (CPA) to account for the random\noccupation of the sites and for chemical disorder. Com-\nparing the total energies at the same lattice parameter,\none finds that the energy of the L21structure with space\ngroupFm3m(Figure1(a)) is 0.5meVlowercomparedto\ntheXstructure with space group F43m(Figure 1(b)).\nThe electronic structure reveals clearly the half-metallic\ncharacterof Mn 1.5V0.5FeAl for both structure types with\nchemical disorder. The gap in the minority DOS is de-\nfined by the states of the Mn atoms located on the 8 c\nand the Fe atoms located on the 8 cor 4cpositions. This\ncoincides with previous calculation for various Heusler\ncompounds, as in most cases, the gap is dominated by\nthe states arising from the atom on the 8 csite [19].\nTABLE I. Site specific magnetic moments in Mn 1.5FeV0.5Al.\nThe calculations were carried out by means of SPRKKR -\nCPA using the L21structure ( F m3m, 225) or the X-type\nstructure ( F43m, 216). All magnetic moments are given in\nµB. The total moments are given per primitive cell. The site\nspecific spin msand orbital mlmagnetic moments are given\nper atom. Note the rounding, the induced moment at Al is\n<0.006µB.\n225 216\nAtom Site msmlSite msml\nMn (8 c) 1.40 0.03 (4 d) 1.38 0.03\nFe (8 c) 0.28 0.02 (4 c) 0.35 0.03\nMn (4 b) -2.79 -0.01 (4 b) -2.86 -0.01\nV (4 b) -0.55 0.01 (4 b) -0.60 0.01\nAl (4 a) -0.01 -0.00 (4 a) -0.01 -0.00\nms,l\ntot 0.003 0.046 0.000 0.058\nmtot 0.05 0.06\nThe calculated magnetic moments of Mn 1.5FeV0.5Al\nare listed in Table I. A summation of the site specific\nmagnetic moments yields a zero total moment, as ex-\npected for a completely compensated ferrimagnet. The\nsignsofthecalculatedmagneticmomentsweresupported\nby XMCD measurements. XMCD spectra were calcu-\nlated for the L21structure with Mn on 8 cmixed with Fe3\nand on 4 bmixed with V. The two different Mn atoms\ncause a zero-crossing with pronounced maximum and\nminimum at the L3edge, that is also clearly revealed\nin the measured spectra. Although the site specific mo-\nment of the Mn atoms could not be determined due to\noverlap of the lines from the Mn atoms at two different\nsites, the total sum moment and moments obtained for\nFe and V from the XMCD measurements well matched\nwith the theoretical values.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s46/s48/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s40/s97/s41/s77 /s40/s84 /s41/s32/s91\n/s66/s93\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s32 /s84 /s32/s91/s75/s93/s32/s32/s48/s46/s49/s32/s84\n/s32/s32\n/s40/s98/s41\n/s32/s48/s46/s53/s32/s84\n/s32/s48/s46/s49/s32/s84/s32/s32/s77 /s40/s84 /s41/s32/s47/s32 /s77 /s40/s48/s41\n/s40/s99/s41\n/s109\n/s73 /s73 /s32/s61/s32 /s50/s42 /s109\n/s73 /s40/s100/s41\n/s109\n/s73 /s73 /s32/s60/s32 /s50/s42 /s109\n/s73 \n/s82/s101/s108/s97/s116/s105/s118/s101/s32/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s32 /s84 /s32/s47/s32 /s84\n/s67\n/s32/s32\n/s32/s77\n/s73 /s73 \n/s32/s124 /s77\n/s116/s111/s116/s124 \n/s32/s77\n/s73 \nFIG. 2. (Color online) Magnetisation of Mn 1.5FeV0.5Al.\n(a) shows the temperature dependence of the magnetiza-\ntionM(T), measured for a completely compensated sample.\n(b) shows the temperature dependence of the magnetization\nmeasured for an overcompensated sample in different fields.\nThe theoretical behaviour of a two sublattice ferrimagnet i n\nthe molecular field approximation is shown in (c) and (d):\n(c) shows the magnetization for a completely compensated\nferrimagnet with |mII|=|2mI|, (d) shows the case with\n|mII|<|2mI|.miare the average magnetic moments of\nthe atoms on the ithsublattice at T= 0.\nFigure 2a shows the temperature dependence of the\nmagnetisation M(T) for a completely compensated sam-\nple. As expected, the magnetization vanishes at 0 K as is\ntypical for a completely compensated ferrimagnet. The\nmagnetisation stays close to Zero up to about 50 K. The\nCurie temperature appears at about 335 K.\nM(T) curves measured for a slightly overcompensated\nsample in different induction fields are shown in Fig-\nure 2b. From the M(T) curves measured in an induction\nfield of 0.1 T a Curie temperature ( TC) of about 308 K is\ndetermined. By decreasingthe temperature the magneti-\nzation first completely reduces to zero at 127 K and then\nincreasesagainbyloweringthe temperaturebelow127K.\nThis type of magnetic behaviour indicates the presence\nof a compensation point of the ferrimagnetic order. Thecompletely compensated behaviour is very sensitive to\nthe composition of the sample as will be shown next.\nThe temperature dependence of the total and the sub-\nlatticemagneticmomentsweresimulatedusingamolecu-\nlarfieldmodel foratwosublatticeferrimagnet. Inpartic-\nular the equations introduced by Stearn [26] for binary\ncompounds (Fe 3Al, Fe 3Si) with Heusler type structure\nwere used. This model may come close to the L21struc-\nture with space group 225. It is assumed that the mag-\nnetic moment mIof the atoms in sublattice I is smaller\n(half) but that twiceasmanyatomsareoccupyinglattice\nI. That is, lattice I describes the 8 csite, whereas lattice\nII corresponds to the 4 bsites with higher magnetic mo-\nmentsmIIbut only half as many atoms ( nI/nII= 2)\ncompared to lattice I. The completely compensated fer-\nrimagnet appears when nImI=nIImII. The exchange\nintegrals Jijshould be largest for interactions between\nthe atomsin sites I and II. Further the exchangeintegrals\nbetween atoms of type I should be much smaller com-\npared to the atoms of type II, with the latter being close\ntothose betweentype I andII atoms. In particularit was\nassumed that JI/JI−II= 1/2andJII/JI−II= 2/3. The\nresults are shown in Figures 2c and 2d that compare the\ncompletelycompensatedcasewithaslightlyovercompen-\nsated case. Figure 2c describes a ferrimagnet where the\ncompensation point appears at T= 0 and the magneti-\nzation stays nearly Zero up to about T/TC≈1/5. For\nmII<|2mI|, a compensation point appears (Figure 2d).\nThe latter is classified as a N´ eel N-type ferrimagnet [27].\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s49/s54/s45/s48/s46/s49/s50/s45/s48/s46/s48/s56/s45/s48/s46/s48/s52/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s77/s97/s103/s110/s101/s116/s105/s115/s97/s116/s105/s111/s110/s32/s32/s32 /s77 /s40/s72 /s41/s32/s91\n/s66/s93\n/s73/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93/s32/s50/s54/s51/s32/s75\n/s32/s32/s32/s32/s32/s50/s32/s75\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49\n/s40/s97/s41/s40/s98/s41/s48/s72\n/s99/s32/s91/s84/s93\n/s84 /s32/s91/s75/s93\nFIG. 3. (Color online) Field dependent magnetisation of\nMn1.5FeV0.5Al.\nShown is the magnetisation M(H) at 2 K and 263 K. Inset\n(a)shows thelowtemperaturebehaviouronan enlargedscale .\nInset(b)shows thetemperature variation ofthe coercive fie ld.\nFigure 3 shows the field dependence of the magneti-\nsation for two different temperatures, at the maximum\nof the magnetisation (263 K) and close to Zero (2 K).\nAt high temperatures the material appears very soft but\nhas a small remanence and coercive field at low tem-4\nperature. The inset (b) shows that the coercive field is\nconstant below 50 K where the magnetisation vanishes\n(compare Figure 2a). Above this critical temperature,\nthe magnetisation softens with increasing temperature.\nThe appearance of a coercive field is a typical effect at\nthe compensation point [28]. In certain cases it is as-\nsumed to diverge at the compensation point, whereas it\nclearly saturates below the critical temperature in the\ncompletely compensated half-metallic ferrimagnet.\nSo far the occurrence and some magnetic properties\nof a completely compensated half-metallic ferrimagnet is\ndemonstrated. The compensation phenomenon is better\nstudied, however,in the slightlyovercompensatedsample\nwith a compensation point at a finite temperature. The\nremaining part is thus devoted to this case.\nTheM(H) loops measured at different temperatures\ndemonstrate the compensation phenomenon (Figure 4a)\nin the overcompensated sample. A nearly linear hystere-\nsis loop with almost zero spontaneous magnetization is\nfound in the vicinity of the compensation temperature\n(127 K). The M(H) loops measured for temperatures\nbelow and above the compensation point exhibit a soft\nmagnetic behaviour. The most important point is that\nbothM(T) andM(H) measurements hint on a satura-\ntion magnetization that is less than 0.1 µBaway from the\ncompensation point. This suggests that the sample vir-\ntually exhibits a nearly compensated magnetic state over\nthe full temperature range.\nIt isseenfromFigure2b thatthe minimum atthecom-\npensation point shifts slightly with increasing induction\nfield. For a deeper understanding of this effect we have\nmeasured Zero field cooled (ZFC) and field cooled (FC)\nM(T) curvesin avery small field of 2 mT (Figure 4b). In\nthiscasetheFCcurve,whichshowsapositivemagnetiza-\ntion at higher temperature, crosses the temperature axis\nat 127 K to give a negative magnetization at low tem-\nperatures. The ZFC curve follows an exactly opposite\nbehaviour to that of the FC curve. The zero-crossing of\nthe magnetization clearly indicates a sublattice magnetic\ncompensation at 127 K. Similar magnetic reversal at the\ncompensation point has been observed in systems with\nspin-orbital compensation [29]. Pulsed magnetic field\nmeasurements at 1.5 K and at the compensation point\n(127 K) show a linear magnetic response with fields up\nto 55 T without any spin-flop transition. This clearly in-\ndicates a strong exchange coupling between the different\nmagnetic sublattices in Mn 1.5V0.5FeAl.\nThe magnetic measurements shown in Figures 2(a)\nand (b) only give an indication of a sublattice spin cross-\ning at the compensation point. Anomalous Hall effect\n(AHE) measurements have been performed at different\ntemperatures, to allow for a direct observation of the\nspin crossing across the compensation point (see Fig-\nure 4). The AHE measured at 50 K and 100 K shows\na negative sign, i.e, negative (positive) value in positive\n(negative) field. At the compensation point the AHE be-/s45/s48/s46/s48/s52/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52\n/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s97/s41\n/s32/s49/s48/s48/s32/s75\n/s32/s49/s50/s55/s32/s75\n/s32/s49/s53/s48/s32/s75/s77 /s32/s91\n/s66/s93\n/s32/s32\n/s40/s98/s41\n/s32/s32\n/s32/s90/s70/s67\n/s32/s70/s67/s32/s32/s40\n/s48/s72 /s32/s61/s32/s50/s32/s109/s84/s41\n/s32/s32\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s32 /s84 /s32/s91/s75/s93/s32/s45/s48/s46/s49/s32/s84\n/s32/s43/s48/s46/s49/s32/s84\n/s32/s49/s48/s48/s32/s75/s32/s49/s53/s48/s32/s75\n/s32/s49/s50/s55/s32/s75/s120/s121/s32/s91 /s99/s109/s93\n/s73/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93/s32\n/s32/s40/s99/s41 /s40/s100/s41\nFIG. 4. (Color online) Magnetic and transport properties of\novercompensated Mn 1.5FeV0.5Al.\n(a) Isothermal magnetization loops, M(H), at different tem-\nperatures. (b) ZFC and FC M(T) curves measured in a small\nfield of 2 mT. (c) Field dependence of the Hall effect mea-\nsured at different temperatures. (d) Temperature dependenc e\nof Hall resistivity measured in ±0.1 T.\ncomes virtually zero. Above the compensation point for\nT= 150 K and 200 K, a positive anomalous Hall effect\nis observed. The change in sign of the AHE can be seen\nin the temperature dependence of the AHE measured in\na field of ±0.1 T (Figure 4d). The AHE changes from\na negative (positive) maximum around 100 K to a pos-\nitive (negative) maximum at 200 K when measured in\na field of 0.1 T (-0.1 T). The two curves cross the zero\nline at about 130 K. Since the AHE is an intrinsic prop-\nerty of ferro- and ferrimagnets, a small uncompensated\nmoment below and abovethe compensation point will re-\nsult in a non-vanishing AHE. The most important point\nis that the AHE changes its sign, which clearly indicates\nthe change of the sublattice magnetic structure across\nthe compensation point. The AHE is an intrinsic man-\nifestation of a Berry curvature, that changes due to the\nchange of the sublattice magnetic moment from spin-up\nto spin-down, resulting in a change of the sign across the\ncompensation point.\nIn conclusion, the existence of a completely compen-\nsated ferrimagnetic state in the half-metallic L21cubic\nHeusler compound Mn 1.5V0.5FeAl has been experimen-\ntally demonstrated. Although there have been several\ntheoretical works regarding realization of a fully com-\npensated magnetic state in the L21cubic Heusler com-\npounds with 24 valence electrons, no successful exper-\nimental attempt has been made until now. This work\nalso establishes the existence of a temperature depen-\ndent sublattice spin crossing in half-metallic ferrimag-\nnets. The compensation temperature can be varied by\nan intentional variation of the stoichiometry. Recently, it\nhasbeendemonstratedthatantiferromagnetsmaybeuti-5\nlized as a principal component in spintronic devices, es-\npecially in tunnel magnetoresistance based devices. The\npresent half-metallic compensated ferrimagnet adds the\nadvantage of nearly 100% spin polarization, which is ex-\ntremely important for spintronics.\nThe authors thank N. Demitri for assistance during\nthe XRD experiment at ELETTRA. This work is funded\nby the Deutsche Forschungs Gemeinschaft (project 1.3-\nA in research unit 1464 ASPIMATT ) and by the ERC\nAdvanced Grant (291472) Idea Heusler . The experi-\nments at the High Magnetic Field Laboratory Dresden\n(HLD) were supported by Euro-MagNET II under the\nEuropean Union contract 228043. 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Rev.\nB66, 161319(R) (2002)." }, { "title": "0812.3897v1.Origin_of_the_Ising_Ferrimagnetism_and_Spin_Charge_Coupling_in_LuFe2O4.pdf", "content": "arXiv:0812.3897v1 [cond-mat.mtrl-sci] 19 Dec 2008Origin of the Ising Ferrimagnetism and Spin-Charge Couplin g in\nLuFe2O4\nH. J. Xiang,1E. J. Kan,2Su-Huai Wei,1M.-H. Whangbo,2and Jinlong Yang3\n1National Renewable Energy Laboratory, Golden, Colorado 80 401, USA\n2Department of Chemistry, North Carolina State University,\nRaleigh, North Carolina 27695-8204, USA\n3Hefei National Laboratory for Physical Sciences at Microsc ale,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, P. R. China\n(Dated: October 8, 2018)\nAbstract\nThe spin ordering and spin-charge coupling in LuFe 2O4were investigated on the basis of den-\nsity functional calculations and Monte Carlo simulations. The 2:1 ferrimagnetism arises from the\nstrong antiferromagnetic intra-sheet Fe3+-Fe3+and Fe3+-Fe2+as well as some substantial antifer-\nromagnetic Fe2+-Fe3+inter-sheet spin exchange interactions. The giant magneto capacitance at\nroom temperature and the enhanced electric polarization at 240 K of LuFe 2O4are explained by\nthe strong spin-charge coupling.\nPACS numbers: 75.80.+q,71.20.-b,77.80.-e,64.60.De\n1Recently, multiferroics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] have attracte d much attention be-\ncause of their potential applications in novel magnetoelectric and m agneto-optical devices.\nAmong the newly discovered multiferroics, LuFe 2O4is particularly interesting due to its\nlarge ferroelectric (FE) polarization [3] and giant magnetocapacita nce at room temperature\n[4]. In the high-temperature crystal structure of LuFe 2O4with space group R ¯3m, layers of\ncomposition Fe 2O4alternate with layers of Lu3+ions, such that there are three Fe 2O4layers\nper unit cell. Each Fe 2O4layer is made up of two triangular sheets (hereafter, T-sheets) o f\ncorner-sharing FeO 5trigonal bipyramids (Fig. 1). Below 320 K ( TCO) LuFe 2O4undergoes a\nthree-dimensional (3D) charge ordering (CO) (2Fe2.5+⇒Fe2++ Fe3+) with the√\n3×√\n3\nsuperstructure in each T-sheet; in each Fe 2O4layer, one T-sheet has the honeycomb net-\nwork of Fe2+ions with a Fe3+ion at the center of each Fe2+hexagon (hereafter, the type\nA T-sheet), while the other T-sheet has an opposite arrangement of the Fe2+and Fe3+ions\n(hereafter the type B T-sheet).\nLuFe2O4, with the novel CO-driven “electronic ferroelectricity”, [3] prese nts several fun-\ndamental questions. First, LuFe 2O4shows strong Ising behavior with the easy axis along c\n[11, 12]. The spin anisotropy of the non-CO state is understandable because the spin down\nelectron of the Fe2.5+ion partially occupies the degenerate ( dx2−y2,dxy) orbitals [5, 13]. How-\never, the Ising behavior below TCOis puzzling because the insulating√\n3×√\n3 CO breaks\nthe 3-fold rotational symmetry hence lifting the degeneracy of th e (dx2−y2,dxy) orbitals [5].\nSecond, LuFe 2O4undergoes a ferrimagnetic spin ordering below 240 K ( TN) [11, 14, 15, 16].\nA number of experimental studies found this spin ordering to be two -dimensional (2D) in\nnature [11, 14, 17]. In contrast, a recent neutron diffraction stu dy observed a finite spin\ncorrelation along cand suggested a 3D spin structure without considering CO [16]. The\nM¨ ossbauer [14] and neutron diffraction [15] studies led to a detailed ferrimagnetic structure\nof LuFe 2O4, in which the majority spin lattice consists of all Fe2+ions plus one-third of the\ntotal Fe3+ions while the minority spin sublattice consists of the remaining Fe3+ions. This\n2:1 ferrimagnetic order was suggested to originate from weak ferr omagnetic (FM) interac-\ntions between the next-nearest neighbor (NNN) Fe sites in the tria ngular antiferromagnetic\n(AFM) Ising lattice [11]. However, using the spin exchange paramete rs estimated from the\nenergy parameters of LaFeO 3, Nakaet al.[18] predicted quite a different spin structure that\nincludes some Fe sites without unique spin direction. Therefore, the detailed ferrimagnetic\nstructure and its origin remain unclear. Third, LuFe 2O4exhibits a giant magnetodielec-\n2tric response at room temperature [4], and a room-temperature d ynamic magnetoelectric\ncoupling was also reported [19]. Furthermore, the FE polarization o f LuFe 2O4was found\nto increase around TN[3]. These observations suggest the occurrence of coupling betwe en\nthe CO and magnetism. The understanding of the spin-charge coup ling is crucial for future\nmagnetodielectric applications of LuFe 2O4.\nIn this Letter, we explore these isuues on the basis of first principle s density functional\ncalculations for the first time. A large spin anisotropy is found along t hecdirection due\nmainly to the Fe2+ions of the B-sheet, the spin ground state of the√\n3×√\n3 CO state has\nthe 2:1 ferrimagnetic spin arrangement proposed by Siratori et al.[15], and there occurs\nstrong spin-charge coupling in LuFe 2O4.\nOur density functional theory calculations employed the frozen-c ore projector aug-\nmented wave method [20] encoded in the Vienna ab initio simulation package [21], and\nthe generalized-gradient approximation (GGA) [22]. To properly des cribe the strong elec-\ntron correlation in the 3d transition-metal oxide, the GGA plus on-s ite repulsion U method\n(GGA+U) [23] was employed with the effective Uvalue (Ueff=U−JwithJ= 0) of\n4.61 eV [5]. It is known experimentally [11, 14, 17] that the interlayer m agnetic interactions\nin LuFe 2O4are weak, which is understandable due to its layered structure. In this work,\ntherefore, we focus on the 2D spin ordering within a single Fe 2O4layer. For the√\n3×√\n3\nCO state of LuFe 2O4, the FE ordering of the Fe 2O4layers will be assumed.\nWe first examine the magnetic anisotropy of the Fe ions by performin g GGA+U cal-\nculations, with spin-orbit coupling (SOC) included, for the FM state o f LuFe 2O4with the\n√\n3×√\n3 CO. As shown in Fig. 1(a), there are two kinds of Fe2+ions and two kinds of\nFe3+ions in the√\n3×√\n3 CO state. We label the Fe2+and Fe3+ions of the type A T-sheet\nas 2A and 3A, respectively, and those of the type B T-sheet as 2B a nd 3B, respectively.\nIn our GGA+U+SOC calculations with spins pointing along several differ ent directions, all\nFe2+and Fe3+spins are kept in the same direction. Our calculations show that the e asy\naxis is along the cdirection, as experimentally observed [11, 12]; the /bardblc-spin orientation is\nmore stable than the ⊥c-spin orientation by 1.5 meV per formula unit (FU). The orbital\nmoments of 2A, 2B, 3A, and 3B for the /bardblc-spin orientation are 0.101, 0.156, 0.031 and 0.035,\nrespectively, which are greater than those for the ⊥c-spin orientation by 0.019, 0.062, 0.015,\nand 0.018 µB, respectively. As expected, the Fe3+(d5) ions have a very small anisotropy,\nHowever, two kinds of the Fe2+ions also have different degree of spin anisotropy. The spin\n3downelectronofthe2BFe2+ionoccupiesthe( dx2−y2,dxy)manifold[5], thereforethe2BFe2+\nion has the largest spin anisotropy along c. Our calculations indicate a non-negligible or-\nbital contribution to the total magnetization, in agreement with th e X-ray magnetic circular\ndichroism result [12].\nTo determine the magnetic ground state of LuFe 2O4in the√\n3×√\n3 CO state, we extract\nits spin exchange parameters by mapping the energy differences be tween ordered spin states\nobtainedfromGGA+Ucalculationsontothecorresponding energyd ifferences obtainedfrom\nthe Ising Hamiltonian [24]:\nH=/summationdisplay\ni,jJijSizSjz, (1)\nwhere the energy is expressed with respect to the spin disorder (p aramagnetic) state, Jijis\nthe spin exchange parameter between the spin sites iandj, andSizis the spin component\nalong the cdirection ( |Sz|= 2 and 2 .5 for Fe2+and Fe3+ions, respectively). We consider\nall 15 possible superexchange (SE) interactions and all 19 super-s uperexchange (SSE) in-\nteractions with the O...O distance less than 3.2 ˚A. The intra- and inter-sheet interactions\nwithin each Fe 2O4layer as well as the SSE interactions between adjacent Fe 2O4layers are\ntaken into account. To evaluate these 34 spin exchange paramete rs reliably, we considered\n111 different ordered spin states leading to 110 energy differences . The 34 spin exchange\nparameters were determined by performing a linear least-square fi tting analysis. The SSE\ninteractions are generally much weaker than the SE interactions wit h the magnitude of all\nSSE interactions less than 1.4 meV. The calculated SE parameters ar e reported in Table I.\nAll intra-sheet SE interactions are AFM, and the strongest intera ctions (∼7.3 meV) occurs\nbetween the 3B Fe3+ions because of the large energy gain of the AFM configuration and a l-\nmost zero FM coupling. The inter-sheet SE interactions are weaker than the the intra-sheet\nSE interactions, and are mostly AFM.\nWith the calculated spin exchange parameters, one can identify the spin ground state of\nthe CO state. The Metropolis Monte Carlo simulation of the Ising mode l is performed to\nsearch for the ground state. Simulations with supercells of severa l different sizes show that\nthe spin ground state has the magnetic structure shown in Fig. 2(a ), which has the same cell\nas the√\n3×√\n3 CO structure. In this state, all Fe2+ions contribute to the majority spin,\nand the Fe3+ions are antiferromagnetically coupled to the Fe2+ions in the type A T-sheet.\nIn the honeycomb lattice of the type B T-sheet, the Fe3+spins are antiferromagnetically\n4coupled. Thus, the spin ground state is ferrimagnetic, as experime ntally observed [11]. This\n2:1 ferrimagnetic structure is the same as the magnetic structure proposed by Siratori et al.\n[15], and differs from the structure proposed by Naka et al.[18].\nThe observed ferrimagnetic ordering can be readily explained in term s of the calculated\nexchangeparameters. Inthehoneycomb networkofthetypeBT -sheet, thenearest-neighbor\n(NN) 3B ions are antiferromagnetically coupled since their SE interac tion is strongly AFM.\nIn the type A T-sheet, the SE interactions between the 2A ions are AFM, and so are those\nbetween the 2A and 3A ions, which leads to spin frustration. As a con sequence, two possible\nspin arrangements compete with each other in the type A T-sheet; the first is the state in\nwhich the coupling between the NN 2A ions are AFM with the spin directio n of the 3A\nion undetermined, and the second is the state in which all 2A ions are a ntiferromagnetically\ncoupledtothe3Aions. Theenergiesofthesetwostates(consider ing onlytheSEinteraction)\nareE1=−4(J2A1,2A2+J2A1,2A4) per 3A ion, and E2=−10(J3A1,2A1+J3A1,2A2+J3A1,2A3)+\n4(J2A1,2A2+J2A1,2A4) per 3A ion, respectively. Due to the relatively strong AFM interact ions\nbetween the 3A and 2A ions (See Table I) and the large spin of the 3A io ns, the second\nstate has a lower energy, i.e., E2< E1. Without loss of generality, we can assume the 2A\n(3A) ions constitute the majority (minority) spin in the second stat e. Now, we examine\nthe spin orientation of the Fe2+ions in the type B T-sheet. The intra-sheet interactions\nof the 2B ion with 3B ions vanish due to the AFM ordering of the 3B ions. As for the\ninter-sheet interactions involving the 2B ions, the dominant one is th e AFM interaction of\nthe 2B ion with the 3A ion ( J3A1−2B1in Table I). Consequently, we obtain the ferrimangetic\nground state shown in Fig. 2(a), in which the spin of the 2B ion contrib utes to the majority\nspin of the Fe 2O4layer. For the stability of the ferrimangetic ground state, the inte r-sheet\ninteraction is essential. This was neglected in the model Hamiltonian st udy of Naka et al.\n[18]. The ferrimangetic state is not due to the FM interactions betwe en NNN Fe ions of the\nT-sheet because they must be vanishingly weak and mostly AFM.\nThe electronic structure of the ferrimangetic state calculated fo r the√\n3×√\n3 CO struc-\nture of LuFe 2O4is shown in Fig. 3. Also shown is the electronic structure calculated fo r\nthe FM state. Both states are semiconducting, and the highest oc cupied (HO) and the\nlowest unoccupied (LU) levels of both states come from the spin-up Fe2+and Fe3+ions,\nrespectively [5]. In addition, the band dispersion from Γ to A is rather small, indicating a\nvery weak interlayer interaction. However, there are some import ant differences. First, the\n5ferrimangetic state has a larger bandgap(1.68 eV) than doesthe F Mstate (0.77 eV). This is\nconsistent with the stability of the ferrimangetic state. Second, t he FM state has an indirect\nband gap with the HO and LU levels located at K and Γ, respectively. In the ferrimangetic\nstate, however, the LU level has the highest energy at Γ and the b and dispersions of the HO\nand LU levels are almost flat fromM to K. This difference comes fromth e orbital interaction\nbetween the spin down ( dx2−y2,dxy) levels of the spin up Fe3+and Fe2+ions.\nTo probe the presence of spin-charge coupling in LuFe 2O4, it is necessary to consider the\nspin ordering in a CO state other than the√\n3×√\n3 CO state. The previous electrostatic\ncalculations [5, 18] showed that the chain CO, in which one-dimensiona l (1D) chains of Fe2+\nions alternate with 1D chains of Fe3+ions in each T-sheet [Fig. 2(b)], is only slightly less\nstable than the√\n3×√\n3 CO, and has no FE polarization. We extract exchange parameters\nby mapping analysis as described above. It is found that the intra-s heet SE between the\nFe3+ions is the strongest ( J= 6.7 meV) as in the√\n3×√\n3 CO case. All intra-sheet SE’s are\nAFM with J(Fe3+-Fe3+)> J(Fe2+-Fe3+)> J(Fe2+-Fe2+). The inter-sheet SE between the\nFe3+ions is very weak ( |J|<0.3 meV), and that between the Fe2+and Fe3+ions is FM with\nJ=−1.4 meV. Interestingly, the inter-sheet SE between the Fe2+ions is rather strongly\nAFM(J= 6.3meV).MonteCarlosimulationsusingthesespinexchangeparamete rsindicate\nthat the spin state shown in Fig. 2(b) is the spin ground state. In th is spin ordering, the\nspins within each chain of Fe2+ions or Fe3+ions are antiferromagnetically coupled. The\nNN chains of Fe2+ions belonging to different T-sheets are coupled antiferromagnetic ally,\nwhereas the corresponding chains of Fe3+are almost decoupled.\nThe above results show that the spin ordering of the chain CO state is dramatically\ndifferent from that of the√\n3×√\n3 CO state. The most important difference is that the\ntotal spin moments are 2.33 µB/FU for the√\n3×√\n3 CO, but 0 µB/FU for the chain CO.\nThis evidences a strong spin-charge coupling in LuFe 2O4. The external magnetic field will\nhave different effects on the two CO states due to the the Zeeman e ffect. It is expected that\nthe magnetic field will further stabilize the ferrimagnetic√\n3×√\n3 CO state. Consequently,\nan external magnetic field will reduce the extent of charge fluctua tion and hence decrease\nthe dielectric constant. This supports our explanation for the gian t magnetocapacitance\neffect of LuFe 2O4at room temperature [5] .\nWithout considering the inter-sheet interactions, Naka et al.[18] suggested that the\ndegeneracy of the spin ground state of the√\n3×√\n3 CO state is of the order O(2N/3)( N is\n6the number of the spin sites), which is much larger than the spin dege neracy [O(2√\nN)] of\nthe chain CO state. Thus, they proposed that spin frustration ind uces reinforcement of the\npolar√\n3×√\n3 CO by a gain of spin entropy. However, our calculations show that t here\nare substantial inter-sheet spin exchange interactions between the 2B1 and 3A1 ions, which\nwould remove the macroscopic degeneracy of the spin ground stat e of the√\n3×√\n3 CO state.\nThe macroscopic degeneracy still persists for the chain CO state. Thus, our work provides\na picture opposite to what Naka et al.proposed. Furthermore, we find that the√\n3×√\n3\nCO state is more favorable for the spin ordering than is the chain CO s tate; with respect to\nthe paramagnetic state, the spin ground state is lower in energy by −78 meV/FU for the\n√\n3×√\n3 CO, but by −57 meV/FU for the chain CO. The model of Naka et al.[18] predicts\nthat the polar√\n3×√\n3 CO state is destabilized and the electric polarization is reduced by\nthe magnetic field, since it will lift the macroscopic spin degeneracy. I n contrast, our work\npredicts that the magnetic field stabilizes the ferrimagnetic√\n3×√\n3 CO state due to the\nZeeman effect, and provides an explanation for why the electric pola rization increases when\nthe temperature is lowered below the Neel temperature [3], becaus e the charge fluctuation\nhas an onset well below TCO[8].\nIn summary, our first principles results explain the experimentally ob served Ising ferri-\nmagnetism, and manifest the spin-charge coupling and magnetoelec tric effect in LuFe 2O4.\nWork at NREL was supported by the U.S. Department of Energy, un der Contract No.\nDE-AC36-08GO28308, and work at NCSU by the U. S. Department o f Energy, under Grant\nDE-FG02-86ER45259.\n7[1] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, an d Y. Tokura, Nature (London)\n426, 55 (2003).\n[2] N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and S-W. Ch eong, Nature (London) 429,\n392 (2004).\n[3] N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami, K. Kaku rai, Y. Murakami, K. Yoshii, S.\nMori, Y. Horibe, and H. Kitˆ o, Nature (London) 436, 1136 (2005).\n[4] M. A. Subramanian, T. He, J. Chen, N. S. Rogado, T. G. Calva rese, and A. W. Sleight, Adv.\nMater.18, 1737 (2006).\n[5] H. J. Xiang and M.-H. Whangbo, Phys. Rev. Lett. 98, 246403 (2007).\n[6] Y. Zhang, H. X. Yang, C. Ma, H. F. Tian, and J. Q. Li, Phys. Re v. Lett.98, 247602 (2007).\n[7] M. Angst, R. P. Hermann, A. D. Christianson, M. D. Lumsden , C. Lee, M.-H. Whangbo,\nJ.-W. Kim, P. J. Ryan, S. E. Nagler, W. Tian, R. Jin, B. C. Sales , and D. Mandrus, Phys.\nRev. Lett. 101, 227601 (2008).\n[8] X. S. Xu, M. Angst, T. V. Brinzari, R. P. Hermann, J. L. Musf eldt, A. D. Christianson, D.\nMandrus, B. C. Sales, S. McGill, J.-W. Kim, and Z. Islam, Phys . Rev. Lett. 101, 227602\n(2008).\n[9] H. J. Xiang and M.-H. Whangbo, Phys. Rev. Lett. 99, 257203 (2007).\n[10] H. J. Xiang, S.-H. Wei, M.-H. Whangbo, and J. L. F. Da Silv a, Phys. Rev. Lett. 101, 037209\n(2008).\n[11] J. Iida, M. Tanaka, Y. 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Solid State Chem. 176, 417 (2003).\n9TABLE I: Calculated superexchange parameters (in meV) in th e√\n3×√\n3 CO state of LuFe 2O4\n(For the spin sites of the 2A, 3A, 2B and 3B ions,see Fig. 1 )\nA-AJ3A1,2A1J3A1,2A2J3A1,2A3J2A1,2A2J2A1,2A4\n3.2 4.0 4.7 1.9 3.6\nB-BJ3B1,3B2J3B1,3B4J2B1,3B1J2B1,3B2J2B1,3B3\n7.0 7.6 1.5 2.8 1.3\nA-BJ3A1,3B1J3A1,2B1J2A1,2B1J2A1,3B2J2A1,3B3\n2.0 1.9 ∼0−0.6 1.2\n10Fe\nOO3A1\n3B43B13B33B2\n2B12A42A1 2A2\n2A3\nFIG. 1: (Color online) Schematic representation of the√\n3×√\n3 CO structure. Large, medium, and\nsmall circles represent the Fe2+, Fe3+, and O2−ions, respectively. The type A (type B) T-sheet has\nthe honeycomb network of Fe2+(Fe3+) ions with a Fe3+(Fe2+) ion at the center of each hexagon.\n2A and 3A (2B and 3B) refer to the Fe2+and Fe3+ions of the type A (type B) T-sheet, respetively.\nThe region enclosed by dashed lines indicates the unit cell o f the CO structure. There is a mirror\nplane of symmetry, which is parallel to the caxis and crosses the 3A1 and 2B1 sites. The inset\nshows an isolated FeO 5trigonal bipyramid.\nFe3+Fe2+3B3Ac\n(a)\n(b)2B2A\nFIG. 2: (Color online) Schematic representations of (a) the spin ground state of the√\n3×√\n3\nCO structure and (b) one of the macroscopic spin ground state s of the chain CO structure. The\narrows denote the spin directions. The region enclosed by th e dashed lines on the bottom T-sheet\nindicates the magnetic unit cell of the spin structure.\n11-2-1012Energy (eV)\nΓ M K AΓ-2-1012Energy (eV)(a) Ferromagnetic\n(b) FerrimagneticFe3+\n3+Fe\nFe2+Fe2+\nFIG. 3: (Color online) Band structures calculated for (a) th e FM state and (b) the ferrimagnetic\nstate of the√\n3×√\n3 CO structure of LuFe 2O4. The solid and dashed lines represent the up-spin\nand down-spin bands, respectively. The√\n3×√\n3×1 hexagonal cell is used in the calculations.\n12" }, { "title": "1412.0450v1.Coexistence_of_superconductivity_and_magnetism_in_spin_fermion_model_of_ferrimagnetic_spinel_in_an_external_magnetic_field.pdf", "content": "arXiv:1412.0450v1 [cond-mat.str-el] 1 Dec 2014Coexistence of superconductivity and magnetism in spin-fe rmion model\nof ferrimagnetic spinel in an external magnetic field\nNaoum Karchev[*]\nDepartment of Physics, University of Sofia, 1164 Sofia, Bulga ria\nA two-sublattice spin-fermion model of ferrimagnetic spin el, with spin-1 /2 itinerant electrons at\nthe sublattice Asite and spin- slocalized electrons at the sublattice Bsite is considered. The\nexchange between itinerant and localized electrons is anti ferromanetic. As a result the external\nmagnetic field, applied along the magnetization of the local ized electrons, compensates the Zeeman\nsplitting due to the spin-fermion exchange and magnon-ferm ion interaction induces spin anti-parallel\np-wave superconductivity which coexists with magnetism. W e have obtained five characteristic\nvalues of the applied field (in units of energy) Hcr1< H3< H0< H4< Hcr2. AtH0the external\nmagnetic field compensates the Zeeman splitting. When Hcr1< H < H cr2the spin antiparallel\np-wave superconductivity with T1uconfiguration coexists with magnetism. The superconductor to\nnormal magnet transition at finite temperature is second ord er when Hruns the interval ( H3,H4).\nIt is an abrupt transition when Hcr1< H < H 3orH4< H < H cr2. This is proved calculating\nthe temperature dependence of the gap for three different val ues of the external magnetic field\nHcr1< H < H 3,H4< H < H cr2andH=H0. In the first two cases the abrupt fall to zero of\nthe gap at superconducting critical temperature shows that the superconductor to normal magnet\ntransition is first order. The Hubbard term (Coulomb repulsi on), in a weak coupling regime, does\nnot affect significantly the magnon induced superconductivi ty. Relying on the above results one can\nformulate a recipe for preparing a superconductor from ferr imagnetic spinel: i) hydrostatic pressure\nabove the critical value of insulator-metal transition. ii ) external magnetic field along the sublattice\nmagnetization with higher amplitude.\nPACS numbers: 75.50.Gg,74.20.Mn,74.20.Rp\nElectron-phonon mechanism of superconductivity [1–\n4] has been developed to explain the pairing in a large\nvariety of materials, from HgandAlto recently discov-\neredMgB2[5]. The discovery of superconductivity in\nLaBaCuO [6] and in other cuprates, as well as discovery\nofsuperconductivityin Fe-basedpnictides[7]established\nanother direction of research in this field. The boson-\nfermion models still remained respected but bosons are\nnot lattice vibrations.\nAlternatively, the possibility of an electronic pairing\nmechanism in systems with rotational invariance was put\nforward in a seminal paper by Kohn and Luttinger [8–\n10]. Although the bare interaction among electrons is\nrepulsive, there is an effective attractive interaction that\narise at higher order of perturbation theory. The Kohn-\nLuttinger instability of a three-dimensional rotationally\ninvariant system results in the formation of a uncon-\nventional superconducting ground state due to the peak\nin the particle-hole susceptibility near zero wave vector.\nThe works [11–15] have made significant progress in our\nunderstanding of superconductivity from repulsive inter-\naction.\nThere are many experiments addressing external mag-\nnetic field induced, enhanced or reentered supercon-\nductivity. Experimentally, an anomalous enhancement\nofHc2(T) was first reported by Fischer et al[16].\nAn increase of about 50 −100kGof the upper crit-\nical field is observed in Sn1.2(1−x)EuxMo6.35S8and\nPb1−xEuxMo6.35S8withrespecttothecompoundswith-\nout Eu. The overall feature of the field-induced super-\nconducting phase is well understood by theory based onthe Jaccarino- Peter (JP) compensation mechanism[17].\nIn a rare earth ferromagnetic metal the conduction\nelectrons are in an effective field due to the exchange\ninteraction with the rare earth spins. It is in general so\nlarge as to inhibit the occurrence of superconductivity.\nFor some systems the exchange interaction have a nega-\ntive sign. This allows for the conduction electron polar-\nization to be canceled by an external magnetic field so\nthat if, in addition these metals possess phonon-induced\nattractive electron-electron interaction, superconductiv-\nity occurs in the compensation region. If the effective\nfield is not large the coexistence of superconductivity end\nmagneticorderispossibleandthe externalmagneticfield\nenhances the superconductivity. The effect can also be\nobserved in a paramagnet since the strong external field\nwill in any case polarize the localized magnetic moments\nat low temperature, and thus produce the necessary fer-\nromagnetic alignment [18].Therefore, superconductivity\ncan occur in two domains: one at the low field, where\nthe pair-breaking field is still small, and one at the high\nfield in the compensation region. The field reentrance of\nsuperconductivity was first reported in [19, 20].\nThe JP compensation mechanism was originally pro-\nposed to explain the superconductivity in some pseu-\ndoternary materials. Recently, the JP effect has been\nproven to be responsible for the magnetic-field-induced\nsuperconductivity in the organic superconductor λ−\n(BETS)2FeCl4[21, 22].\nThe superconductivity in the Jaccarino- Peter theory\nis induced by phonon fluctuations and spin fluctuations\n(magnons) weaken the spin singlet Cooper pairing. In2\nthe present paper we consider magnon induced super-\nconductivity based on the compensation mechanism[17].\nWe study the conditions for the coexistence of supercon-\nductivity and magnetism in a spin-fermion system which\nis a prototype model of itinerant ferrimagnetic spinel. A\ntwo-sublatticesystem is defined ona body centeredcubic\nlattice, with spin-1 /2 itinerant electrons at the sublattice\nAsite and spin- slocalized electrons at the sublattice B\nsite. The subtle point is the exchange between itinerant\nand localized electrons which is antiferromanetic and ap-\nplyinganexternalmagneticfieldalongthemagnetization\nof the localized electrons one can compensate the Zee-\nman splitting due to the spin-fermion exchange. Then,\nmagnon-fermion interaction induces spin anti-parallel p-\nwave superconductivity, with T1uconfiguration, which\ncoexists with magnetism. We have studied the supercon-\nducting gap as a function of applied magnetic field and\ntemperature. The Coulombrepulsion, inaweakcoupling\nregime, does not affect significantly the magnon induced\nsuperconductivity.\nRelying on the above results one can formulate a\nrecipe for preparing a superconductor from ferrimagnetic\nspinel: i) hydrostatic pressure above the critical value\nof insulator-metal transition. ii) external magnetic field\nalong the sublattice magnetization with higher ampli-\ntude. In favor of this recipe one can mention that met-\nallization in magnetite Fe3O4is found under a pressure\nabove 8GPa[23–25]. While the model under considera-\ntion does not match well the Fe3O4system we expect to\nfind superconductivity applying external magnetic field\nalong sublattice B magnetization, when the hydrostatic\npressure is above the critical one.\nOn the other hand, there are spinel compounds\nwell known as superconductors at ambient pressure\nCuRh 2Se4,CuRh 2S4[26–31]. The results of the present\npaper inspire that applying external magnetic field one\ncan expect an enhancement of the superconducting tran-\nsition temperature Tsc. This is quite specific phe-\nnomenon for the spinel superconductivity and it deserves\nto be experimentally verified.\nThe Hamiltonian of the spin-fermion model of ferri-\nmagnetic spinel defined on a body centered cubic lattice\nis\nh=−t/summationdisplay\n≪ij≫A/parenleftbig\nc+\niσcjσ+h.c./parenrightbig\n−µ/summationdisplay\ni∈Ani\n−JB/summationdisplay\n≪ij≫BSB\ni·SB\nj+J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj(1)\n−H/summationdisplay\ni∈ASzA\ni−H/summationdisplay\ni∈BSzB\ni,\nwhereSνA\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 Asite ,SB\niis the spin of the localized electrons\nat the sublattice Bsite,µis the chemical potential, and\nni=c+\niσciσ. The sums are over all sites of a body cen-\ntered cubic lattice, /angbracketlefti,j/angbracketrightdenotes the sum overthe nearestneighbors, while ≪ij≫Aand≪ij≫Bare sums over\nall sites of sublattice AandBrespectively. The Heisen-\nberg term ( JB>0) describes ferromagnetic Heisenberg\nexchangebetweenlocalizedelectronsand J >0is the an-\ntiferromagnetic exchange constant between localized and\nitinerant electrons. H >0 is the Zeeman splitting energy\ndue to the externalmagnetic field (magnetic field in units\nof energy).\nTo proceed we use the Holstein-Primakoff repre-\nsentation of the spin operators of localized electrons\nSB\nj(a+\nj,aj), where a+\nj, ajare Bose fields. In terms of\nthese fields and keeping only the quadratic terms, the\nHamiltonian Eq.(1) is a sum of three terms\nh=hb+hf+hbf, (2)\nwhere\nhb=sJB/summationdisplay\n≪ij≫B(a+\niai+a+\njaj−a+\njai−a+\niaj)\n+H/summationdisplay\ni∈Ba+\niai\nhf=−t/summationdisplay\n≪ij≫A/parenleftbig\nc+\niσcjσ+h.c./parenrightbig\n−µ/summationdisplay\ni∈Ani(3)\n+(4Js−H)/summationdisplay\ni∈A1\n2c+\niστ3\nσσ′ciσ′\nhbf=/radicalbiggs\n2J/summationdisplay\n/angbracketleftij/angbracketright/parenleftBig\nc+\ni↓ci↑aj+c+\ni↑ci↓a+\nj/parenrightBig\nIn momentum space representation, the Hamiltonian\nreads\nhb=/summationdisplay\nk∈Brεka+\nkak\nhf=/summationdisplay\nk∈Brσεkσc+\nkσckσ (4)\nhbf=4J√\n2s√\nN/summationdisplay\nkqp∈Brδ(p−q−k)coskx\n2cosky\n2coskz\n2\n×/parenleftBig\nc+\np↓cq↑ak+c+\nq↑cp↓a+\nk/parenrightBig\n,\nwith bose εkand fermi εkσdispersions\nεk= 2sJB(3−coskx−cosky−coskz)+H(5)\nεk↑=−2t(coskx+cosky+coskz)−µ+4sJ−H\n2\nεk↓=−2t(coskx+cosky+coskz)−µ−4sJ−H\n2\nThe two equivalent sublattices A and B of the body cen-\nter cubic lattice are simple cubic lattices. Therefor the\nwave vectors p,q,krun over the first Brillouin zone of a\ncubic lattice Br.\nLet us average in the subspace of Bosons ( a+,a) ( to\nintegrate the Bosons in the path integral approach). In\nstaticapproximationoneobtainsaneffectivefermionthe-\nory with Hamiltonian heff=hf+hint, wherehfis the3\nfreefermionHamiltonianEq.(4)andthemagnon-induced\nfour-fermion interaction is\nhint=−1\nN/summationdisplay\nkipi∈Brδ(k1−k2−p1+p2)\n×Vk1−k2c+\nk1↓ck2↑c+\np2↑cp1↓ (6)\nwith potential\nVk=32sJ2/parenleftBig\ncoskx\n2cosky\n2coskz\n2/parenrightBig2\n2sJB(3−coskx−cosky−coskz)+H(7)\nFollowing standard procedure one obtains the effective\nHamiltonian in the Hartree-Fock approximation\nhHF=/summationdisplay\nk∈Br/bracketleftBig\nεkσc+\nkσckσ+∆kc+\n−k↓ck↑+∆+\nkck↑c−k↓/bracketrightBig\n,\n(8)\nwith gap function\n∆k=1\nN/summationdisplay\np∈Br< c−p↑cp↓> Vp−k (9)\nThe Hamiltonian can be written in a diagonal form by\nmeansofBogoliubovexcitations α+,α,β+,β, which have\nthe following dispersions:\nEα\nk=1\n2/bracketleftbigg\nεk↑−εk↓+/radicalBig\n(εk↑+εk↓)2+4|∆k|2/bracketrightbigg\n(10)\nEβ\nk=1\n2/bracketleftbigg\n−εk↑+εk↓+/radicalBig\n(εk↑+εk↓)2+4|∆k|2/bracketrightbigg\n.\nIn terms of the new excitations the gap equation reads\n∆k=−1\nN/summationdisplay\np∈BrVk+p∆p/radicalbig\n(εp↑+εp↓)2+4|∆p|2\n×/parenleftbig\n1−< α+\npαp>−< β+\npβp>/parenrightbig\n,(11)\nwhere< α+\npαp>and< β+\npβp>are fermi functions for\nBogoliubov fermions.\nStraightforward calculations show that equation (11)\nhas not spin-singlet ∆ −k= ∆ksolutions. Having\nin mind that sublattices are simple cubic lattices and\nfollowing the classifications for spin-triplet gap func-\ntions ∆ −k=−∆k, we obtained that A1ustate ∆ k=\n∆sinkxsinkysinkzis not solution of the equation (11)\ntoo. The gap function with T1uconfiguration\n∆k= ∆(sin kx+sinky+sinkz)) (12)\nis a solution of the gap equation for some values of the\nexternal magnetic field and temperature.\nIt follows from equations (5), that the external mag-\nnetic field (in units of energy) compensates the Zeeman\nsplitting, due tothe spin-fermionexchange, at H=H0=\n4sJ. We calculate the gap parameter ∆, from Eq.(11),\nas a function of H/H0, setting the chemical potential µ/s48/s44/s57/s48 /s48/s44/s57/s53 /s49/s44/s48/s48 /s49/s44/s48/s53 /s49/s44/s49/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/s49/s44/s54/s49/s44/s56/s50/s44/s48\n/s47/s116\n/s72/s47/s72\n/s48\nFIG. 1: (Color online) Dimensionless gap ∆ /tas a function\nofH/H0. The critical values are Hcr1/H0= 0.927 and\nHcr2/H0= 1.062. The red lines correspond to H3/H0=\n0.944 and H4/H0= 1.045. When H3/H0< H/H 0< H4/H0\nthe thermal superconductor to normal magnet transition is\nsecond order. In other cases it is abrupt.\nequal to zero. The last means that in normal phase the\ndensity of sublattice A itinerant electrons is n= 1. We\nhave obtained four characteristic values of the applied\nfieldHcr1< H3< H0< H4< Hcr2. WhenHcr1< H <\nHcr2the spin antiparallel p-wave superconductivity with\nT1uconfiguration coexists with magnetism. The thermal\nsuperconductortonormalmagnettransitionissecondor-\nder when Hruns the interval ( H3,H4). It is an abrupt\ntransition when Hcr1< H < H 3orH4< H < H cr2.\nThe dimensionless gap ∆ /tas a function of H/H0at\nzero temperature is depicted in Fig.(1) for parameters\nJ/t= 4/0.3 andJB/J= 1/15. The critical values of\nthe external magnetic fields are Hcr1/H0= 0.927 and\nHcr2/H0= 1.062. The red vertical lines in Fig.(1) corre-\nspond to the H3/H0= 0.944 and H4/H0= 1.045.\nTo demonstrate the nature of the thermal\nsuperconductor-normal magnet transition, we have\ncalculated the gap ∆ /tas a function of the temperature\nT/tfor three different values of the external magnetic\nfield:H/H0= 0.89,H/H0= 1 and H/H0= 1.09. The\nresult is shown in Fig.(2). The black line represents ∆ /t\nas a function of T/tforH=H0. The second order\nphase transition demonstrates itself through the smooth\ndecrease of the gap up to zero at critical temperature\nTsc= 1.393t. The other two lines, blue H= 0.938H0\nand redH= 1.051H0, demonstrateabrupt fall ofthe gap\nat superconducting critical temperatures Tsc= 0.825t\nandTsc= 0.53trespectively.\nTo account for the Coulomb repulsion one has to add\nthe Hubbard term to the Hamiltonian Eq.(1)\nh→h+U/summationdisplay\ni∈Ac+\ni↑ci↑c+\ni↓ci↓ (13)\nWhen the coupling U/tis strong enough the model de-\nscribes a system of localized electrons on sublattice A4\n/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/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/s116\n/s84/s47/s116/s32/s72/s47/s72/s111/s61/s49\n/s32/s72/s47/s72/s111/s61/s49/s46/s48/s53/s49\n/s32/s72/s47/s72/s111/s61/s48/s46/s57/s51/s56\nFIG. 2: (Color online) Dimensionless gap ∆ /tas a function\nof dimensionless temperature T/t. The black line represents\nthe function for H=H0, the blue line for H= 0.938H0and\nthe red line for H= 1.051H0.\nsites. Under the pressure, the charge screening increases\n(the Coulomb repulsion Udecreases) and overlap of the\nwave functions of electrons increases (the hopping pa-\nrameter tincreases). As a result the coupling constant\nU/tdecreases and we can treat the Hubbard term in a\nweak coupling regime.\nThe contribution of the first order term in U/texpan-\nsion to the magnon induced superconductivity changes\nthe potential Eq.(7)\nVk→Vk+U. (14)\nIt is known [15] that this term do not contribute to\nany unconventional channel of superconductivity. To as-\nsess the suppression of the superconductivity due to the\nfirst term we calculate the superconducting critical tem-\nperature as a function of U/tforH=H0,J/t= 10\nandJB/J= 0.05. When the Zeeman splitting is com-\npensated ( H=H0) the thermal superconductor-normal\nmagnet transition is a second order and we can use the\nlinearized gap equation to determine the critical temper-\natureTscas a function of the Coulomb repulsion U\n1 =1\n31\nN2/summationdisplay\nkp∈BrΓk(Vk−p+U)Γp\nEptanhEp\n2Tsc.(15)\nIn equation (15) Γ k= sinkx+ sinky+ sinkzandEp=\n2t|cospx+cospy+cospz|.\nTheresultshowsthatthecontributionofthefirstorder\nterm inU/texpansion to the magnon induced supercon-\nductivity is unessential. For example for U/t= 0 the\ncritical temperature is Tsc/t= 1.393, while for U/t= 0.8\nit isTsc/t= 1.39.\nThe higher order terms in a weak coupling expansion\ncontribute to the superconductivity through the Kohn-\nLuttinger mechanism. The results show[14] that the ef-\nfect on the p-wave superconductivity with T1uconfigura-\ntion is weak. This permits to conclude that the Coulombrepulsion, in a weak coupling regime, does not impact\nsignificantly the magnon induced superconductivity and\nwe can drop it.\nFinally, we consider the effect of the chemical ma-\nnipulation. To this end we study the critical tempera-\ntureTscas a function of the density of states of itiner-\nant electrons in normal phase for the same parameters\nof the system as above. The equation for the critical\ntemperature Tscis the equation (15) with U= 0 and\nEp=/radicalbig\n[2t(cospx+cospy+cospz)+µ]2, whereµis the\nchemical potential. The table shows that decreasing the\ndensity of itinerant electrons the superconducting criti-\ncal temperature Tsc/tslowly decreases. This is true if\nthe electrons are delocalized. If the sublattice A elec-\ntrons are localized the deviation from half-filling is a way\nto delocalize them and we expect an opposite tendency.\nn10.90.80.70.60.5\nTsc/t1.3931.38731.37611.35341.31691.2823\nIn summary, we have proposed a method of prepara-\ntion of superconducting ferrimagnetic spinel. We have\nstudied a two-sublattice spin-fermion model of ferrimag-\nnetic spinel, with spin-1 /2 itinerant electrons at the sub-\nlatticeAsite and spin- slocalized electrons at the sub-\nlatticeBsite in an external magnetic field, applied along\nthe magnetization of the localized electrons. Magnon in-\nduced superconductivity is predicted when the Coulomb\nrepulsion is small (the system is under hydrostatic pres-\nsure) and the external magnetic field compensates the\nZeeman splitting due to the spin-fermion exchange.\nThere are two methods of preparation of spinels.\nIf, during the preparation, an external magnetic field\nas high as 300 O¨ e is applied upon cooling the mate-\nrial is named field-cooled (FC). If the applied field is\nabout 1O¨ e the material is zero-field cooled (ZFC). The\nmagnetization-temperature [32] and magnetic suscepti-\nbility [33, 34] curves for these materials display a pro-\nnounced bifurcation below N´ eel TNtemperature. The\n(ZFC) curve exhibits a maximum and then a mono-\ntonic decrease upon cooling from TN, while the (FC)\ncurve increases steeply, shows a dip near the temper-\nature at which the (ZFC) curve has a maximum and\nfinally increases monotonically[34]. The magnetization-\ntemperaturecurveisclosetothereferencecurveobtained\nfrom contribution of localized spins on the one of the\nsublattices. Hence, in (FC) materials the electrons on\nthe other sublattice have dispersion with approximately\ncompensated Zeeman splitting. This permits to think\nthat at high hydrostaticpressure these material will have\na superconducting state.\nThe Zeeman splitting energy H0can be obtained from\nthe external magnetic field used for the preparation of\n(FC) material. For MnV2O4the field is as high as 300\nO¨ e[34].\nThe model we have considered is a prototype model\nof itinerant ferrimagnetic spinel. It is prototype model\nbecause the sublattice A sites are occupied, usually, by5\nmore than one electron. But this is not a toy model,\nbecause it capture all physical relevant properties of the\nspinel system and the existence of more than one elec-\ntrons on sublattice A sites will not change the conclusion\nthat the magnon induced superconductivity exist upon\nsome conditions.\nFinally, we have not considered the s-band elec-trons because the spinel magnetism is determined by d-\nelectrons. The goalofthe paperis to study the formation\nof Cooper pairs of delocalized sublattice A d-electrons\n(the system is under hydrostatic pressure) in external\nmagneticfield. Thes-electronsareaccountedforthrough\ntherenormalizationoftheparametersofthe spin-fermion\nmodel (1).\n[1] J. Bardeen, Phys. Rev. 79, 167 (1950).\n[2] H. Fr¨ ohlich, Phys. Rev. 79, 845 (1950).\n[3] L. N. Cooper, Phys. Rev. 104, 1189 (1956).\n[4] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys.\nRev.108, 1175 (1957).\n[5] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani\nand J. 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Phys.: Condens. Matter 20, 135218\n(2008)\n[*] Electronic address: naoum@phys.uni-sofia.bg" }, { "title": "1411.2461v1.Temperature_evolution_of_the_effective_magnetic_anisotropy_in_the_MnCr__2_O__4__spinel.pdf", "content": "Temperature evolution of the effective magnetic anisotropy in the\nMnCr 2O4spinel\nDina Tobia1,2, Julián Milano1,2, María Teresa Causa1and Elin L. Winkler1,2\n1Centro Atómico Bariloche , CNEA, 8400 S.C. de Bariloche, Río Negro, Argentina\n2Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina\nAbstract\nIn this work we present a study of the low temperature magnetic phases ofpolycrystalline\nMnCr2O4spinel through dc magnetization and ferromagnetic resonance spectroscopy\n(FMR). Through these experiments we determined the main characteristic temperatures: T C\n~ 41 K and T H~ 18 K corresponding, respectively, to the ferrimagnetic order and to the low\ntemperature helicoidal transitions. The temperature evolution of the system is describe dby\na phenomenological approach that considers the different terms that contribute to the free\nenergy density. Below the Curie temperature the FMRspectra were modeled by a cubic\nmagnetocrystalline anisotropy to the second order, with K1andK2anisotropy constants that\ndefine the easy magnetization axis along the <110> direction . At lower temperatures, the\nformation of ahelicoidal phase was considered by including uniaxial anisotropy axis along\nthe]011[\npropagation direction o f the spiral arrange, with a Kuanisotropy constant . The\nvalues obtained from the fittings at 5 K are K1=-2.3x104erg/cm3,K2= 6.4x104erg/cm3and\nKu= 7.5x104erg/cm3.\n1. IntroductionThe cubic spinels AB 2O4, where the tetrahedral A -sites are occupied by non -magnetic ions\nand the octahedral B -sites are occupied by Cr ions, are model systems to study magnetic\nfrustration [1, 2, 3]. In these compounds the main magnetic interaction isthestrong J CrCr\nantiferromagnetic direct exchange between the nearest neighbors ions[4, 5]. However, the\ngeometrical arrangement of thesemagnetic ions in a pyrochlore -like array prevents the\nmagnetic order till very low temperature, ascompared to the Curie temperature ,CW[2, 6,\n7]. Several authors have proposed that trough the magnetoelastic coupling the strong\nmagnetic frustration could be released and the system could develop a magnetic transition\n[8,9]; in fact the low temperature magnetic ordered state is usually accompanied by\nstructural distortions. [ 10, 11] Instead, when the tetrahedral A -site is occupied by a\nmagnetic ion, the magnetic frustration is partially relieved by the J ACrsuperexchange\ninteraction. [1 2] In this case the system presents nearly degenera ted ground states and it\ndevelops complex low temperature magnetic order.\nIn particular in the MnCr 2O4the competing Cr -Cr, Cr-Mn and Mn -Mn exchange\ninteractions prevent the development of ferrimagnetic order till to TC~41 K,even\nconsidering the importan t exchange energies observed ( CW/TC>10)[4].Neutron\ndiffraction studies reported that bel ow TCthe system presents long -rangeferrimagnetic\norder with an easy axis parallel to the <1 10> direction [13 -15], when the temperature\ndecreases belowTH~18 K,this magnetic phase coexists with short -rangespiral order. In the\nspiral arrange two positions can be distinguished for the Cr, and the magnetic moment s\ndescribea cone on each sublattice, with helicoidal propagation vector in the ]011[\ndirection.\nThe complex low temperature order, where the spin rotation axis does not coincide with the\nhelicoidal propagation vector, positioned this material as a good candidate to presentmagnetodiel ectric coupling [16 -18]. Recently, Mufti and collaborat ors [19,20] have\nreported that the dielectric and magnetic properties are coupled below THin powder\nMnCr2O4oxide.In addition ,recent FMR results on frustrated spinels [ 21] have related the\nunusual FMR temperature dependence to phase separation. In thiscomplex scenario the\nferromagnetic resonance (FMR) spectroscopy emerges as a suitable technique because it\nprovides microscopic information related to the exchange and magnetic anisotropy and\nallows extending the knowledge of the nature of the long-rangeferrimagnetic order and the\nspiralshort-rangestate. In this context we present a study of the low temperature magnetic\nphasesin a cubic chromium spinel with A=Mn by magnetic and FMR measurements. We\nfollow the temperature evolution of the parameters tha t characterize the FMR spectra in a\npolycrystalline sample. We describe the evolution of the FMR spectra by a\nphenomenological model that takes into account the different terms that contribute to the\nmagnetic anisotropy of the system.\n2. Experimental\nSingle phase polycrystalline samples of MnCr 2O4were fabricated by solid state reaction of\nMnO and Cr 2O3powders, as described elsewhere [4].This system has a normal cubic spinel\nstructure, belonging to the Fd -3mspace group. The magnetic properties were in vestigated\non loosely packed powdered samples in the 5 –90 K temperature range, with applied fields\nup to 5 T, using a commercial superconducting quantum interference device (SQUID,\nQuantum Design MPMS-5S) magnetometer. The temperature dependence of the\nferromagnetic resonance ( FMR) spectra was recorded by a Bruker ESP300 spectrometer\noperating in the conventional absorption mode at 2~24 GHz (K -band), for temperaturesranging from 4 K to 300 K. Magnetic -field scans were performed in the range 0 –15000\nOe.Care was taken in order to avoid cavity detuning effects, as are usually present in\nspectra of strongly magnetic compounds. For that purpose, the MnCr 2O4powder was\nthoroughly milled and mixed with a non -absorbing KCl salt. No noticeable changes in the\nquality factor (Q) of the cavity w ereregistered in the whole set of experiments.\n3. Results and discussion\n3.1 Magnetic properties\nFigure 1 presents the magnetization vs. temperature measurements, M(T), under zero -field-\ncooling (ZFC) and field -cooling(FC) conditions, with an applied field of 50 Oe. Near 41\nK, a sudden jump is observed, consistent with the ferrimagnetic transition ( TC). As the\ntemperature is further lowered, other anomalies are manifested at TH~18 K and Tf~14 K,\ncorresponding, respectively, to the helicoidal order temperature and to the “lock -in”\ntransition at which the spiral becomes fully developed, as it was determined from neutron\ndiffraction experiments [ 13-15]. The inset in figure 1 exhibits the M(T) ZFC -FC curves\nmeasured with an applied field of 8 kOe, where it can be observed that the TCvalue\nincreases and the transition becomes broader. Also, when the applied magnetic field is\nenhanced, both low -temperature anomalies become less defined, as it was previously\nreported b y Mufti et al. [ 19,20].01020304050607080900246810Tf\nTHM (emu/g)\nT (K)TC0102030405060708090051015202530M (emu/g)\nT (K)\nFigure 1. Temperature dependence of the ZFC (solid symbols) and FC (open symbols)\nmagnetization measured in a field of 50 Oe. The arrows signal the ferrimagnetic transition\n(TC),the helicoidal order temperature (TH)and the “lock -in” transition where the spiral\ncomponent is fully developed (Tf). The inset shows the M(T) ZFC -FC curves measured\nwith an applied field of 8 kOe.\nFigure 2 shows the magnetization as a function of the applied magnetic field acquired at\ndifferent temperatures. As the temperature descends below ~45 K the magnetization\npresents a n important increase that starts near 2.5 kOe . The spontaneous magnetization of\nMnCr2O4at 5 K was estimated to be ~1.1 Bper unit formula in agreement with the value\npreviously reported [19,20,22 ].Noticeable, a linear increase of the high field magnetization\nis clearly observed for temperatures below 30 K .This lineal contribution signals a non -\ncollinear spins arrangement of the MnCr 2O4ferrimagnet .As is stated in references [23,24]in non-collinear configuration the applied magnetic field exerts a torque that could change\nthe angles between the canted magnetic moments; as a result the magnetization increases\nlinearly with the magnetic field. By neu tron diffraction studies non -collinear order was\nfound below T~18 K where short -range spiral arrangement is developed [13,14].In order\nto shed light onto this complex behavior we have performed ferromagnetic resonance\nmeasurements.\n0 10 20 30 40 50051015202530\n30K\n35K\n40K\n45K\n50KM (emu/g)\nH (kOe) 5K\n15K\n20K\n25K\nFigure 2. Magnetizati on versus applied field at different temperatures near and below TC.\n3.2Ferromagnetic resonance\nTheFMRspectroscopy is a very sensitive technique to detect magnetic transitions as well\nas changes in the magnetic ani sotropy of local -moment systems [25, 26],which are usually\ndifficult to measure by other techniques, particularly in polycrystalline samples. Figure 3\n(a) and (b) exhibit representative FMRspectra measured at different temperatures in theT, <111> and <100> directions of the crystal,\nrespectively , for all temperatures .It is noteworthy that ,if another relation between K1and\nK2is chosen (resulting in different medium and hard magnetization directions), thisleads to\nqualitatively different spectra features, where secondary absorption peaks are localized in\nthe g<2 higher field region. We also want to remark that the aforementioned choice ofparameters is consistent with the magnetization easy axis direction r eported from neutron\ndiffraction and magnetization studiesperformed on single crystal samples[13,14,31]and\ndiffersfromthe results reported by [32] where different orientation of the easy\nmagnetization direction was found.\nRegarding Ku, this parameter takes into account the propagation dir ection of the helicoidal\norder[14,15], that breaks the cubic symmetry imposed by the crystalline structure. This\nkind of magnetic ordering is observed when several comparable exchange interactions are\npresent and the description in terms of sublattices is interdicted. This is the case, for\nexample, when the step of the spiral is not commensurate with the lattice parameter [33 ].\nFrom the magnetic free energy, equations (1) to (4), the angular derivative s\n) and , (2 2 22 2 E E E , evaluated at the equilibrium angles for the\nmagnetization for each orientation of the magnetic field, were calculated. The FMR\nresonance condition was obtained evaluating the Smit -Beljers equation [ 27,28]:\n.\nsin122\n22\n22\n22\n02\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n EEE\nM(5)\nHereis the angular frequency GHz,is the gyromagnetic ratio and M0is the\nsaturation magnetization value measured at kOe (Figure 2). As we measured a\npolycrystalline sample we assume that the absorption line corresponds to thesum of\nLorentzian lineshape resonances with a homogeneous angular distribution of the\nanisotropies ax es related to the magnetic field. For simplicity no angular variation of the\nresonance linewidth was considered. Furthermore ,for temperatures near and be lowTCthe\nlines present an additional asymmetry that could be attribut ed to a dispersive component[34,35]as we are going to discuss later . Consequently, in this range we have also included\nin the simulat ed spectra a dispersive term , determining a linesh ape of the form: (1 -)\nAbsorption + Dispersion, where 0< <1[35-36].We solved the Smit -Beljers equation\n(Eq. 5) in a self -consistent way, with g K1,K2andKuas adjusted parameters, and we have\nobtained a numerical simulation for the FMR resonance absorption at each temperature.\nThe gyromagnetic factor obtained from the fittings in all the T ≤TCrange isg~2.05(2). The\ncalculated spectraarepresented in straightlines in figure 3 , wheregood agreement between\nthe spectral lines and the model is observed in all the studiedtemperature range. Notice that\nthe calculated spectra reproduce well the general features of the lineshape, as the resonant\nfield, the field positions of the satellite peaks and the linewidth, even for T270K. An evolution of the interaction between Ni and Re magnetic sublattices in\nthis geometrically frustrated fcc perovskite structure, i s revealed as a function of temperature and\nmagnetic field, through the critical behaviour and thermal e volution of microscopic and macroscopic\nphysical quantities.\nI.Introduction\nIn the research on multifunctional materials, oxides\nwith a double perovskite structure are continuously at-\ntracting our interest due to their structural malleabil-\nity and a multitude of complex physical properties that\narise through their magnetically frustrated geometry\n[1,2]. Properties such as a magnetoelectric effect, mag-\nnetocaloric effect, magnetoresistance or superconductiv-\nity may appear in the various phases of perovskite oxides\n[3–5]. To be able to implement this class of material in fu-\nture magnetoelectronic and/or spintronic applications, i t\nis of high importance to reveal and understand the static\n∗Electronic address: yasmine.sassa@chalmers.seand dynamic processes driven from spin reorientation.\nThe general composition of of double perovskite com-\npounds, illustrated in Fig. 1a, isAA′BB′O6whereA,A′\nare alkaline earth or lanthanide cations, and B,B′are\n3d, 4dor 5dtransition metals (TMs) in various oxidation\nstates. The AA′andB,B′1:1 ratios provide an order-\ning ofBO6,B′O6edge-sharing octahedra which form two\ncrystallographically distinct sublattices [ 6,7]. The struc-\nture is flexible to expand/contract, distort or tilt the oc-\ntahedra due to the Jahn-Teller effect. These distortions\nare responsible for changes in character of superexchange\ninteractions (ferromagnetic-antiferromagnetic) [ 8,9] or\nthe appearance of Dzyaloshinski-Moriya (DM) interac-\ntions [ 10], which significantly alter the physical prop-\nerties of the perovskite [ 11]. The magnetic phases of\nthese compounds are controlled both by the magnetic\nand non-magnetic cations in the crystal structure. More2\nspecifically, the sublattice symmetries, the cation’s nom-\ninal spin and its spin-orbit coupling (SOC) as well as\nexchange interactions, are degrees of freedom that deter-\nmine the magnetic ground state [ 12].\nA particular interesting double perovskite compound is\nthe LaCaNiReO 6. This material is the sister compound\nof LaSrNiReO 6[13], and contrary to other double per-\novskite compound [ 14], the AA’ positions in the crys-\ntal structure are occupied randomly by either La3+or\nCa2+. The BB’ positions are also occupied by two dif-\nferent cations, Ni2+and Re5+that, at variance with the\nA-cations, form an ordered, rock salt-type network or\ncorner-sharing octahedra [Fig. 1a]. In double perovskite\ncompounds, the nearest neighbour TM cations are con-\nnected by oxygen sites and the superexchange interac-\ntion and controls the sign and magnitude of the different\nmagnetic couplings, which depend on both, the TM or-\nbital filling and the B-O-B’ bond angle. According to\nGoodenough-Kanamori (GKA) rules, the interaction is\nthe strongest at 180◦and is antiferromagnetic if the vir-\ntual electron ( e−) transfer is between overlapping orbitals\nthat are each half filled, but ferromagnetic if the virtual\ne−transfer is from a half filled to an empty orbital or\nfrom filled to half filled orbital. At 90◦the interaction is\nthe weakest and it is favoured to be ferromagnetic [ 15].\nFor LaCaNiReO 6, the B-O-B’ bonds are bent to an angle\nof152◦at1K [8]. Between the half filled egorbitals of\nNi2+and partially filled t2gorbitals of Re5+, the interac-\ntion should favour an antiferromagnetic coupling between\nnearest neighbours (NN) Ni-Re. In addition there ex-\nists a competition between these NN interaction and the\nlonger range, next nearest neighbours (NNN) Ni-Ni and\nRe-Re. For the 180◦NNN bonds, according to GKA,\na ferromagnetic coupling is possible [ 16]. Neutron pow-\nder diffraction (NPD) refinement reflects a ground state\nthat is determined by two interacting magnetic sublat-\ntices where the Ni spins orient antiparallel to the Re\nspins, while the moments of each individual sublattice\norder along the same direction [ 8].\nIn this study we employed both muon spin rota-\ntion/relaxation/resonance ( µ+SR) and magnetometry\ntechniques to scrutinize the phases of magnetic order-\ning in this compound. Muons as local probes can ex-\ntract space and time dependent information of the ma-\nterial’s intrinsic magnetic environment in zero applied\nfields. A muon spin precession frequency is a direct indi-\ncation of an ordered state, while the signal relaxation pro-\nvides information of the ordering range and fluctuations\n[17]. Since the time window of muons is typically 10−12-\n10−6s , these results are correlated to AC and DC mag-\nnetization measurements that provide a bulk view of the\nsample in a 10−6-1s time window, taking into account\nfast and slow magnetic fluctuations that may occur.\nOur results show a ferrimagnetic transition to a long-\nrange, commensurate and dynamically ordered state be-\nlowTN= 103 K. The ferrimagnetic ordering has been\nproposed in previous studies [ 8], however, our results sug-\ngest that this transition results to a metastable phase.The system is found to undergo a phase transition from\na slow to a high dynamics phase between 30−70K.\nThe collective relaxation of the spin lattice towards a\nthermodynamically stable state is characterized by an\nintrinsic time rate. This defines the magnetization of the\nsystem due to an externally applied magnetic field with\nspecific period and amplitude. Above TN, the (µ+SR)\nfits and magnetization measurements support the exis-\ntence of two co-existing magnetic phases from 103K up\nto230K, while from 270K up to room temperature the\nparamagnetic phase takes over.\nII.Experimental Details\nA polycrystalline sample of LaCaNiReO 6was prepared\nin stoichiometric ratio by solid state synthesis. La 2O3,\nSrCO3, CaCO 3, NiO, Re 2O7and Re metal were used\nas the starting materials, forming the final compound\nthrough mixing, grinding, pelleting and sintering cycles.\nX-ray and neutron diffraction patterns were recorded at\n300,125and1K to verify the composition. The P21/n\nstructural model fits the NPD data at all temperatures,\nmanifesting the absence of a structural phase transition\ngoing across the magnetic ordering, based on the resolu-\ntion of the given NPD data. More information about the\nsynthesis and structure characterization can be found in\n[8].\nAC-DC magnetization measurements over a 5-300K\ntemperature range were carried out in a zero-field cooled\n(ZFC), field-cooled while cooling (FCC) and field-cooled\nwhile warming (FCW) process. Magnetic field scans were\nalso performed in a −60 kOe−+60kOe magnetic field\nrange, at various temperatures. The instruments used\nwere a Quantum Design Physical Property Measurement\nSystem (PPMS) with a vibration sample magnetometry\nsetup (VSM) as well as a Magnetic Property Measure-\nment Systems (MPMS) Superconducting Quantum In-\ntereference Device (SQUID).\nTheµ+SR experiments were performed at the surface\nmuon beamline M20 in the TRIUMF facilities. A 100%\npolarized, continuous, positive muon beam was targeted\nonto an aluminium coated mylar envelope of approxi-\nmately 1 cm2surface, filled with ∼1 g of the powdered\nsample. The sample was inserted in a4He cryostat and\nreached a 2K base temperature. The software package\nmusrfit was used to analyze the data [ 18]. Measure-\nments were performed in the zero field (ZF), longitudi-\nnal field (LF) and transverse field (TF) geometry, with\nrespect to the initial muon spin polarization. Muons are\nimplanted one at a time into the sample, and come to\nrest at an interstitial site. There, muons interact with\nthe local magnetic field, undergoing a Larmor precession\nand finally emitting a positron, with a high probability\nin the direction of the spin orientation before decay [ 17].\nOur data sets consist of ∼10 million positron counts for\nthe TF and LF measurements, and ∼35 million counts\nfor the ZF measurements.3\na= 5.498Å\nb=5.5828Å\nc=7.8054Å=90°\n=90.135°\n=90°Volume\n239.60Å^3Space group\nP21/n\nα\nβ\nγ\n(a)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s90/s70/s67\n/s70/s67/s67\n/s32/s32/s40/s32/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s32/s32\n/s90/s70/s67\n/s70/s67/s67\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s109/s111/s108/s47/s101/s109/s117 /s41\n/s32/s32\n(b)\n/s45/s54/s48 /s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48/s45/s48/s46/s55/s53/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\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s45/s48/s46/s55/s53/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/s32/s53/s75\n/s32/s55/s53/s75\n/s32/s57/s53/s75\n/s32/s49/s48/s53/s75\n/s32/s32/s77\n/s68/s67/s32/s40\n/s66/s47/s102/s46/s117/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41\n/s32/s32/s77\n/s68/s67/s32/s40\n/s66/s47/s102/s46/s117 /s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41\n(c)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s57/s46/s57/s32/s72/s122\n/s57/s57/s46/s57/s32/s72/s122\n/s57/s57/s55/s32/s72/s122\n/s32/s32/s32/s40/s49/s48/s45/s51/s32\n/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s57/s46/s57/s32/s72/s122\n/s32/s32/s32/s40/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s109/s111/s108 /s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n(d)\nFIG. 1: (a) The crystal structure of LaCaNiReO 6, a geometrically frustrated, face-centered cubic, monocl inic crystal lattice\nthat accomodates alternating Ni-O and Re-O octahedra. (b) T emperature dependence of DC magnetic susceptibility in ZFC\nand FCC sequences. The inset presents the inverse magnetic s usceptibility. (c) Magnetic hysteresis loops for 5,75,95,105K.\nThe inset focuses around the remanence and coercive field reg ions. (d) AC susceptibility versus temperature for differen t\nfrequencies. The evolution of susceptibility at 9.9 Hz is al so shown independently for clarity.\nIII.Results\nA.Magnetic Susceptibility\nTemperature dependent DC magnetic susceptibility\nwas measured at 5-300K with an applied field of 100Oe,\nas shown in Fig. 1b. The FCC and FCW (not displayed)\ncurves were identical meaning that there was no quench-\ning of magnetic moments during the cooling. The bifur-\ncation between the FCC and ZFC curves reveals a change\nin the magnetic response of the system around 213K.\nThe FCC susceptibility at low temperatures reveals a\nferromagnetic component and from its first derivative we\ndetermined a Curie temperature TC= 100.3(3)K. To ex-\ntend the results of the previous study [ 8] below the Curie\ntemperature, we show explicitly that the ZFC susceptibil-\nity reaches the value of 0.08emu/mol at 5K for an 100Oeapplied field. The ZFC susceptibility increases with in-\ncreasing temperature and obtains a maximum close to\nthe derived Curie temperature. The ZFC and FCC in-\nverse susceptibility is presented in Fig. 1b. The region\nabove transition could not be fitted with a hyberbola\naccording to Neel’s molecular field model [ 19]. Conse-\nquently, we fitted the two linear regions (highlighted red\nregions) at 150 K< T <220 K and270 K< T <300 K\nto the Curie-Weiss law1\nχ=T−θ\nC[20]. The high temper-\nature fit resulted to a Curie constant of Ch= 0.94(2) K\nand Curie temperature θh= 60(4) K, while the low tem-\nperature fit to Cl= 1.06(3) K andθl= 100(2) K. The\nZFC-FCC curve types below transition and the calcu-\nlated Curie temperature values are characteristic feature s\nof a ferrimagnetic ordering. The effective magnetic mo-\nments were then calculated from µeff= 2.83(Cm\nZ)1\n2µB,\nwhereCmthe molar Curie constant, Zthe formula unit\nper unit cell and µBthe Bohr magneton [ 21]. The re-4\n(a)\n(b)\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s49/s50/s51/s52/s53/s84/s70/s40 /s115/s45/s49\n/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n(c)\nFIG. 2: (a) Weak transverse field (TF = 50Oe) time spectra\nrecorded at T= 45, 107, 125, 200 and 289 K. The solid lines are\nfits obtained from Eq.( 1).The oscillation amplitude clearly in-\ncreases with increasing temperature, although the maximum\nasymmetry is fully recovered at 289K. (b) The asymmentry\n[Eq.(1)] versus temperature shows a magnetic transition at\n104.3(4)K taking place in two stages, between 130-90K and\n90-50K. A second transition appears also above 230K. (c)\nThe relaxation of the oscillatory component exhibits a peak\naround the transition temperature while a shoulder appears\nat75K.spective effective moments are µeff/h= 0.63µBand\nµeff/l= 0.67µB. If we consider unquenched or par-\ntially quenched orbital moments for the Re and Ni ions\nin the octahedral complexes, the total µeffis given by\nµ2\neff=xg2J(J+ 1)µ2\nB, where xthe fraction of mag-\nnetic ions per formula unit, gtheir gyromagnetic factor\nandJthe total angular momentum. Assuming that the\nNi2+and Re5+magnetic systems have the same order-\ning temperature through superexchange coupling [ 22,23],\nwe calculate µeff=/radicalbig\nµ2\neff.Ni−µ2\neff.Re= 1.03µB, where\nµeff.Ni= 1.81µBandµeff.Re= 1.49µB. This theo-\nretical value is an overestimation to the experimentally\nextracted values from our magnetization measurements,\nbut still in good agreement with the values obtained from\npowder neutron diffraction experiments [ 8]. The preci-\nsion of the spin determination of the magnetic moment\nis limited since SOC in Re ions, Jahn-Teller distortions of\nRe-O octahedra, and the possible canted magnetic struc-\nture affect the spontaneous magnetization of the com-\npound.\nIsothermal field dependent DC magnetization mea-\nsurements were performed at 5,75,95,105and300K for\nmagnetic fields from - 60kOe to + 60kOe. The evolution\nof the magnetization is presented in figure Fig. 1c(the\nparamagnetic, linear behaviour at 300K is not shown).\nMagnetisation saturation is never reached for ferrimag-\nnets, however at the maximum field of 60kOe we consider\nan effective magnetic moment of µeff5K= 0.60µBat5K,\nfollowing the previous experimental values. A long range\norder is evident and a very large coercive field ( ≈7kOe)\nis measured. High coercivity and remanence has been\nalso observed in other Re-based double perovskites [ 24–\n26] and is attributed to an intrinsic anisotropy of these\ncompounds. Taking into account a strong SOC of Re\nions, first-principle calculations [ 27] predict a large un-\nquenched orbital moment, which is thought to be the\norigin of the magnetic anisotropy [ 28]. In the 5K hys-\nteresis loop, shoulders appear at zero field which indicate\nspin reorientation [ 29,30]. In the recorded loops at 75K\nand above, these shoulders disappear, while the rema-\nnence and coercivity approach zero at 105K.\nAC magnetic susceptibility measurements were also\nperformed at three ac field frequencies, to probe the mag-\nnetic relaxation in the two sublattices spin system as\nshown in Fig. 1d. The ordering temperature does not de-\npend on frequency, which excludes a spin glass behaviour.\nWe observe for the first time that a second peak in the\nsusceptibility curve rises at 50K as the frequency is tuned\nfrom 1000 Hz to 10 Hz [see inset Fig. 1d]. This can be an\neffect of reordering of spin domains with a subtle equi-\nlibrium, that respond to an ac field. This fluctuation is\nanother indication of reorientation and freezing of trans-\nverse spin components [ 31].5\nB.Muon Spin Rotation\nTheµ+SR study consists of weak TF, ZF and LF mea-\nsurements. We can extract the local field distribution\nfrom the time spectra of the muon decay asymmetries\nbetween the surrounding detectors [ 32,33]. From the\nTF spectra we extract the initial and baseline asymme-\ntry which are used as constants when treating the ZF\ntime spectra. The ZF and LF measurements follow in\norder to gain detailed information about the conditions\nof magnetic ordering, as well as the intrinsic magnetic\nfields, arising from nuclear and electronic moments.\n1.Weak transverse field (TF)\nMuon depolarization spectra were recorded for TF =\n50Oe in the temperature range 2 K≤T≤300 K . We\npresent the evolution of the TF spectra at selected tem-\nperatures, above and below the magnetic phase transition\nin Fig. 2a. Following the oscillation that is driven from\nthe applied external magnetic field, the TF time spectra\nwere fitted using an oscillatory component multiplied by\na non-oscillatory depolarizing component according to:\nA0PTF(t) =ATFcos(2πft+φ)e−(λt)+Abg(1)\nwhereA0is the initial asymmetry and PTFis the muon\nspin polarisation function. ATF,f,φandλare the asym-\nmetry, frequency, relative phase and depolarisation rate\nof the implanted muons under applied TF, while Abgac-\ncounts for background muon depolarization. We observe\na dumping of the oscillation with decreasing temperature,\ntowards a magnetically ordered state, with an internal\nfield that also takes part, together with the applied field,\nin the muon spin precession. The oscillation frequency\nis proportional to the applied magnetic field while the\nasymmetry ATFcorresponds to the externally magne-\ntized fraction of the sample. The transverse field asym-\nmetry is displayed as a function of temperature in figure\nFig.2band is fitted with a Boltzmann sigmoid function\nwhich outputs a transition temperature Tc= 104.3(4)K.\nA step appears below transition, between 100-50K, in-\ndicating that the static magnetic ground state is not yet\nreached. Above transition, the sample enters the param-\nagnetic phase and full asymmetry is recovered only above\n230K, a behaviour that corroborates the dc susceptibil-\nity results. The depolarization rate λTFis presented in\nfigure Fig. 2c. At high temperatures above transition,\nthe damping is approaching zero as the fluctuation of lo-\ncal magnetic moments increases resulting to a motional\nnarrowing effect. When temperature is decreased, λTF\npeaks around the magnetic phase transition, at the crit-\nical slowing down of magnetic fluctuations. At 75K a\nshoulder appears, pointing to the dynamic phase tran-\nsition also observed in AC susceptibility. Eventually, a\nstatic, commensurate magnetic ordering appears to be\nachieved at base temperature./s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/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/s48/s46/s50/s56\n/s48 /s50 /s52 /s54 /s56 /s49/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/s50/s75/s32 /s57/s53/s75/s32 /s49/s48/s53/s75/s32 /s50/s48/s48/s75/s32/s32/s65\n/s48/s80\n/s90/s70\n/s84/s105/s109/s101/s32/s40 /s115/s41\n/s32/s32\n(a)\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s32/s32/s82/s101/s97/s108/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s77/s72/s122/s41\n(b)\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53\n/s32/s32/s102/s32/s40/s77/s72/s122/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s102/s40/s84/s41/s61/s102\n/s48/s91/s49/s45/s40/s84/s47/s84\n/s99/s41 /s93\n/s102\n/s48/s32/s32/s61/s32/s51/s51/s46/s50/s57/s32 /s48/s46/s49/s50\n/s84\n/s99/s32/s61/s32/s49/s48/s51/s46/s50/s49/s32 /s48/s46/s52/s57\n/s61/s32/s50/s46/s56/s57/s32 /s32/s48/s46/s49/s55\n/s32/s32/s61/s32/s48/s46/s52/s50/s32 /s32/s48/s46/s48/s50/s56\n(c)\nFIG. 3: (a) Zero field time spectra at 2,95,105and200K\nfitted with Eq. ( 2). The left and right window present the\nshort and long time domain of the ZF time spectrum, respec-\ntively. A damped oscillation is observed up to 0.20µs while\nthe longer time domain displays a exponential relaxation. ( b)\nThe real part of the Fourier transform which exhibits the os-\ncillation frequency at 2 K. (c) The frequency component as a\nfunction of temperature, fitted with the power law function\n3. A magnetic transition appears at TN= 103.2(5)K where\nthe oscillation of the signal takes effect.6\n2.Zero field (ZF)\nZF measurements were carried out in the temperature\nrange2 K≤T≤200 K . We present a selection of muon\nspin depolarization spectra at temperatures above and\nbelow the magnetic transition in Fig. 3a. In the short\ntime scale, the onset of a precession and a fast relaxation\nof the muon spin ensemble appears below 100K. This\ndamped oscillation is the effect of a static internal field\ndistribution, perpendicular to the muon spin. At base\ntemperature T= 2K an oscillation appears with a\nsingle precessing frequency at 33.2 MHz up to 0.1µs,\nwhich denotes a commensurate magnetic ordering. In\nthe long time scale, a slower relaxation describes the\nspin dynamics which correspond to the longitudinal field\ncomponents to the muon spin.\na. Below the transition temperature.\nThe time spectrum is fitted with an internal field\n[34] , oscillating function and a Lorentzian Kubo-Toyabe\n(LKT) function [ 35] up to105K:\nA0PZF(t) =AIF/bracketleftbig\nξcos(2πft+ϕ)e−λTt+(1−ξ)e−λLt/bracketrightbig\n+ALKT/bracketleftbigg1\n3+2\n3(1−∆t)e−(∆t)/bracketrightbigg\n(2)\nwhereAIFis the asymmetry, fthe frequency which\nis a measure of the sublattice magnetization, and φthe\nrelative phase which is set to zero. The λTandλLare\nthe transverse and longitudinal depolarization rate ap-\nplied to the fast precessing part and slow relaxing tail\nrespectively. In the KT function each orthogonal compo-\nnent of the magnetic field at the muon site is represented\nby a probability distribution width ∆, the corresponding\ndistribution width.\nFrom DFT calculations on the sister compound\nLaSrNiReO 6two possible muon sites were predicted [ 13].\nFor the LaCaNiReO 6compound one frequency deemed\nappropriate to reconstruct and analyse the data. A\nBessel function or the use of two oscillators did not pro-\nduce a consistent fitting. The use of a single oscillation\nfrequency implies that the muon sites appear to be equiv-\nalent below the transition temperature. This precession\nfrequency is indicated in the real part of the Fourier\ntransform of the time spectrum, as shown in Fig 3b. In\nFig.3cthis frequency, as extracted from the ZF data, is\npresented against temperature. The frequency is propor-\ntional to the sample magnetization and the experimental\ndata can be fitted by a power-law function [ 11,17,36]\nthat considers both spontaneous magnetization and spin-\nwave excitations at low temperatures, but also the mag-\nnetic anisotropy that becomes significant near the Curie\ntemperature:\nf(T) =f0/bracketleftbigg\n1−/parenleftbiggT\nTN/parenrightbiggα/bracketrightbiggβ\n(3)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s65\n/s84/s65\n/s76/s65\n/s76/s75/s84/s65\n/s71/s75/s84/s65\n/s76/s75/s84\n/s32/s32/s65\n/s90/s70\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n(a)\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s52/s48/s53/s48/s54/s48/s55/s48\n/s84 \n/s76\n/s32/s32/s40 /s115/s45/s49\n/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s97/s84/s50\n/s108/s110/s40/s84/s41\n/s97/s61/s49/s46/s56/s53/s40/s49/s56/s41/s32 /s49/s48/s45/s54\n(b)\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s32/s76/s75/s84/s40 /s115/s45/s49\n/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n(c)\nFIG. 4: Fit results for the zero field data fitted with Eq. ( 2).\nAll fit parameters are presented as a function of temperature :\n(a) Asymmetries for the depolarization contributions to th e\nmuon ensembles. (b) The fast λTand slow λLdepolarization\nrates. (c) Field distribution width of the Lorentzian KT.7\n/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48/s54/s53\n/s32\n/s71/s75/s84\n/s32\n/s71/s75/s84\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40 /s115/s45/s49\n/s41\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s40 /s115/s45/s49\n/s41\n(a)\n/s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s49/s50/s51/s52/s53/s54\n/s32\n/s76/s75/s84\n/s32\n/s76/s75/s84\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40 /s115/s45/s49\n/s41\n/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\n/s32 /s40 /s115/s45/s49\n/s41\n(b)\nFIG. 5: The temperature dependencies of the dynamic Kubo-\nToyabe parameters of Eq. ( 4) in the temperature range 102.5−\n200K. The field distribution width and field fluctuation rate\n(a) for the Gaussian KT and (b) for the Lorentzian KT.\nwheref0is proportional to the spontaneous magnetiza-\ntion at base temperature. The critical exponent αcor-\nresponds to the low temperature properties, and βde-\ntermines the asymptotic behaviour near the transition\ntemperature. The fit results to a critical temperature\nTN= 103.2(5)K. This is the actual critical temper-\nature of the sample since the ZF measurement probes\nthe intrinsic magnetic ordering without excitations. The\ncritical exponent β= 0.42(3) matches to the mean-field\nmodel, while the low temperature exponent α= 2.89(17)\nfollows the T5/2power law according to the Dyson for-\nmalism [ 37]. This behaviour describes a system of two\nspin waves interacting in a ferromagnet. In case of a\nferrimagnet, the magnon excitation includes transverse\nfluctuations of both sublattice spins and the dispersion\ncurve consists of two branches [ 38,39].\nThe parameter ξin Eq.( 2) is used to regulate the\nAT=2AIF\n3andAL=AIF\n3ratio, describing the per-\npendicular (precessing) and the parallel (relaxing) tailcomponents. Below transition both AT,ALand the\ncontribution from the KT function are constant as illus-\ntrated in figure Fig. 4a. TheALKTcorresponds to a 6%\nvolumic fraction of the sample and the field distribution\nwidth narrows down with lower temperatures. This\ncontribution is attributed to randomly oriented dilute\nspins. Both depolarization rates λT,λLincrease as the\ntemperature approaches the transition, revealing an\nincrease in dynamics. The transverse component has a\nhigh value down to base temperature since it depends on\nthe static field distribution [Fig. 4b]. The longitudinal\ncomponent on the other hand slowly approaches zero as\nthe spin dynamics weaken but are still existent [Fig. 4c].\nBothλTandλLdenote a dynamic phase between\n30 K< T <100 K in agreement with the TF results\nand the bulk susceptibility measurements. A reasonable\ninterpretation can be to associate these features with the\nslowing down of small, canted components of the spin\nsublattices. In this region the spin-lattice is relaxing\nuntil it reaches a static, commensurate phase below\n30K. The evolution of λLrate, a measure of spin\nlattice relaxation, can be described as a function of\ntemperature aT2ln(T)belowTN. This dependence has\nbeen identified to be a signature of muon depolarization\nfrom a two magnon excitation [ 40,41], consistent with\nthe power law magnetization model.\nb. Above the transition temperature.\nFrom102.5K to200K the ZF- µ+SR spectra were\nfitted with a combination of a dynamic Gaussian Kubo-\nToyabe function ( Gdyn\nKT) and a dynamic Lorentzian Kubo-\nToyabe function ( Ldyn\nKT) [42]:\nA0PZF(t) =AGGdyn\nKT(∆,ν,t)+ALLdyn\nKT(∆,ν,t)(4)\nAbove the transition, the spectrum consists of a\nrapidly and slowly relaxing part as shown in Fig. 6a, in\nthe short and long time domain of the zero field time\nspectra. The fast relaxing part is related to magnetic\ndomains still present in the sample, revealed also by the\nM(H)measurement. The slow relaxing part describes\na paramagnetic state enriched with dilute magnetic mo-\nments. The amplitudes AGandALare a measure of\nthe volume fractions of the magnetic and paramagnetic\nregions respectively. With increasing temperature the\nmagnetic regions are diminished and the paramagnetic\nstate is eventually prevalent [Figs. 4a,2b].\nIn Figs. 5aand5b, we present the field distribution\nwidth (∆GKTand∆LKT) and field fluctuation rate ( νGKT\nandνLKT) of the Gaussian and Lorentzian KT func-\ntions respectively. An increase in dynamics appears,\nas expected, close to the transition temperature. It is\nnoteworthy that ∆GKTexhibits a new local maximum\natTf= 117 K, a phenomenon similar to the one ob-\nserved in the sister compound LaSrNiReO 6[13]. Possibly\nthis maximum indicates the presence of another phase\ntransition that is invisible to bulk magnetization mea-\nsurements. A dynamic glass-like crossover through local8\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48\n/s32/s32/s65\n/s48/s80\n/s90/s70\n/s84/s105/s109/s101/s32/s40 /s115/s41/s49/s48/s55/s46/s53/s75/s32/s32/s65\n/s48/s80\n/s90/s70\n/s84/s105/s109/s101/s32 /s40 /s115/s41\n(a)/s48/s46/s49/s52/s48/s46/s49/s54/s48/s46/s49/s56/s48/s46/s50/s48/s48/s46/s50/s50\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s49/s52/s48/s46/s49/s54/s48/s46/s49/s56/s48/s46/s50/s48/s48/s46/s50/s50\n/s50 /s52 /s54 /s56 /s49/s48/s32/s32\n/s90/s70/s50/s48/s79/s101\n/s32/s32\n/s51/s52/s79/s101\n/s32/s32\n/s84/s105/s109/s101/s32/s40 /s115/s41/s65\n/s48/s80\n/s76/s70\n/s54/s48/s79/s101\n/s32/s32\n(b)\nFIG. 6: (a) Zero field time spectra at T= 107.5K fitted with Eq.( 4). The inset shows the depolarization function at early\ntimes. (b) The zero and longitudinal field depolarization sp ectra for LF = 20,34,60Oe, atT= 175 K, also fitted with Eq.( 4).\nfreezing of the components of a random dense magnetic\nstate described by the Gaussian distribution, which in\nthis case is too broad for any oscillatory signal to ap-\npear.\nThis complex magnetic arrangement was further ex-\namined by measuring the longitudinal field dependence\nof theµ+SR spectra at 175K. The ZF- and LF- µ+SR\nspectra under fields of 20,34and60Oe are presented in\nFig.6b. The LF- µ+SR spectra are found to be success-\nfully fitted with the same Eq.( 4) as for the ZF- µ+SR spec-\ntra. Focusing on the slow end we observe a clear decou-\npling behaviour, an indication of a static but random field\ndistribution in this domain. Indeed the lorentzian KT pa-\nrameters presented in Fig. 7show an arguably static field\ndistribution. If we now turn our focus on the fast end of\nthe spectra, the fast relaxing component persists. Over-\nall, a large distribution width and its fluctuation rate\n(∆GKTandνGKT) suggest that 25%of the volume is\ncharacterized by electronic moments in a dynamic state.\nThis is roughly consistent with the wTF- µ+SR result, in\nwhich about 20%of the sample is in a magnetic state at\ntemperatures between 150 and 250K [see Fig. 2b].\nIV.Discussion\nBoth conventional bulk susceptibility and microscopic\nµ+SR experiments were used to complete the picture of\nthe magnetically diverse double perovskite LaCaNiReO 6\ndown to a long-range ferrimagnetic order established be-\nlowTN= 103 K. The underlying superexchange inter-\nactions between first, second or higher order neighbours\ndefine the magnetic ground state. Alterations to the elec-\ntron configuration, as a result of spin orbit coupling, the\nJahn-Teller effect as well as the composition of the crystal\nlattice, will result to a frustrate structure. In our case, e x-\nchange for a relatively smaller alkaline earth metal in oursample results to a different ground state realised in its\nsister compound LaSrNiReO 6, for which an incommen-\nsurate magnetic ground state was suggested [ 13]. These\ntwo compounds are a characteristic example proving that\nfrustration together with superexchange interaction play\na definitive role in the magnetic ordering of perovskite\noxides.\n/s48 /s50/s48 /s52/s48 /s54/s48/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48/s32\n/s32\n/s76/s75/s84\n/s32\n/s76/s75/s84/s40 /s115/s45/s49\n/s41\n/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s32\n/s32\n/s71/s75/s84\n/s32\n/s71/s75/s84\n/s48/s72\n/s76/s40/s79/s101/s41/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32 /s40 /s115/s45/s49\n/s41\nFIG. 7: Field distribution width and depolarization rate fo r\nthe Gaussian and Lorentzian KT of Eq. 4as a function of the\napplied longitudinal field.\nAs GKA rules predict, for a 152◦Ni-O-Re angle, there\nexists an antiferromagnetic (AF) exchange interaction\nbetween Ni2+and Re5+ions, a signature of geometri-\ncal frustration. The long range Ni-Ni and Re-Re interac-\ntions in the two sublattices are also significant, as they\nwere described in a similar case for LaSrNiRuO 6[16].\nThe NNN couplings are expected to be of ferromagnetic\nnature, leading to ferromagnetically ordered sublattices\nthat are antiferromagnetically coupled. In parallel, the9\nstrong spin-orbit coupling of Re atoms, in comparison to\nNi ones, reduces the total magnetic moment of the for-\nmer but also facilitates a DM exchange interaction and\nantisymmetric superexchange coupling [ 43], leaving open\nthe possibility to observe frozen spin components perpen-\ndicular to the ferrimagnetic order [ 44,45].\nIn our experiments we followed the evolution of mag-\nnetic phases with temperature as sublattices order and\nconsequently AF bonds are introduced. Based on the\nZF-µ+SR result [Figs. 3aand3c], we conclude that mag-\nnetic domains appear at TN. The depolarization line-\nshape consists of a highly damped oscillation and a slowly\nrelaxing part which correspond to the static magnetic\ndomains and the dynamically arranged magnetic frac-\ntion respectively. The TF measurements [Figs. 2aand2b]\nrevealed that the full asymmetry is not recovered until\naboveT= 270 K where the sample enters the paramag-\nnetic state. In the 103 K< T <230 K region a mixed\nmagnetic phase appears. This evolution is corroborated\nby the inverse dc susceptibility [Fig. 1b] behaviour until\nwe reach this high temperature region described by the\nCurie-Weiss law. Both results hint towards non-single\nphase sample although the P21/nspace group fits the\nNPD patterns at 300, 125 and 1K. De Teresa et alin the\nstudy of the magnetically similar Ca 2FeReO 6suggested\na mesoscopic phase separation in two monoclinic phases\nthat coexist in a wide temperature range, with different\nmagnetic properties [ 24]. This picture is compatible with\nour results above and below TNand the appearance of\ntwo phases can be justified as a result of magnetostriction\ndue to changes in the distance and angle of the B-O-B’\nbonds.\nWe now focus on the magnetic ordering below TN. In\nthe recorded hysteresis loop at 5 K, shoulders appear at\nremanent magnetisation. The origin of this phenomenon\nis thought to be an ordering of transverse spin compo-\nnents due to DM interaction. The monoclinic structure\nfacilitates DM exchange coupling below the ferrimagnetic\nspin ordering at TN[46,47]. The strength of this inter-\naction is proportional to the spin orbit coupling constant\nwhich will be significant for the heavy transition metal,\nsuch as, Re. This interplay may result to a portion of\nthe spin components ordering perpendicular to the ferri-\nmagnetic order, as predicted for magnetically frustrated\nsystems [ 44].\nTo substantiate this claim we have used the ZF- µ+SR\ntechnique which is capable to unambiguously separate\nstatic from dynamic signatures compared to bulk mag-\nnetization measurements. Below TNthe Ni and Re sub-\nlattices order [ 8,48,49] and the critical exponents ex-\ntracted from this transition follow the mean field model.\nFor a ferrimagnet, this model consists of two alternating\nsublattices with unequal and antiparallel magnetic mo-\nments. The molecular field consists of three coefficients,\ntwo ferromagnetic for each sublattice and one antiferro-\nmagnetic for their interaction. In the 30 K< T <100 K\nregion the spin lattice is relaxing towards a static com-\nmensurate order. The ZF- µ+SR depolarization data canbe understood by a spin wave treatment below TN, where\nboth inter- and intra-sublattice exchange is considered,\nresulting in a two magnon process [Figs. 4b]. An internal\nfield function was used to describe the ZF muon depolar-\nization spectra and an oscillation is clearly observed up to\n0.10µsat base temperature T= 2K. The corresponding\nfrequency and asymmetry components display a transi-\ntion to a commensurate magnetic order. This result is in\nagreement with the DC and AC susceptibility measure-\nments. The dynamic relaxation rates typically display\na maximum as the dynamics increase around transition\nand decay as the dynamics slow down with decreasing\ntemperature. However both λLandλTexhibit a pecu-\nliar behaviour between 30 K< T <100 K . This may be\nanother indicator for a dynamic phase transition as dis-\ncussed above, a signature of canting of spin components\n[44]. In addition to this evolution of the depolarization\nrate components, we observe a transition of the TF asym-\nmetry in two stages between 50 and 130K, and the ap-\npearance of a frequency dependent peak at 50K in the AC\nsusceptibility. These are characteristic features of a spi n\ncanted magnetic structure at low temperatures, originat-\ning from DM antisymmetric superexchange [ 50,51]. In\nthat case the spins of the two sublattices will be corre-\nlated through a canting angle θthat will give the ratio\nof the total to sublattice magnetization [ 43,52].\nAboveTN, the ZF- and LF- µ+SR spectra are described\nby a dynamic and a static field distribution function. The\nfast relaxing component persists at increasing tempera-\nture and longitudinal field. We interpret this behaviour\nas the footprint of a dynamic, random, dense magnetic\nstate. A peak in the distribution width [Figs. 5a] may\nsignify that spin clusters locally freeze out in this phase.\nThe slow relaxing component prevails as temperature in-\ncreases. A static field distribution is expected in this\ndomain. The origin of this distribution is most likely a\nnuclear magnetic field created by the nuclear magnetic\nmoments of the isotopes. A nuclear magnetic field is\ntypically described by a Gaussian distribution, however\nin our case a GKT function or a combination of distribu-\ntions did not produce an acceptable fit. A LKT was used\ninstead, assuming randomly distributed nuclear magnetic\nmoments as well as some very dilute Ni and/or Re elec-\ntronic moments.\nV.Conclusions\nWe have utilised magnetometry and muon spin spec-\ntroscopy to elucidate the magnetic properties of the dou-\nble perovskite compound LaCaNiReO 6. As the Ni and\nRe sublattices mutually order, magnetic phases appear as\nearly asT <230K. With decreasing temperature these\nphases evolve and finally a transition into a commensu-\nrate ferrimagnetic order occurs below TN= 103 K. As a\nresult of geometrical frustration of the crystal structure ,\nwe also find combined microscopic and bulk evidence of\na dynamic phase of spin arrangements for 30 K< T <10\n100 K . A canting of spins, which does not compromise\nthe static ferrimagnetic order down to base temperature\nat2K, is a probable scenario. It is a suggestive obser-\nvation, that although both LaCa xSr1−xNiReO 6,x= 1,0\nshare a common dense and dilute magnetic phase above\ntheir magnetic order transition and up to 230K, the\nsubstitution of a larger with a smaller diameter alkaline-\nearth drastically facilitates or hinders the formation of\nmagnetic order at low temperatures.\nAcknowledgments\nWe thank Dr. J.-C. Orain for experimental support.\nThis research was supported by the European Com-\nmission through a Marie Skłodowska-Curie Action and\nthe Swedish Research Council - VR (Dnr. 2014-6426\nand 2016-06955) as well as the Carl Tryggers Founda-\ntion for Scientific Research (CTS-18:272). J.S. acknowl-\nedge support from Japan Society for the Promotion Sci-\nence (JSPS) KAKENHI Grant Nos. JP18H01863 andJP20K21149. Y.S. is funded by the Swedish Research\nCouncil (VR) through a Starting Grant (Dnr. 2017-\n05078) and E.N. the Swedish Foundation for Strate-\ngic Research (SSF) within the Swedish national gradu-\nate school in neutron scattering (SwedNess). Y.S. and\nK.P. acknowledge funding a funding from the Area of\nAdvance- Material Sciences from Chalmers University\nof Technology. D.A. acknowledges partial financial sup-\nport from the Romanian UEFISCDI Project No. PN-III-\nP4-ID-PCCF-2016-0112. 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Izumi, Journal of Applied Crystallog-\nraphy44, 1272 (2011)." }, { "title": "1910.01405v2.Current_driven_domain_wall_dynamics_in_ferrimagnetic_strips_explained_by_means_of_a_two_interacting_sublattices_model.pdf", "content": "Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model AIP/123-QED\nCurrent driven domain wall dynamics in ferrimagnetic strips explained by means of a\ntwo interacting sublattices model\nEduardo Mart \u0013 \u0010nez,1V \u0013 \u0010ctor Raposo,1and \u0013Oscar Alejos2\n1)Dpto. F\u0013 \u0010sica Aplicada, Universidad de Salamanca, 37008 Salamanca,\nSpain\n2)Dpto. Electricidad y Electr\u0013 onica. Universidad de Valladolid. 47011 Valladolid,\nSpaina)\n(Dated: 23 October 2019)\nThe current-driven domain wall dynamics along ferrimagnetic elements are here the-\noretically analyzed as a function of temperature by means of micromagnetic simula-\ntions and a one dimensional model. Contrarily to conventional e\u000bective approaches,\nour model takes into account the two coupled ferromagnetic sublattices forming the\nferrimagnetic element. Although the model is suitable for elements with asymmet-\nric exchange interaction and spin-orbit coupling e\u000bects due to adjacent heavy metal\nlayers, we here focus our attention on the case of single-layer ferrimagnetic strips\nwhere domain walls adopt achiral Bloch con\fgurations at rest. Such domain walls\ncan be driven by either out-of-plane \felds or spin transfer torques upon bulk current\ninjection. Our results indicate that the domain wall velocity is optimized at the an-\ngular compensation temperature for both \feld-driven and current-driven cases. Our\nadvanced models allow us to infer that the precession of the internal domain wall\nmoments is suppressed at such compensation temperature, and they will be useful to\ninterpret state-of-the art experiments on these elements.\na)Electronic mail: oscar.alejos@uva.es.\n1arXiv:1910.01405v2 [physics.app-ph] 22 Oct 2019Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\nI. INTRODUCTION\nA great e\u000bort is being devoted to the \fnding of optimal systems permitting fast displace-\nment of domain walls (DWs) along racetrack elements.1As recent experiments demonstrate,\nDW velocities in the order of 1km\nscan be achieved along ferrimagnetic (FiM) strips,2,3with\na linear relationship between DW velocities and the magnitude of applied stimuli.2{4\nHere we provide a theoretical description of DW dynamics in FiM strips based on an\nextended collective coordinates model (1DM).5,6Di\u000berently from other approaches, based\non e\u000bective parameters, our model considers such elements as formed by two ferromagnetic\nsublattices, and coupled by means of an interlattice exchange interaction. Full micromag-\nnetic (\u0016M) simulations have been performed also to back up those drawn by the 1DM.\nImportantly, our approaches allow to infer results not achievable from e\u000bective models, and\nto provide insights and interesting predictions of the current-driven dynamics of DWs along\nFiM \flms.\nFig.1.(a) schematizes the local orientation of magnetic moments in the ferrimagnet.\n~ mi(i= 1;2) represent the orientations of the respective magnetic moments of each ferro-\nmagnetic sublattice. The magnetization of each sublattice is temperature dependent, so that\nmagnetization of each sublattice vanishes at Curie temperature ( TC), with a magnetization\ncompensation temperature TM, as it is shown in Fig.1.(b). The temperature dependence\ncan be described by the analytical functions: Ms;i(T) =M0\ns;i\u0010\n1\u0000T\nTC\u0011ai,M0\ns;ibeing the\nrespective magnetizations at zero temperature, and aibeing dependent on the sublattice\ncomponents.\nThe model can be applied to two di\u000berent architectures. As a \frst architecture (Fig.1.(c)),\na FiM strip on top of a heavy metal (HM) can be considered. The FiM/HM interface\npromotes interfacial asymmetric exchange, resulting in N\u0013 eel type DWs and current driven\ndomain wall motion (CDDWM) due to spin orbit torques (SOT), with rigid DWs. At the\nangular momentum compensation temperature ( TA), di\u000bering from TMdue to the distinct\nLand\u0013 e factors gifor each sublattice, DW magnetic moments keep aligned with the current,\nleading to a linear increase of DW velocities. Thus, DW velocities are maximized at TA.\nThis \frst architecture has already been adequately discussed from both the experimental3\nand theoretical3,6points of view, in particular, by using the model to be here recalled6.\nIn the second architecture (Fig.1.(d)), the FiM does not lie on a HM, and so interfacial\n2Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\n(a)\nX YZ\n𝑚2\n𝜓1𝑚1 𝜓2\n𝜓1>0𝜓2>0 (b)\nTC\nMs,i\nT emp eratureMs,1(T)\nMs,2(T)\nTMTA\n(c)\nX\nYZ\n\u0004\u0005\u0006\u0007 \n(d)\nX\nYZ\n\u0004\u0005\u0006\u0007 \nFIG. 1. Two sublattices constitute the FiM: (a) magnetizations are represented by the unit vectors\n~ m1and~ m2, with in-plane orientation angles 1and 2, respectively, (b) temperature dependence\nof the magnetization of each sublattice, (c) magnetic DW of N\u0013 eel type, and (d) magnetic DW of\nBloch type amidst two domains oriented out of plane (the strip width wis here shown).\nasymmetric exchange vanishes. CDDWM is dominated by the spin transfer torques (STT),\nand DW precessional regimes emerge, due to reduced magnetostic interactions, resulting in\nDW velocities proportional to current magnitudes. Again, DW velocities have been found\nto maximize at TA, when precession freezes, leading to a CDDWM characterized by rigid\nDWs, what is to be shown along this text.\nII. TWO-SUBLATTICE MODEL OF FERRIMAGNETS\nThe description of the DW dynamics by means of a 1DM starts from the application of\nvariational principles to the \u0016M equation, i.e, the Landau-Lifshitz-Gilbert (LLG) equation.7,8\nThis procedure is then augmented to study the magnetization dynamics in FiMs by posing\ntwo coupled LLG equations, that is, a two-sublattice model (TSLM). Details on the deriva-\n3Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\ntion of the 1DM equations for the TSLM are given in Ref.6, so here we will only recall the\nrequired model parameters.\nWithin the model, the respective Gilbert constants of each sublattice are represented\nby the values \u000bi. The e\u000bective \felds are the sum of the external \feld, the demagnetizing\n(magnetostatic) \felds, the anisotropy \felds, the isotropic exchange \felds and the asymmetric\nexchange \felds. The external \feld have components ( Bx;By;Bz). The demagnetizing term\npossesses out-of-plane and in-plane components, given by the e\u000bective anisotropy constants\nKeff;iandKsh;i. The asymmetric exchange provides a chiral character to some magnetic\ntextures, whereas the isotropic one can be reduced on \frst approach to the sum of an\nintra-sublattice exchange \feld, given by the exchange sti\u000bness Ai, and an inter-sublattice\ninteraction due to the misalignment of both sublattices. The latter is accounted for by\na parameter B12>0 (<0), which promotes the antiparallel (parallel) alignment of the\nsublattices. Finally, LLG equations also include the torques due to spin polarized currents,\ni.e., the STT7and the SOT8. Here, we focus our attention on the STT, consisting of\nadiabatic interactions and their non-adiabatic counterparts. The adiabatic interactions are\nde\fned by values ui, proportional to the electric density current Jx\rowing along the element,\nand calculated as ui=1\n2gi\u0016BP\neMs;iJx, with\u0016Bbeing Bohr's magneton, ethe electron charge,\nandPthe degree of polarization of the spin current. The non-adiabatic interactions are\nproportional to the adiabatic ones by factors \fi.\nThe derivation of the 1DM requires the DW pro\fle to be described in terms of the DW\npositionq, width \u0001 and transition type Q. In the TSLM, the DW is considered to be\ncomposed of two transitions, one for each sublattice, which share the same q, and the same\n\u0001 (see Fig.1.(c) and (d)), but Qi=\u00061 establishes the transition type for each sublattice.\nQi= +1 ( \u00001) means up-down (down-up) transition. Due to the antiferro coupling between\nsublattices, it follows that Q1=\u0000Q2.\nIII. RESULTS AND DISCUSSION\nWhen FiMs, such as GdFeCo or Mn 4N, are grown on top of certain substrates, the ab-\nsence of interfacial asymmetric exchange2,9results in the formation of achiral DWs. The\norientation of DW internal moments at rest is then dependent on purely geometrical as-\npects. In particular, for thin strips su\u000eciently wide, magnetostatic interactions determine\n4Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\nthe formation of Bloch-type walls. Importantly, due to the low net magnetization of FiMs as\ncompared with ferromagnets, the magnetostatic interactions are rather low. If some paral-\nlelism between ferro- and ferrimagnets is made, Walker breakdown in FiMs is then expected\nto occur for rather low applied \felds10or currents11,12in the temperature range around TM.\nConsequently, the DW dynamics for moderate \felds or currents is ruled by the precession\nof DW magnetic moments.\nThe case of the \feld-driven DW dynamics in ferrimagnetic GdFeCo alloys can be recalled\nat this point. This has been the subject of recent experimental work,2where fast \feld-driven\nantiferromagnetic spin dynamics is realized in FiMs at TA. This behavior has been found\nto be reproducible with the TSLM. Our simulations have been carried out with a set of\nparameters similar to those considered in previous works,3,6but adapted as to take into\naccount the absence of interfacial asymmetric exchange and SOTs. The parameters are:\nAi= 70pJ\nm,Keff;i\u0019Ku;i= 1:4MJ\nm3,Ku;ibeing the magnetic uniaxial anisotropy constant\nof the FiM sublattices. With these parameters, DW width is \u0001 \u00196nm. Besides, \u000bi=\n0:02. Due to the low net magnetization in the temperature range of interest, Ksh;i\u0019\n0. The antiferromagnetic coupling is accounted for by the parameter B12= 9MJ\nm3.13The\ngyromagnetic ratios ( \ri=gi\u0016B\n~) are di\u000berent due to distinct Land\u0013 e factors: g1= 2:2 and\ng2= 2:0.2The Curie temperature is set to TC= 450K, and M0\ns;1= 1:4MA\nmandM0\ns;2=\n1:71MA\nm, witha1= 0:5 anda2= 0:76. According to these values, TM\u0019241:5K, and\nTA\u0019305K. The dimensions of the FiM strips are w\u0002tFiM= 512nm \u00026nm.\nFig.2.(a) presents the dependence of the DW terminal velocity, computed as vst=\nq(\u0001t)\u0000q(0)\n\u0001t, with \u0001t= 2ns, on the out-of-plane applied \feld Bzat di\u000berent temperatures.\nIn agreement with experiments,2vstincrease linearly with Bz, and the slope reaches a max-\nimun atTA. This fact is made clear in Fig.2.(b) where terminal velocity is represented as\na function of temperature with Bzas a parameter. In all shown cases, no dynamics occurs\natTMsince the net magnetization vanishes, whereas the highest speeds are found close to\nTA. The clue for this behavior can be found in DW precession, represented as a function\nof temperature in Fig.2.(c). Precession frequencies are obtained as \u0017=_ i(\u0001t)\n2\u0019(i= 1;2),\nsince _ 1(\u0001t)\u0019_ 2(\u0001t). The results demonstrate that during the dynamics, DW magnetic\nmoments precess except at temperatures around TMandTA, where precession freezes and\nthe orientation of DW magnetic moments during the whole dynamics holds.\nPrevious \feld-driven analysis serves as a starting point to also understand the CDDWM in\n5Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\n(a)\n050100150200250300350\n0 20 40 60 80 100 120\nvst/parenleftbig\nm ·s−1/parenrightbig\nBz(mT)T= 200K\nT= 241 .5K\nT= 270K\nT= 305K\nT= 340K (d)\n020406080100120\n0 0 .5 1 1 .5 2\n|vst|/parenleftbig\nm·s−1/parenrightbig\nJx/parenleftbig\nTA·m−2/parenrightbigT= 241 .5K\nT= 270K\nT= 300K\nT= 340K (g)\n050100150200\n0 0 .5 1 1 .5 2\n|vst|/parenleftbig\nm·s−1/parenrightbig\nJx/parenleftbig\nTA·m−2/parenrightbigT= 241 .5K\nT= 270K\nT= 300K\nT= 350K\n(b)\n050100150200250300350\n220 240 260 280 300 320 340 360\nvst/parenleftbig\nm ·s−1/parenrightbig\nT(K)Bz= 20mT\nBz= 40mT\nBz= 80mT\nBz= 120mTTM TA (e)\n020406080100120\n220 240 260 280 300 320 340 360\n|vst|/parenleftbig\nm·s−1/parenrightbig\nT(K)Jx= 0.5TA·m−2\nJx= 1.0TA·m−2\nJx= 1.5TA·m−2\nJx= 2.0TA·m−2TM TA (h)\n050100150200250\n220 240 260 280 300 320 340 360\n|vst|/parenleftbig\nm·s−1/parenrightbig\nT(K)Jx= 0.5TA·m−2\nJx= 1.0TA·m−2\nJx= 1.5TA·m−2\nJx= 2.0TA·m−2TM TA\n(c)\n−1−0.500.511.522.5\n200 250 300 350 400\nν(GHz)\nT(K)sublattice 1\nsublattice 2\nTMTA (f)\n−1−0.8−0.6−0.4−0.200.20.40.60.8\n220 240 260 280 300 320 340 360\nν(GHz)\nT(K)sublattice 1\nsublattice 2\nTM TA (i)\n−2−1.5−1−0.500.511.5\n220 240 260 280 300 320 340 360\nν(GHz)\nT(K)sublattice 1\nsublattice 2\nTM TA\nFIG. 2. Field-driven an current-driven dynamics in a FiM strip: (a) terminal velocity as a function\nofBzwith temperature as a parameter, (b) terminal velocity with Bzas a parameter and (c)\nprecessional frecuencies of DWs for Bz= 40mT as functions of temperature, (d) and (g) terminal\nvelocity as a function of Jxwith temperature as a parameter, (e) and (h) terminal velocity with\nJxas a parameter and (f) and (i) precessional frecuency of DWs for Jx= 1TA\nm2as functions of\ntemperature. \fi=\u000bifor (d), (e) and (f), whereas \fi= 2\u000bifor (g), (h) and (i). Dots and\ncontinuous lines correspond respectively to full \u0016M simulations and the 1DM results.\nthese elements. This dynamics is purely governed by STT because DWs move contrary to the\ncurrent direction.9Fig.2.(d) and (g) present the dependence of the absolute terminal velocity\nas a function of the current Jxwith the temperature as a parameter. The polarization has\nbeen set to P= 0:7, and the non-adiabatic transfer torque parameters have been chosen\nas (d)\fi=\u000bi(also for \fgures (e) and (f)), and (g) \fi= 2\u000bi(also for \fgures (h) and (i)).\nDi\u000berently from the results obtained in the \feld-driven case, the CDDWM at TMis not\nnull, since the STT pushes the transitions in each sublattice in the same direction (and not\nin opposite directions as it occurs in the \feld-driven case). However, the maximum slope is\nagain found at TA, when the precessional frequency vanishes.\nTo show in more detail this behavior, Fig.3 presents the snapshots of the CDDWM at\ntwo representative temperatures, for the case \fi=\u000bi. The two sublattices composing the\nFiM are presented superposed, as to simplify the view, so one sublattice is on top of the\n6Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\n(a)T < T A (b)T=TA\n(b) up -down , \n0 ns 𝑡(ns)\n𝑥\n0.5 ns\n1 ns\n𝐽𝑥=1TA\nm2\n𝑚𝑥𝑚𝑦\n(b) up -down , \n0 ns 𝑡(ns)\n𝑥\n0.5 ns\n1 ns\nFIG. 3. Snapshots of the CDDWM in a FiM strip with \fi=\u000biat (a)T 𝐵c)−𝑅H(𝐵z<−𝐵c) with coerciv e field 𝐵c, are summarized in Fig. 1(b). The Δ𝑅H shows a sign change at 𝑇~165.5 K, \nindicating that the 𝑇M is approximately 165.5 K [11, 12 ]. \nThe DW dynamics above the 𝑇M next can be investigate d, where 𝑇A is expected to \nappear [12]. To measure the DW speed, we adopt the real -time DW measurement technique as \ndescribed in the following [13, 14]. The GdFeCo microstrip is first saturated by a sufficiently \nlarge magnetic field ( B = -150 mT) to the downward direction and then, a magnetic field (𝐵𝑧), \nwhich is smaller than the coercive field (𝐵C) but is larger than propagation field ( 𝐵𝑃) is applied \nto the upward direction . Note that the 𝐵𝑧 does not create DWs nor reverse the magnetization \nbecause the 𝐵𝑧 is smaller than 𝐵C. Subsequently, a current pulse ( 12–16 V and 5–30 ns) is \ninjected into the left vertical electrode as shown in Fig.1(a), which creates a DW near the \nelectrode . Once the DW is created, the DW is immediately moved by 𝐵𝑧, because the 𝐵𝑧 is \nlarger than the 𝐵𝑃. When the DW passes through the Hall cross, the Hall voltage drop is \nrecorded in the oscilloscope , from which we obtain the arrival time t. The DW speed 𝑣 is then \ncalculated by the travel length 𝑙 and the arrival time 𝑡. The DW speed was determined from \n10 times repeated measurements for each 𝐵𝑧. The temperature ranging from 200 K to 300 K \nis examined using the low temperature probe station. \nTo define the dynamic regime of DW, we investigate the magnetic field dependence \nof DW speed 𝑣. Figure 2(b) shows the 𝑣 - 𝐵𝑧 relation obtained at 𝑇= 260 K . Threshold \nmagnetic field ( 𝐵𝑧𝑡ℎ~ 30 mT) is clearly observed, suggesting that the thermally activated creep \nDW motion can appear near 𝐵𝑧𝑡ℎ [15–22]. For larger magnetic field ( 𝐵z> 40 mT ), on the \nother hand, the D W velocity shows linear increase by satisfying 𝑣=𝜇𝐵z. Here 𝜇 is the DW \nmobility. This suggests that DW motion belongs to the flow regime in the higher magnetic field \n[13, 20, 23 –25]. Therefore , the magnetic field dependence allows us to investigate the DW \ndynamics in two different dynamic regimes. We confirmed that the DW speed shows a similar field dependence at all temperatures examined. \nThe flow regime is first investigated. Figure 2 (b) shows 𝑣 with respect to 𝑇 for \n𝐵z=50 mT. The 𝑣 exhibits a maximum at 𝑇~240 K as indicated by the blue arrow. This \nresult is in line with the recent observation that the DW speed become s maximized at the \nangular moment compensation temperature 𝑇A due to the pure antiferromagnetic spin \ndynamics at 𝑇A [12]. Therefore, we can conclude that the 𝑇A~240 K in our GdFeCo \nmicrostrip . \nAn important outstanding question is whether the DW speed exhibit s sharp increase at \n𝑇A even in the creep regime. To check this, we perform the experiment near 𝐵𝑧𝑡ℎ for T>𝑇A. \nFigure 3(a) shows the log𝑡 with respect to temperature for several magnetic fields. Blue and \nred symbols correspond to the data in creep (𝐵z< 40 mT ) and flow regime (𝐵z> 40 mT ). \nHere, the reason why we plot the log𝑡 instead of log𝑣 is that it is hard to define the creep \nvelocity due to stochasticity (that is, measured 𝑡 may not be the arrival time but be the \ndepinning time ). The result shows that the temperature dependence of t is clearly different \ndepending on the dynamic regime. To clearly see the difference, we define a slope of log(𝑡)-\n𝑇 as 𝛽 (≡log (𝑡)/𝑇). Figure 3(b) summarizes 𝛽 with respect to 𝐵z. It is clear that 𝛽 is \npositive in the flow regime ( 𝐵z> 40 mT ). This means that , in the flow re gime , the DW \nvelocity decreases with increasing the temperature for T>𝑇A, as observed in Fig. 2(b). \nContrary to this, 𝛽 has a negative value in the creep regime ( 𝐵z< 40 mT ). That is, in the \ncreep regime, the higher the temperature, the shorter ( faster ) the DW depinning time ( speed ). \nThis result is consistent with the thermal activat ion process , in which the depinning time \ndecreases with increasing temperature due to the assistance of the thermal energy . This means \nthat the unique antiferromagnetic DW dynamics observed at 𝑇A is not relevant in the DW \ncreep regime. Instead, the thermal activation over energy barrier s dominates the DW motion in the creep regime. Our results therefore imply that the identification of the dynamic regime is \nimportant for ferrimagnet -based spintronic applications [26–29]. \nIn conclusion, we have investigat ed the motion of ferrimagnetic DW near the angular \nmom entum compensation temperature in two dynamic regimes , creep and flow regimes . We \nfound a distinct temperature dependence of the DW speed between two dynamic regime s. The \nDW speed shows a peak at 𝑇A in the flow regime , where as it increases monotonically with \nincreasing temperature in the creep regime. These observation s imply that the DW dynamics \nis governed by the total angular momentum in flow regime, whereas it is dominated by the \nthermal activation process in creep regime. 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The red dot ted line represents the best linear fit base d on 𝒗=𝝁𝑩𝐳. (b) DW \nspeed 𝒗 as a function of 𝑻 for 𝑩𝐳= 50 mT . The red arrow represents 𝑻𝐌 and the blue \narrow indicates 𝑻𝐀. \nFigure 3 . (a) The measured 𝐥𝐨𝐠(𝒕) as a function of 𝑻 for several 𝑩𝐳. (b) The slope in \nFig. 3(a) which is defined as 𝜷(≡𝐥𝐨𝐠 (𝒕)/𝑻) with respect to 𝑩𝐳. The dot ted lines guide \nthe eye . Acknowledgements \nThis work was partly supported by JSPS KAKENHI Grant Numbers 15H05702, 26870300, \n26870304, 26103002, 25220604, 2604316 Collaborative Research Program of the Institute for \nChemical Research, Kyoto University, and R & D project for ICT Key Technology of MEXT \nfrom the Japan Society for the Promotion of Science (JSPS). D.-H.K. was supported from \nOverseas Researcher under Postdoctoral Fellowship of JSPS (Grant Number P16314). KJK \nwas supported by the National Research Foundation of Korea (NRF) grant funded by the Korea \ngovernment (MSIP) (No. 2017R1C1B2009686) and by the DGIST R&D Program of the \nMinistry of Science, ICT and Future Planning (17 -BT-02). \n \nFig. 1 \n \n \n \nFig. 2 \n \n \n \nFig. 3 \n" }, { "title": "2009.10005v1.Unusual_effects_of_magnetic_dilution_in_the_ferrimagnetic_columnar_ordered___mathrm_Sm_2MnMnMn__4_x_Ti_xO__12____perovskites.pdf", "content": "Unusual e\u000bects of magnetic dilution in the ferrimagnetic columnar ordered\nSm2MnMnMn 4{xTixO12perovskites\nAnuradha M. Vibhakar,1, 2Dmitry D. Khalyavin,3Pascal Manuel,3Ran\nLiu,4, 5Kazunari Yamaura,4, 5Alexei A. Belik,4and Roger D. Johnson1, 2\n1Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom\n2Department of Physics and Astronomy, University College London,\nGower Street, London, WC1E 6BT, United Kingdom\n3ISIS facility, Rutherford Appleton Laboratory-STFC, Chilton, Didcot, OX11 0QX, United Kingdom\n4International Center for Materials Nanoarchitectonics (WPI-MANA),\nNational Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan\n5Graduate School of Chemical Sciences and Engineering, Hokkaido University,\nNorth 10 West 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan\n(Dated: September 22, 2020)\nPowder neutron di\u000braction experiments have been employed to establish the e\u000bects of site-selective\nmagnetic dilution in the Sm 2MnMnMn 4{xTixO12A-site columnar ordered quadruple perovskite\nmanganites ( x= 1,x= 2 andx= 3). We show that in all three compositions the Mn ions adopt a\ncollinear ferrimagnetic structure below 27 K, 62 K and 34 K, respectively. An unexpected increase\nin the ordering temperature was observed between the x= 1 andx= 2 samples, which indicates a\nconsiderable departure from mean \feld behaviour. This result is corroborated by large reductions\nin the theoretical ground state magnetic moments observed across the series, which indicate the\npresence of spin \ructuations and or disorder. We show that long range magnetic order in the x= 3\nsample, which occurs below the percolation threshold for B-B exchange, can only be understood\nto arise if magnetic order in Sm 2MnMnMn 4{xTixO12is mediated via both A-B and B-B exchange,\nhence con\frming the importance of A-B exchange interactions in these materials. Finally we show\nthat site-selective magnetic dilution enables the tuning of a ferrimagnetic compensation point and\nthe introduction of temperature-induced magnetization reversal.\nI. INTRODUCTION\nThe simple perovskite manganites (general chemical\nformula ABO 3, B=Mn) are canonical examples of cor-\nrelated electron systems in which charge, spin and or-\nbital order can be tuned to give rise to exotic electronic\nground states. This is most famously demonstrated in\nLa1{xA0xMnO 3(A0= Ca2+, Sr2+, and Ba2+) where a\ntransition from an antiferromagnetic insulating phase to\na ferromagnetic metallic phase, driven by dopant concen-\ntration, leads to the emergence of the technologically im-\nportant property of colossal magnetoresistance [1]. More\nbroadly, when the A-site is occupied by a rare-earth ion\nor yttrium (R) a variety of B-site magnetic structures\nhave been found. For example, A-type antiferromagnetic\norder was observed experimentally for R = La →Gd\n[2, 3], and for compositions with smaller R3+ionic radii\n(Ho→Yb) E-type antiferromagnetic order is stabilised\n[3]. Furthermore, in the mixed valence manganites, such\nas La 0.5Ca0.5MnO 3, ziz-zag spin chains of the CE type\nmagnetic structure arise as a result of the charge and or-\nbital order associated with a checkerboard arrangement\nof B-site Mn3+and Mn4+ions [4]. Despite considerable\ndepartures from the ideal 180 °Mn-O-Mn bonding geom-\netry of up to 40 °(due to octahedral tilts), these mag-\nnetic structures can be well understood in terms of dom-\ninant B-B magnetic exchange interactions described by\nthe Goodenough-Kanamori-Anderson (GKA) rules [5].\nRare-earth magnetism can play an important role via\nA-A and A-B interactions, but typically only at low tem-peratures [6] owing to the much weaker exchange between\nf\u0000fandf\u0000delectrons, respectively, compared to that\nofd\u0000delectrons.\nIn the AMn 7O12quadruple perovskite manganites a\nlargea+a+a+octahedral tilting pattern (in Glazer nota-\ntion [7]) introduces an ordered arrangement of Mn ions\nonto the A sites of the perovskite framework such that\nthese systems also incorporate A-A and A-B d\u0000dex-\nchange pathways. Compared to the simple perovskites,\ncompetition between B-B exchange and these additional\ninteractions can lead to new and complex paradigms in\nfrustrated magnetism, especially because the B-B ex-\nchange is diminished due to the large octahedral tilt-\ning pattern (Mn-O-Mn bond angles reduced by \u001840-45°\naway from the ideal 180 °[8, 9]). For example, when R =\nLa, Ce, Nd, Sm and Eu a collinear ferrimagnetic struc-\nture was observed that cannot be explained by domi-\nnant B-B interactions alone [10, 11]. For A = Ca, Sr,\nCd, Pb and Na 1{xCaxMn7O12more complex magnetic\nstructures were observed; a constant moment magnetic\nhelix with a modulated spin helicity, and an incommen-\nsurate pseudo CE-type phase, respectively [12{14]. Both\nincommensurate structures arise from a balance between\ncompeting A-A, A-B, and B-B exchange interactions.\nRecently, a family of A 2A0A00B4O12A-site columnar\nordered quadruple perovskite manganites have been syn-\nthesised [15]. In these manganese oxides a large a+a+c\u0000\noctahedral tilting pattern gives rise to three crystallo-\ngraphically distinct A sites, and creates a unique set\nof exchange pathways not found in either the simplearXiv:2009.10005v1 [cond-mat.str-el] 21 Sep 20202\nor quadruple perovskite manganites described above.\nHere, both A-B and B-B d\u0000dsuper-exchange inter-\nactions are present, while the a+a+c\u0000tilts remove A-\nAd\u0000dsuper-exchange pathways, which are then re-\nduced to super-super-exchange. The relative strength of\nA-Ad\u0000dexchange has recently been demonstrated in\n(NaDy)MnMnTi 4O12[16], where in the absence of A-B\nand B-B interactions (B = Ti4+) antiferromagnetic order\ndevelops on the A-site sublattices only below \u001812 K. To\nthe contrary, in Tm 2MnMnMn 4O12ferrimagnetic order\nappears below 74 K, with a ferromagnetic B-site sublat-\ntice that is in direct contradiction to the C-type antiferro-\nmagnetic B-site sublattice theoretically predicted by the\nGKA rules | a state that is instead thought to arise as\na result of A-B exchange dominating over both A-A and\nB-B exchange [17]. Indeed, it was shown that when A-\nB exchange is weakened by substituting Cu2+for Mn3+\non the A0sites (R 2CuMnMn 4O12, R = Dy and Y) anti-\nferromagnetic spin canting is introduced onto the B-site\nsublattice [18]. Hence, it has become clear that the A-\nsite columnar ordered quadruple perovskite manganites\npresent a \rexible framework in which novel frustrated\ngeometries of A-A, A-B, and B-B exchange interactions\nmay be tuned through chemical substitution to give rise\nto unconventional magnetic states.\nIn this paper, we report the tuning of magnetic ex-\nchange interactions in the Sm 2MnMnMn 4{xTixO12A-\nsite columnar ordered quadruple perovskite manganites\nby chemical substitution of non-magnetic Ti4+for B-site\nMn3+forx= 1,x= 2 andx= 3. This series interpo-\nlates between the A-site only, antiferromagnetic structure\nof (NaDy)MnMnTi 4O12[16], and the full ferrimagnetic\nstructure of Tm 2MnMnMn 4O12[17], discussed above.\nSm2MnMnMn 4{xTixO12has a tetragonal crystal struc-\nture with space group P42=nmc , where the charge and\norbital order responsible for the lower orthorhombic sym-\nmetry of the R 2MnMnMn 4O12manganites [15, 17] has\nbeen suppressed by the B-site chemical disorder [19]. The\nthree symmetry inequivalent A sites, labelled A(Sm1),\nA0(Mn1) and A00(Mn2), are occupied by 10-fold coordi-\nnated Sm3+, square planar coordinated Mn2+or Mn3+,\ndepending on the amount of Ti4+substituted on the\nB sites, and by tetrahedrally coordinated Mn2+respec-\ntively. The B sites, labelled Mn3/Ti1, are octahedrally\ncoordinated and occupied by 75% Mn3+and 25% Ti4+\nfor thex= 1 sample, 50% Mn3+and 50% Ti4+for the\nx= 2 sample and 25% Mn2+and 75% Ti4+for thex= 3\nsample. The two cations are disordered across the B sites\nwith an average oxidation state of between +3.25 and\n+3.5. Published DC magnetometry measurements show\nthat samples with x= 1,x= 2 andx= 3 undergo a\nsingle magnetic phase transition at temperatures of Tc=\n27 K, 62 K and 40 K respectively [19]. Here we perform a\nquantitative neutron di\u000braction study and show that all\ncompositions undergo a transition to a long range ordered\nmagnetic phase below Tcin which the Mn ions adopt a\ncollinear ferrimagnetic structure. Remarkably these re-\nsults demonstrate an unusual increase in the orderingtemperature between the x= 1 andx= 2 samples, de-\nspite an increase in the amount of non-magnetic Ti4+on\nthe B-site sublattice { contrary to what is observed in\nnumerous other magnetically dilute systems [20] and is\nsuggestive of a departure from mean \feld physics. In the\nx= 2 sample we demonstrate that site-selective magnetic\ndilution has enabled the tuning of ferrimagnetic compen-\nsation and the introduction of temperature-induced mag-\nnetization reversal. We show that in the x= 3 sample,\nlong-range B-site magnetic order must percolate via both\nA-B and B-B exchange, demonstrating the importance\nof A-B exchange in mediating magnetism in the A-site\ncolumnar ordered quadruple perovskites.\nThe paper is organised as follows. In Sec. II we present\nthe experimental details, followed by the results of the\nneutron powder di\u000braction re\fnement, crystal electric\n\feld and magnetic anisotropy calculations and percola-\ntion calculations in Sec. III A, III B and III C respec-\ntively. A discussion on the variation of ordering temper-\nature and size of the magnetic moments with Ti4+con-\ntent is given in Sec. IV and a summary of our \fndings\nin Sec. V.\nII. EXPERIMENTAL DETAILS\nPolycrystalline samples of Sm 2MnMnMn 4{xTixO12\nwere synthesised under high-temperature and high-\npressure conditions from stoichiometric mixtures of\nMn2O3, Mn 3O4, TiO 2and Sm 2O3forx= 1, from\nMn2O3, MnO, TiO 2and Sm 2O3forx= 2, and MnO,\nTiO 2and Sm 2O3forx= 3, as detailed in Ref. 19. Neu-\ntron powder di\u000braction experiments were performed us-\ning the time of \right di\u000bractometer, WISH at ISIS [21].\nx= 1,x= 2 andx= 3 samples of mass 0.8g, 0.9g, and\n0.7g, respectively, were loaded into 3mm vanadium cans\nand cooled to 1.5 K. Di\u000braction data with good counting\nstatistics were collected on warming from a base tempera-\nture of 1.5 K up to 35 K in 5 K steps for the x= 1 sample,\nup to 78 K in 6 K steps for the x= 2 sample, and up to 60\nK in 6 K steps for the x= 3 sample. Measurements with\nhigh counting statistics were made in the paramagnetic\nphase (80 K for x= 1 andx= 3, 90 K for x= 2) and at\n1.5 K for all three samples. Symmetry analysis was per-\nformed using the isotropy software suite [22], and the\ndi\u000braction data were \ft using fullprof [23]. Magnetic\nmeasurements were performed on a SQUID magnetome-\nter (Quantum Design, MPMS-XL-7T) between 2 and 400\nK in 100 Oe under both zero-\feld-cooled (ZFC) and \feld-\ncooled on cooling (FCC) conditions. Isothermal magne-\ntization measurements were performed between \u000070 and\n70 kOe at 5 K.3\nFIG. 1. Neutron powder di\u000braction data collected on the time of \right di\u000bractometer WISH at ISIS of (a)-(b) the x = 1 sample\n(c)-(d) the x = 2 sample and (e)-(f) the x = 3 sample, at temperatures representative of the paramagnetic phase and at 1.5\nK. The data are given by the red circles, the \ft of the data by the solid black line, and their di\u000berence as a blue line. The\ngreen tick marks are the nuclear and magnetic re\rections from top to bottom of Sm 2MnMnMn 4{xTixO12labelled (1) and an\nimpurity phase labelled (2).\nIII. RESULTS\nA. Crystal and Magnetic Structures\nTheP42=nmc crystal structure model previously re-\nported for Sm 2MnMnMn 4{xTixO12[19] was re\fned\nagainst the neutron powder di\u000braction data measured in\nthe paramagnetic phase of each of the samples. For the x\n= 1 and x = 3 samples weak di\u000braction peaks that could\nnot be indexed by the P42=nmc model were found to\noriginate in an MnCO 3impurity phase (0.11 wt. %) and\nSm2Ti2O7impurity phase (0.25 wt%) respectively. For\nall three samples the thermal parameter of Mn1 re\fned\nto unphysically large values when the ion was constrained\nto the centre of the square planar coordination (Wycko\u000b\nposition 2a). Hence, we adopted a split-atom model rep-\nresentative of site disorder above and below the square\nplanar coordination (Wycko\u000b position 4c), as employed\nin Ref. 19, which reduced the thermal parameter to phys-\nical values. Sm was free to substitute Mn on the A0and\nA00sites and Mn was free to substitute Sm on the A\nsites, while imposing full site occupations. An excellent\n\ft was achieved in all samples and the re\fned stoichiome-\ntries were as follows: Sm 2Mn(Mn 0.84Sm0.16)Mn 3TiO 12\n(R= 2:72%,wR = 2:90%,RBragg = 4:06% at\n80 K), Sm 2Mn(Mn 0.93Sm0.07)Mn 2Ti2O12(R= 2:71%,\nwR = 2:87%,RBragg = 5:05% at 90 K) and\nSm2MnMnMnTi 3O12(R= 2:03%,wR = 2:20%,\nRBragg = 5:69% at 80 K). Note that the RBragg values are\nlarge in all samples due to strong absorption from Sm3+\nwhich results in much weaker di\u000bracted intensities.\nA gradual growth of the (200), (002), (112) and (110)\nre\rections was observed for all three samples below Tcin\naccordance with changes in the sample's magnetization\n(Fig. 4), and hence these peaks were identi\fed as mag-\nnetic in origin. For all three samples, the magnetic peaksare sharp and of a similar width to the nuclear peaks,\nindicating that the magnetic structure is well correlated.\nThe above peak positions were consistent with a \u0000-point,\nk= (0;0;0), magnetic propagation vector. The mag-\nnetic \u0000-point representation for the Wycko\u000b positions\nof the Sm and Mn sublattices decomposes into six 1-\ndimensional irreducible representations (irreps) (\u0000\u0006\n1, \u0000\u0006\n2,\n\u0000+\n3and \u0000+\n4) and two 2-dimensional (2D) irreps (\u0000\u0006\n5),\nwhich together de\fne a total of 12 di\u000berent magnetic\nsymmetries. The rapid increase of the magnetic suscep-\ntibility below Tc(Fig. 4) indicated the presence of ferro-\nmagnetic sublattices, which are only consistent with the\n\u0000+\n3or \u0000+\n5irreps. Furthermore, the observation of \fnite\nmagnetic intensity on the (002) re\rection indicates that\nthe magnetic moments are not aligned parallel to the\nc-axis (as magnetic di\u000braction intensity is proportional\nto the magnetic moment components perpendicular to\nthe scattering vector), which uniquely identi\fes that the\nmagnetic structure transforms by the \u0000+\n5irrep.\nThe \u0000+\n5symmetry adapted basis functions for the A,\nA0,A00and B sites are summarised in Tables III, IV, and\nV. Magnetic structure models constructed from linear\ncombinations of these basis functions were exhaustively\ntested by re\fnement against the neutron di\u000braction data\nmeasured at 1.5 K from all samples. A model in which\nthe B-sites moments were oriented collinearly in the ab\nplane and ferromagnetically coupled, and the A0andA00\nsites collinear but anti-aligned with respect to the B site\nsublattice, gave the best \ft to the data ( RMag = 17.0%\nfor x = 1, 5.39% for x= 2 and 4.34% for x = 3), and is\nshown in Fig. 2a. The re\fned moment magnitudes are\ngiven in Table I, which also lists the net ferrimagnetic\nmoment per formula unit. The strong dependence of the\nnet ferrimagnetic moment on composition, x, is re\rected\nin the remnant magnetisation, Mr, evaluated by extrap-\nolating the powder-averaged bulk magnetisation to zero4\n\feld (see Figure 3). Scaled extrapolated values are listed\nin Table I, and their relative magnitudes correspond well\nto the net ferrimagnetic moment per formula unit ob-\ntained from neutron di\u000braction, except for the x= 2\nsample. This discrepancy cannot be explained by the\nproposed magnetic structure model.\nFIG. 2. (a) Experimentally determined magnetic structure\nof Sm 2MnMnMn 4{xTixO12upto a direction in the abplane.\n(b) Theoretically proposed spin con\fguration of the Sm sites\nas determined by point charge calculations. The Asites are\ncolored in green, the A0sites in black, and the A00sites in\nred. The B sites are colored in blue, and for clarity we show\nthe scenario in which all B-site ions of a magnetic unit cell\nare Mn3 { a more accurate depiction of the unit cell would\nconsist of statistically disordered Ti1 and Mn3 ions on the B-\nsites. The magnetic space group is Ccc0a0(No. 68.516) with\nbasis vectors (1,1,0),(-1,1,0),(0,0,1) and an origin of (1,1/2,1)\nwith respect to the paramagnetic space group.\nFIG. 3. Plots of the magnetization, M, as a function of\napplied \feld, H, taken at a temperature of 5 K for each of\nthe samples. A magnetic \feld range of 22.5 to 70 kOe was\nused to extrapolate the magnetization to zero \feld and the\nextrapolated remnant magnetization is shown by the red line.\nThe re\fned magnetic structure is consistent\nwith the collinear ferrimagnetic phase found in\nTm2MnMnMn 4O12[17] and the FI phase of\nR2CuMnMn 4O12[18]. Moment orientations of\nmjj[x;0;0],mjj[x;x;0] ormjj[x;y;0] correspond to\ndi\u000berent magnetic symmetries described by the three\ndistinct order parameter directions of the \u0000+\n5irrep (see\nTables III, IV and V), however these three cases cannot\nbe di\u000berentiated by our neutron di\u000braction data due to\npowder averaging in the tetragonal crystal symmetry. In\nthe following we adopt a model with mjj[x;x;0], which\nis consistent with Sm crystal electric \feld calculations\n(Sec. III B).The temperature dependences of all symmetry-\ninequivalent moment magnitudes, plotted in Fig. 4, were\nevaluated by \ftting the magnetic structure model to vari-\nable temperature neutron powder di\u000braction data mea-\nsured from all three samples. The re\fnements were con-\nstrained such that each magnetic ion had a moment no\nlarger than its theoretically predicted maximum, given in\nTable I, and with the ratio of the Mn1:Mn2 moments cho-\nsen to be consistent with their nominal oxidation states.\nThe re\fnement was insensitive to a Sm1 moment at all\ntemperatures, so it was set to zero. For the x= 2 sam-\nple the B-site moments grow rapidly as the temperature\ndecreases, but quickly saturate in comparison to the the\nA-site moments which steadily grow with decreasing tem-\nperature. As a result, the net magnetisation (the sum\nover all magnetic moments in the unit cell with the A\nsublattices antialigned with respect to the B sublattice)\nreverses direction with decreasing temperature, giving\nrise to a compensation point at 45 \u00063 K. Magnetisation\nreversal was also observed in the temperature dependent\nmagnetic susceptibility FCC data of the x= 2 sample, as\nshown in the inset of Fig 4e, for which the compensation\ntemperature was determined to be 46 \u00061 K in excellent\nagreement with the neutron di\u000braction results. We note\nthat the absolute value of the negative, low temperature\nmagnetic susceptibility is not re\rective of the full net\nferrimagnetic moment, as the coercive \feld of the sample\nmay come close to the 100 Oe applied \feld.\nTABLE I. Absolute values of the 1.5 K experimental\n(Exp.) and theoretical (Calc.) ionic magnetic moments of\nSm2MnMnMn 4{xTixO12. The net magnetisation, M, per for-\nmula unit is determined from the experimental moment val-\nues, where the A sites align antiparallel to the B sites. The\nremnant magnetization, M r, extrapolated from magnetome-\ntry data was scaled by a factor of 2 to account for powder\naveraging. All moments are given in units of \u0016B.\nIon x = 1 x = 2 x = 3\nExp. Calc. Exp. Calc. Exp. Calc.\nSm1 0.00 0.71 0.00 0.71 0.00 0.71\nMn1 1.70(7) 4.0 3.72(7) 5.0 4.09(8) 5.0\nMn2 2.03(9) 5.0 3.72(7) 5.0 4.09(8) 5.0\nMn3 0.82(6) 4.0 2.54(6) 4.0 2.1(1) 5.0\nM per f.u 1.3(1) 2.4(1) 6.0(2)\nMrper f.u 1.006(7) 1.05(1) 5.68(3)\nTc 27 K 62 K 34 K\nB. Magnetic Anisotropy\nThe Jahn-Teller active Mn3+ions present in the\nx= 1 andx= 2 samples are expected to impose an\naverage weak single-ion anisotropy jjc[17, 24], while\nall other Mn ions are isotropic Mn2+. The three\nSm2MnMnMn 4{xTixO12samples have a magnetization\ndirection in the abplane, perpendicular to the easy axis of\nthe Mn3+ions, suggesting that the magnetic anisotropy5\nFIG. 4. (a)-(c) The temperature dependence of ionic mo-\nment magnitudes given for each of the symmetry inequivalent\nmagnetic ions. The moments of Mn1 and Mn2 were \fxed to\nbe the same in the x= 2 andx= 3 samples to be consistent\nwith their equivalent nominal oxidation states. (d)-(f) ZFC\nand FCC magnetization measurements under an applied DC\n\feld of 100 Oe. The inset in (e) shows the ZFC and FCC\nmeasurements for the x= 2 sample on a reduced scale.\nof Sm3+may instead play an important role despite our\ndata being insensitive to an ordered moment on the Sm\nsites (the predicted Sm3+moment of\u00180.7\u0016B is feasibly\nbelow the sensitivity of our neutron di\u000braction experi-\nment). We note that in Tm 2MnMnMn 4O12the compe-\ntition between the Mn3+and RE magnetic anisotropies\nwas found to result in a spin reorientation phase tran-\nsition. If the magnetic anisotropy is indeed determined\nby Sm3+, then the absence of a spin-reorientation phase\ntransition in the Sm 2MnMnMn 4{xTixO12compounds\nimplies signi\fcant f\u0000dexchange coupling immediately\nbelowTc(N.B. in the orthoferrites strong f\u0000dexchange\ncoupling is also present in SmFeO 3and leads to a phase\ntransition approx. 300 K - 400 K above all other rare-\nearth orthoferrites [25]).\nPoint charge approximations to the Sm3+crystal elec-\ntric \feld (CEF) were used to calculate the ground\nstate magnetic anisotropy of Sm3+, and hence estab-\nlish whether f\u0000dexchange coupling of rare-earth mo-\nments to the manganese sublattices might be responsi-\nble for the abplane magnetization direction observed in\nSm2MnMnMn 4{xTixO12. Sm3+has electronic con\fgu-\nration [Xe]4 f5, and Hunds rules give S = 5/2, L = 5\nand J = 5/2 for the lowest-energy 6-fold multiplet of\nstates. The local crystal electric \feld of Sm3+has the\npoint group symmetry 2 mm, which lifts the multiplet\ndegeneracy to form 3 Kramers doublets. All non-zero\ncrystal electric \feld parameters, Bm\nn[26], were calculated\nfor Sm3+in its two ionic positions, (1\n4,1\n4,z) and (1\n4,1\n4,z+1\n2), labelled Sm11 and Sm12 respectively (Table II).\nIonic positions re\fned for the x= 2 compound at 1.5 K\nwere used for reference, and similar values were obtained\nusing the ionic positions of the x= 1 andx= 3 samples.\nSm11 and Sm12 are symmetry equivalent, and related by\na 42screw, therefore all values of Bm\nnare the same across\nthe two sites, except the B2\n2andB2\n4terms which change\nsign upon 90\u000erotation about c.\nWe consider the following Hamiltonian for a single\nSm3+ion;\nH=X\nnnX\nm=\u0000nBm\nnOm\nn+gJ\u0016BJ\u0001Bex (1)\nwhereOm\nnare the Stevens operators [26], gJis the Land\u0013 e\ng-factor, J= (Jx;Jy;Jz) is the vector total angular mo-\nmentum operator, and Bexis an e\u000bective exchange \feld\noriginating in f\u0000dexchange interactions with the man-\nganese sublattices.\nThe magnetic anisotropy of Sm11 and Sm12 was deter-\nmined by repeatedly solving Equation 1 for a 1 T \feld ap-\nplied in di\u000berent directions over a full hemisphere. Eigen-\nfunctions were found through diagonalisation of H, and\nmagnetic moment components mx,my, andmzwere cal-\nculated by summing the expectation values of hJxi,hJyi\nandhJziover all eigenfunctions, multiplied by gJ, and\nweighted by Boltzmann statistics evaluated at T = 10 K.\nThe total Sm11 and Sm12 moments for the di\u000berent\n\feld directions are shown in Fig. 5 as a stereographic\nprojection. A strong Ising like anisotropy was found jjy\nfor Sm11 andjjxfor Sm12. The easy axis rotates be-\ntween x and y for Sm3+sites in accordance with the\n42screw symmetry that relates the two sites. We note\nthat this perpendicular alignment of Sm11 and Sm12 mo-\nments parallel to yandx, respectively, is described by\nthe B1 symmetry adapted mode associated with the (a,a)\norder parameter direction of \u0000+\n5(see Table III). Within\nthis symmetry, the Sm moments only couple to the Mn\ncollinear ferrimagnetic structure if the Mn moments lie\njj[x;x;0]. Hence, not only can the Sm ions impose an ab-\nplane magnetic anisotropy, they can also determine the\ndirection of the magnetization in the plane (as shown in\nFig. 2b).\nTABLE II. Non-zero crystal electric \feld parameters for\nSm3+in Sm 2MnMnMn 2Ti2O12evaluated by the point charge\nmodel, and given in units \u0016eV.\nIon Frac. coords. B0\n2B2\n2B0\n4B2\n4B4\n4\nSm11 0.25, 0.25, 0.209 2430 5204 -11.65 40.94 41.85\nSm12 0.25, 0.25, 0.709 2430 -5204 -11.65 -40.94 41.85\nC. Percolation Calculations\nFirst, we consider the percolation of long-range\nmagnetic order via B-B exchange alone. In6\nFIG. 5. The magnetic susceptibility of Sm11 and Sm12\nplotted as a stereographic projection over a full hemisphere\nunder an applied \feld of 1T.\nSm2MnMnMn 4{xTixO12B-B exchange pathways span\na three dimensional framework. Each B site has six\nnearest-neighbour B sites, such that for every non mag-\nnetic Ti4+ion substituted onto this sublattice, six near-\nest neighbour magnetic B-B exchange pathways are re-\nmoved. The site-percolation threshold (the probablility,\np, of Mn B site occupation required to establish a per-\ncolating cluster that spans the entire lattice between two\nopposing faces) for a simple cubic lattice representative\nof the B-B framework is 31.16% [27]. Hence, B-B ex-\nchange is expected to play no role in establishing long-\nrange magnetic order in the x= 3 sample, for which the\nB site Mn occupation is 25%.\nTo explore the origin of long range magnetic order in\nthex= 3 sample we calculated the percolation thresh-\nold if magnetic order were to percolate via both A-B and\nB-B exchange. Two interpenetrating simple cubic lat-\ntices were used to represent the A 2A0A00B4O12unit cell,\nwhere the Mn A0andA00sites were set to be fully oc-\ncupied, and the Sm A-sites were set to be unoccupied to\nexclude interactions with the RE ions. Mn ions were ran-\ndomly placed on the B sites according to a probability of\noccupation, p. For a given p, the sites were sorted into\nclusters, where a cluster is composed of a continuous net-\nwork of nearest neighbour Mn ions, separated from other\nclusters by Ti ions. The percolation threshold was eval-\nuated as the minimum value of p for which there exists\na cluster that percolates through the entire lattice along\nthe c-axis.\nWe found that in this case the percolation threshold\nwas reduced from 31.16% to 12.4(1)%, as shown in Fig. 6,\ndemonstrating that A-B exchange is crucial to mediating\nlong range magnetic order in the x= 3 sample.\nWe note that A-A exchange could also perco-\nlate long range magnetic order, however measure-\nments of (NaDy)Mn 2Ti4O12demonstrated that it does\nso at temperatures of \u001812 K [16] which is signi\f-\ncantly below the ordering temperatures of all measured\nSm2MnMnMn 4{xTixO12samples.\nFIG. 6. A three dimensional illustration of a cubic lattice,\nspanning a hundred A 2A0A00B4O12unit cells in each of the\nthree Cartesian directions and showing only clusters of B-\nsite ions when magnetic order is percolated by both A-B and\nB-B exchange, given for nine di\u000berent values of the lattice\noccupation, p. In each \fgure, only voxels that intersect the\nsurface are shown, and a quarter of the cubic lattice has been\nremoved to reveal the interior. The largest Mn cluster is given\nin red, all other sites occupied by Mn are given in white and\nsites occupied by Ti4+are in grey.\nIV. DISCUSSION\nDiluting a magnetic sublattice with non magnetic\nions is predicted to decrease the magnetic ordering\ntemperature [20, 28], and at the mean \feld level\none expects an approximately linear dependence in\nthe transition temperature on magnetic ion concentra-\ntion [29{31]. Remarkably, the ordering temperature\nof the Sm 2MnMnMn 4{xTixO12solid solution does not\nsmoothly decrease upon increasing x(Figure 7a), which\nindicates a signi\fcant departure from mean \feld physics\nin this system [32] that is also re\rected in the moment\nmagnitudes. In insulating materials the magnetic mo-\nment of a given ion should reach its full ghJzi\u0016Bvalue\nin the ground state. The ground state magnetic mo-\nments re\fned for Sm 2MnMnMn 4{xTixO12are compared\nto their theoretical values in Table I, and plotted in terms\nof a relative reduction factor in Figure 7b. A consid-\nerable departure from the full moment is observed for\nall Sm 2MnMnMn 4{xTixO12samples, and the relative re-\nduction in moment magnitude (Figure 7b) correlates well\nwith the non-linear variation observed in Tc(Figure 7a).\nTaken together, these results demonstrate the presence\nof strong spin \ructuations and/or static disorder in the\nx= 1 andx= 3 samples, which are weakened in the\nx= 2 sample.\nIdeally we would be able to perform the same analysis\non ax= 0 sample, which has a fully magnetic B-site sub-7\nlattice, and a x= 4 sample, which has a non-magnetic\nB-site sublattice. However, the former can only be grown\nin very small quantities [15], and the latter requires the\nA-site Mn ions to adopt an unfeasible +1 oxidation state.\nInstead, we have used the moment magnitudes and or-\ndering temperature for Tm 2MnMnMn 4O12, as represen-\ntative of the x= 0 sample, and the ordering tempera-\nture and moment magnitude of (NaDy)MnMnTi 4O12as\nrepresentative of the x= 4 sample, plotted in Fig. 7\nas un\flled circles. Importantly, these data further sup-\nport the correlation between the reduction in the relative\nmoment magnitudes and the ordering temperature - an\ne\u000bect that appears to be prevalent in the broader family\nof materials.\nFIG. 7. The dependence of (a) the ordering temperature\nTcand (b) the reduction in magnetic moment relative to\nthe fullghJzi\u0016Bground state value, on the concentration of\nnon-magnetic Ti4+(labelled by x). The data for the x =\n0 and x = 4 samples are from Tm 2MnMnMn 4O12[17] and\n(NaDy)MnMnTi 4O12[16]. The grey arrow in (a) is used to\nillustrate the trend of the mean \feld dependence of the or-\ndering temperature with x.\nIn Sm 2MnMnMn 4{xTixO12static disorder of the mag-\nnetic structure may originate in at least one of two forms\nof crystallographic disorder inherent to these compounds;\ni) mixed cation occupation and ii) displacements of Mn1\nions above and below their coordination plane. All nu-\nclear neutron di\u000braction peaks were found to be sharp\nand limited by the instrumental resolution, indicating\nthat either form of crystallographic disorder statistically\ne\u000bected every unit cell equally, giving a crystal structure\nthat is well correlated on average. The magnetic di\u000brac-\ntion peaks had widths similar to those of the nuclear\npeaks, indicating that the magnetic structures were also\nwell correlated on average, such that any static disorder\nof the magnetic structure must similarly a\u000bect all unit\ncells equally (i.e. local inhomogeneities are not preva-\nlent).\nBy comparison, spin \ructuations can coexist with well\ncorrelated magnetic order [33], and have been shown to\nreduce moment magnitudes even at low temperature [34].The prevalence of spin \ructuations is emerging in the\nwider family of A-site columnar ordered quadruple per-\novskites that support competing exchange pathways. For\nexample, in R 2CuMnMn 4O12(R = Y or Dy), B site\nMn3+moments appear to saturate at3\n4of their theoreti-\ncal value, and only on cooling through a low temperature\nphase transition, proposed to originate in the softening of\nlow energy magnons, do they recover their full moment\n[18].\nV. CONCLUSIONS\nAll three magnetically dilute x= 1,x= 2 andx= 3\nSm2MnMnMn 4{xTixO12samples adopt long range mag-\nnetically ordered phases below 27 K, 62 K and 34 K,\nrespectively. The Mn ions were empirically found to\nadopt a collinear ferrimagnetic structure upto a direc-\ntion in the abplane, and we propose that the moment\ndirection within the plane is likely determined by f-dex-\nchange interactions between the Mn and Sm sublattices.\nWe show that the introduction of 50% magnetic dilution\nonto the B-site sublattice gives rise to a ferrimagnetic\ncompensation point and magnetization reversal. Percola-\ntion calculations demonstrated that long range magnetic\norder in the x= 3 sample can only occur if it percolates\nvia both A-B and B-B exchange, hence demonstrating\nthe importance of A-B exchange in the A-site colum-\nnar ordered quadruple perovskite manganites. Finally we\nshowed that the unusual variation in transition tempera-\nture that occurred upon magnetic dilution was re\rected\nin the reduction of ground state magnetic moments ob-\nserved on all Mn sites and in all samples. Together, these\nresults suggest the presence of spin \ructuations and or\ndisorder leading to a departure from mean \feld physics.\nIn future studies it would be interesting to perform inelas-\ntic neutron scattering experiments to measure the Sm3+\nCEF energy levels and further re\fne our model of the\nmagnetic anisotropy, as well as probe the presence of spin\n\ructuations at the lowest measured temperatures.\nVI. ACKNOWLEDGEMENTS\nR. D. J. acknowledges \fnancial support from the\nRoyal Society. K. Y. and A. A. B. acknowledge JSPS\nKAKENHI Grant No. JP20H05276, a research grant\n(40-37) from Nippon Sheet Glass Foundation for Ma-\nterials Science and Engineering, and Innovative Sci-\nence and Technology Initiative for Security (Grant No.\nJPJ004596) from Acquisition, Technology, and Logistics\nAgency (ATLA), Japan.\n[1] S. Jin, T. H. Tiefel, M. McCormack, R. Fastnacht,\nR. Ramesh, and L. Chen, Science 264, 413 (1994).[2] E. O. Wollan and W. C. Koehler, Phys. Rev. 100, 545\n(1955).8\n[3] J.-S. Zhou and J. B. Goodenough, Phys. Rev. Lett. 96,\n247202 (2006).\n[4] P. G. Radaelli, D. E. Cox, M. Marezio, and S.-W.\nCheong, Phys. Rev. B 55, 3015 (1997).\n[5] J. B. Goodenough, Phys. Rev. 100, 564 (1955).\n[6] A. Mu~ noz, M. T. Cas\u0013 ais, J. A. Alonso, M. J. Mart\u0013 \u0010nez-\nLope, J. L. Mart\u0013 \u0010nez, and M. T. Fern\u0013 andez-D\u0013 \u0010az, Inor-\nganic Chemistry 40, 1020 (2001).\n[7] A. M. Glazer, Acta Crystallographica Section B 28, 3384\n(1972).\n[8] A. A. Belik, Y. S. 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O'Kee\u000be, Acta Crys-\ntallographica Section B 47, 192 (1991),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1107/S0108768190011041.\nAppendix A: Irreducible representations and their\nsymmetry adapted basis functions\nWe give the basis functions of the \u0000+\n5irreducible rep-\nresentation used to describe the magnetic structures of\nSm2MnMnMn 4{xTixO12for theA,A0andA00, and B\nsites in Tables III, IV and V respectively.\nAppendix B: Crystal structure parameters of\nSm2MnMnMn 4{xTixO12\nWe give the crystal structure parameters of\nSm2MnMnMn 4{xTixO12forx= 1,x= 2 and\nx= 3 in Tables VI, VII and VIII respectively.9\nTABLE III. The \u0000+\n5symmetry adapted basis functions given for the Asite Wycko\u000b position in Sm 2MnMnMn 4{xTixO12, listed\nseparately for the (a,0), (a,a) and (a,b) order parameter directions. Moment components of symmetry-equivalent ions have\nthe same magnitude. Note that the modes and irreps are listed according to the labelling scheme adopted in the isotropy\nsoftware suite [22].\nO. P. Mode 0.25, 0.25,z 0.75, 0.75,\u0000z 0.25, 0.25,z+1\n20.75, 0.75,\u0000z+1\n2\n(a,0) B2(a) [1, 0, 0] [1, 0, 0] [0, 0, 0] [0, 0, 0]\nB1(a) [0, 0, 0] [0, 0, 0] [1, 0, 0] [1, 0, 0]\n(a,a) B2(a) [1, 0, 0] [1, 0, 0] [0, 1, 0] [0, 1, 0]\nB1(a) [0, 1, 0] [0, 1, 0] [1, 0, 0] [1, 0, 0]\n(a,b) B2(a) [1, 0, 0] [1, 0, 0] [0, 0, 0] [0, 0, 0]\nB2(b) [0, 0, 0] [0, 0, 0] [0, 1, 0] [0, 1, 0]\nB1(a) [0, 0, 0] [0, 0, 0] [1, 0, 0] [1, 0, 0]\nB1(b) [0, 1, 0] [0, 1, 0] [0, 0, 0] [0, 0, 0]\nTABLE IV. The \u0000+\n5symmetry adapted basis functions given for the A0andA00site Wycko\u000b positions in\nSm2MnMnMn 4{xTixO12, listed separately for the (a,0), (a,a) and (a,b) order parameter directions. Here we use the Wycko\u000b\nposition 2a of the A0site, such that it is constrained to the centre of the square planar coordination. Moment components\nof symmetry-equivalent ions have the same magnitude. Note that the modes and irreps are listed according to the labelling\nscheme adopted in the isotropy software suite [22].\nO. P. Mode 0.75, 0.25, 0.75 0.25, 0.75, 0.25\n(a,0) E(a) [1,0,0] [1,0,0]\n(a,a) E(a) [1,1,0] [1,1,0]\n(a,b) E(a) [1,0,0] [1,0,0]\nE(b) [0,1,0] [0,1,0]\nTABLE V. The \u0000+\n5symmetry adapted basis functions given for the B site Wycko\u000b position in Sm 2MnMnMn 4{xTixO12, listed\nseparately for the (a,0), (a,a) and (a,b) order parameter directions. Moment components of symmetry-equivalent ions have\nthe same magnitude. Note that the modes and irreps are listed according to the labelling scheme adopted in the isotropy\nsoftware suite [22].\nO. P. Mode 0.0, 0.0, 0.0 0.5, 0.0, 0.0 0.0, 0.5, 0.0 0.5, 0.5, 0.0 0.0, 0.0, 0.5 0.5, 0.0, 0.5 0.0, 0.5, 0.5 0.5, 0.5, 0.5\n(a,0) Ag 1(a) [1, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg2(a) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 1, 0] [0, -1, 0] [0, -1, 0] [0, 1, 0]\nAg3(a) [0, 1, 0] [0, -1, 0] [0, -1, 0] [0, 1, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg4(a) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0]\nAg5(a) [0, 0, 1] [0, 0, -1] [0, 0, 1] [0, 0, -1] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg6(a) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 1] [0, 0, -1] [0, 0, 1] [0, 0, -1]\n(a,a) Ag 1(a) [1, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0] [0, 1, 0] [0, 1, 0] [0, 1, 0] [0, 1, 0]\nAg2(a) [1, 0, 0] [-1, 0, 0] [-1, 0, 0] [1, 0, 0] [0, 1, 0] [0, -1, 0] [0, -1, 0] [0, 1, 0]\nAg3(a) [0, 1, 0] [0, -1, 0] [0, -1, 0] [0, 1, 0] [1, 0, 0] [-1, 0, 0] [-1, 0, 0] [1, 0, 0]\nAg4(a) [0, 1, 0] [0, 1, 0] [0, 1, 0] [0, 1, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0]\nAg5(a) [0, 0, 1] [0, 0, -1] [0, 0, 1] [0, 0, -1] [0, 0, 1] [0, 0, 1] [0, 0, -1] [0, 0, -1]\nAg6(a) [0, 0, 1] [0, 0, 1] [0, 0, -1] [0, 0, -1] [0, 0, 1] [0, 0, -1] [0, 0, 1] [0, 0, -1]\n(a,b) Ag 1(a) [1, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg1(b) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 1, 0] [0, 1, 0] [0, 1, 0] [0, 1, 0]\nAg2(a) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 1, 0] [0, -1, 0] [0, -1, 0] [0, 1, 0]\nAg2(b) [1, 0, 0] [-1, 0, 0] [-1, 0, 0] [1, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg3(a) [0, 1, 0] [0, -1, 0] [0, -1, 0] [0, 1, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg3(b) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [1, 0, 0] [-1, 0, 0] [-1, 0, 0] [1, 0, 0]\nAg4(a) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0]\nAg4(b) [0, 1, 0] [0, 1, 0] [0, 1, 0] [0, 1, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg5(a) [0, 0, 1] [0, 0, -1] [0, 0, 1] [0, 0, -1] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]\nAg5(b) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 1] [0, 0, 1] [0, 0, -1] [0, 0, -1]\nAg6(a) [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 1] [0, 0, -1] [0, 0, 1] [0, 0, -1]\nAg6(b) [0, 0, 1] [0, 0, 1] [0, 0, -1] [0, 0, -1] [0, 0, 0] [0, 0, 0] [0, 0, 0] [0, 0, 0]10\nTABLE VI. Crystal structure parameters of Sm 2Mn(Mn 0.84Sm0.16)Mn 3TiO 12(Z= 2, space group P42=nmc ) re\fned at 80\nK. The lattice parameters were determined to be a= 7:4068(1) \u0017A andc= 7:9275(2) \u0017A. Excellent reliability parameters of\nR= 2:72%,wR= 2:90%,RBragg = 4:06% were achieved in the re\fnement. Bond valence sums (BVS) were calculated using\nthe bond valence parameters, R0(Sm3+) = 2.01(1), R0(Mn3+) = 1.76(1), R0(Mn2+) = 1.79(1), R0(Ti4+) = 1.82(1), B = 0.37\n[35]. N.B the Wycko\u000b position of the Mn1 ions is listed for the split site with an occupation of 0.5, and the high symmetry\nposition is given in square brackets with an occupation of 1.\nIon Site Sym. x y z U iso(\u0017A2) B.v.s. ( jej)\nSm1 4d 2mm: 0.25 0.25 0.211(3) 0.010(9) 3.2(2)\nMn1 4c[2a] 2mm: 0.75 0.25 0.720(1) 0.009(4) 2.34(4)\nMn2 2b \u00164m2 0.75 0.25 0.250 0.008(5) 1.74(4)\nMn3/Ti1 8e \u00161 0 0 0 0.023(2) 3.7(1)/3.23(9)\nO1 8g :m: 0.25 0.0597(4) -0.0370(3) 0.028(2) -\nO2 8g :m: 0.25 0.5356(4) 0.5770(4) 0.028(2) -\nO3 8f :: 2 0.4355(2) -0.4355(2) 0.25 0.029(2) -\nTABLE VII. Crystal structure parameters of Sm 2Mn(Mn 0.93Sm0.07)Mn 2Ti2O12(Z= 2, space group P42=nmc ) re\fned at 90\nK. The lattice parameters were determined to be a= 7:5506(1) \u0017A andc= 7:7273(3) \u0017A. Excellent reliability parameters of\nR= 2:71%,wR= 2:87%,RBragg = 5:06% were achieved in the re\fnement. Bond valence sums (BVS) were calculated using\nthe bond valence parameters, R0(Sm3+) = 2.01(1), R0(Mn2+) = 1.79(1), R0(Mn3+) = 1.76(1), R0(Ti4+) = 1.82(1), B = 0.37\n[35]. N.B the Wycko\u000b position of the Mn1 ions is listed for the split site with an occupation of 0.5, and the high symmetry\nposition is given in square brackets with an occupation of 1.\nIon Site Sym. x y z U iso(\u0017A2) B.v.s. ( jej)\nSm1 4d 2mm: 0.25 0.25 0.209(3) 0.022(4) 3.2(2)\nMn1 4c[2a] 2mm: 0.75 0.25 0.789(1) 0.022(4) 1.78(5)\nMn2 2b \u00164m2 0.75 0.25 0.250 0.022(4) 1.71(4)\nMn3/Ti1 8e \u00161 0 0 0 0.014(2) 3.7(1)/ 3.22(9)\nO1 8g :m: 0.25 0.0611(4) -0.0343(3) 0.019(2) -\nO2 8g :m: 0.25 0.5406(4) 0.5705(4) 0.022(2) -\nO3 8f :: 2 0.4436(2) -0.4436(2) 0.25 0.023(2) -\nTABLE VIII. Crystal structure parameters of Sm 2MnMnMnTi 3O12(Z= 2, space group P42=nmc ) re\fned at 80 K. The lattice\nparameters were determined to be a= 7:6290(1) \u0017A andc= 7:6932(3) \u0017A. Excellent reliability parameters of R= 2:03%,\nwR= 2:20%,RBragg = 4:21% were achieved in the re\fnement. Bond valence sums (BVS) were calculated using the bond\nvalence parameters, R0(Sm3+) = 2.01(1), R0(Mn2+) = 1.79(1), R0(Mn3+) = 1.76(1), R0(Ti4+) = 1.82(1), B = 0.37 [35]. N.B\nthe Wycko\u000b position of the Mn1 ions is listed for the split site with an occupation of 0.5, and the high symmetry position is\ngiven in brackets with an occupation of 1.\nIon Site Sym. x y z U iso(\u0017A2) B.v.s. ( jej)\nSm1 4d 2mm: 0.25 0.25 0.214(3) 0.023(3) 3.0(2)\nMn1 4c[2a] 2mm: 0.75 0.25 0.797(1) 0.023(3) 1.58(2)\nMn2 2b \u00164m2 0.75 0.25 0.250 0.023(3) 1.81(2)\nMn3/Ti1 8e \u00161 0 0 0 0.022(2) 3.6(1)/3.36(9)\nO1 8g :m: 0.25 0.0641(4) -0.0339(4) 0.027(2) -\nO2 8g :m: 0.25 0.5441(4) 0.5719(3) 0.024(2) -\nO3 8f :: 2 0.4448(3) -0.4448(3) 0.25 0.028(2) -" }, { "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). Simulation details for different \nsamples are summarized in Supplementary Note 4 and used parameters are listed in Supplementary Table \nS3. \nData Availability. \nThe datasets generated during and/or analyzed during the current study are available in Supplementary \nMaterials and also available from the corresponding author s on reasonable request. \n 11 \nReference s \n1. Torrejon, J. et al. Neuromorphic computing with nanoscale spintronic oscillators. 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J., Wisser, J. J., Emori, S. & Suzuki, Y. Ultra -low magnetic damping in epitaxial \nLi0.5Fe2.5O4thin films. Appl. Phys. Lett. 117, 092407 (2020). \n50. Zheng, X. Y. et al. Spin -orbit torque switching in ultra -thin epitaxial spinel ferrite heterostrucutres \nwith low critical current density. Under Rev. (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": "2006.00777v1.Enhancement_in_Thermally_Generated_Spin_Voltage_at_Pd_NiFe__2_O__4__Interfaces_by_the_Growth_on_Lattice_Matched_Substrates.pdf", "content": "Enhancement in Thermally Generated Spin Voltage at Pd/NiFe 2O4\nInterfaces by the Growth on Lattice-Matched Substrates\nA. Rastogi,1,\u0003Z. Li,1, 2,\u0003A. V . Singh,1S. Regmi,1, 2T. Peters,3P. Bougiatioti,3D. Carsten né\nMeier,3J. B. Mohammadi,1, 2B. Khodadadi,1, 2T. Mewes,1, 2R. Mishra,4J. Gazquez,5A.\nY . Borisevich,6Z. Galazka,7R. Uecker,7G. Reiss,3T. Kuschel,3, †and A. Gupta1, ‡\n1Center for Materials for Information Technology,\nThe University of Alabama, Tuscaloosa, Alabama 35487, USA\n2Department of Physics & Astronomy,\nThe University of Alabama, Tuscaloosa, Alabama 35487, USA\n3Center for Spinelectronic Materials and Devices,\nDepartment of Physics, Bielefeld University,\nUniversitätsstraße 25, 33615 Bielefeld, Germany\n4Department of Mechanical Engineering and Materials Science,\nand Institute of Materials Science and Engineering,\nWashington University in St. Louis, St. Louis, Missouri 63130, USA\n5Institut de Ciència de Materials de Barcelona,\nCampus de la UAB, 08193, Bellaterra, Spain\n6Materials Science and Technology Division,\nOak Ridge National Laboratory, TN 37831, USA\n7Leibniz-Institut für Kristallzüchtung, Max-Born-Str. 2, 12489 Berlin, Germany\n(Dated: June 2, 2020)\n1arXiv:2006.00777v1 [cond-mat.mtrl-sci] 1 Jun 2020Abstract\nEfficient spin injection from epitaxial ferrimagnetic NiFe 2O4thin films into a Pd layer is demonstrated\nvia spin Seebeck effect measurements in the longitudinal geometry. The NiFe 2O4films (60 nm to 1 µm)\nare grown by pulsed laser deposition on isostructural spinel MgAl 2O4, MgGa 2O4, and CoGa 2O4substrates\nwith lattice mismatch varying between 3.2% and 0.2%. For the thinner films ( \u0014330 nm), an increase in the\nspin Seebeck voltage is observed with decreasing lattice mismatch, which correlates well with a decrease in\nthe Gilbert damping parameter as determined from ferromagnetic resonance measurements. High resolution\ntransmission electron microscopy studies indicate substantial decrease of antiphase boundary and interface\ndefects that cause strain-relaxation, i.e., misfit dislocations, in the films with decreasing lattice mismatch.\nThis highlights the importance of reducing structural defects in spinel ferrites for efficient spin injection.\nIt is further shown that angle-dependent spin Seebeck effect measurements provide a qualitative method to\nprobe for in-plane magnetic anisotropies present in the films.\n2I. INTRODUCTION\nEfficient conversion of heat to electric energy in thermo-electric materials is an active field of\nresearch. Recent studies on the interaction between electron spin and heat flow have created a new\narea of research in spintronics that is commonly referred to as spin caloritronics [1–7]. The spin\nSeebeck effect (SSE), which involves generation of spin current through heat flow, is one of the\nmost promising phenomena in the emerging field of spin caloritronics. One approach to efficiently\ngenerate spin current is the implementation of a temperature gradient across a magnetic thin film\nthat is perpendicular to the magnetization [8–12]. The spin current is generated parallel to the\ntemperature gradient via the so-called longitudinal spin Seebeck effect (LSSE). It can be injected\ninto a normal metal (Pt, Pd, Au, etc.) electrode and converted into a charge current due to the\ninverse spin Hall effect (ISHE) [13–15]. The electric field ( EISHE) generated by the spin current\nin a normal metal is described by the relationship [11]\nEISHE =qSHrJs\u0002\u001b; (1)\nwhere qSHis the spin-Hall angle, ris the electrical resistivity of the normal metal, Jsis spin\ncurrent density, and \u001bis spin-polarization vector, collinear with the magnetization M.\nUsing magnetic insulators as a source of spin current has advantages over magnetic metals\nbecause unintended effects such as the anomalous Nernst effect can be neglected due to the absence\nof conduction electrons [16]. In magnetic insulators, magnons, the quanta of spin waves, are the\ncarriers of the generated spin current.\nYttrium iron garnet (YIG) is the most widely studied insulating ferrimagnetic material for LSSE\nexperiments because of its low magnetic coercivity and an extremely low Gilbert damping [17].\nNickel ferrite (NiFe 2O4, NFO) is also a promising candidate for high frequency applications as\nits saturation magnetization is much higher than YIG [18]. The use of NFO has further advan-\ntages such as the tuning of electrical properties by temperature [12] or by oxygen content [19, 20].\nHowever, so far there have been only few reports of LSSE using NFO thin films. The NFO films\nused in previous studies were deposited by either chemical vapor deposition method [12, 21–23]\nor reactive co-sputtering [19, 24–26] on MgAl 2O4substrate that has a large lattice mismatch of\n\u00183.2%, resulting in the formation of antiphase boundaries (APBs) and interface defects, such as\nmisfit dislocations [27], which limits their usability for device applications. Nevertheless, recent\nnonlocal magnon spin transport experiments [26] based on the SSE in sputter-deposited NFO on\n3MgAl 2O4show that the magnon spin diffusion length is \u00183 µm, which is in the same range as for\nYIG [28]. We have recently shown that with appropriate choice of substrates and growth condi-\ntions, NFO thin films can exhibit a saturation magnetization as high as its bulk value, with damping\nconstant and coercivity values comparable to that of YIG [29]. Moreover, Pd is another metal with\nhigh spin Hall angle besides Pt, which shows strong potential for spintronics applications [30–32].\nIn this work, we report on a systematic study of enhancement in the thermally generated ISHE\nvoltage for Pd/NFO films on different (001)-oriented isostructural spinel substrates: MgAl 2O4\n(MAO), MgGa 2O4(MGO), and CoGa 2O4(CGO) with decreasing lattice mismatch of \u00183.2%,\n0.8%, and 0.2% with NFO, respectively. The overall microstructure and the interface between\nthe films and substrates have been investigated by high resolution scanning transmission electron\nmicroscopy (STEM), which shows a substantial decrease of APBs and misfit dislocations with de-\ncreasing lattice mismatch. For thinner films ( \u0014330 nm), the obtained LSSE results correlate well\nwith the damping parameters as determined by ferromagnetic resonance measurements (FMR).\nThe thermally generated spin voltage signal increases with decreasing lattice mismatch, whereas\nthe damping parameter decreases.\nII. EXPERIMENTAL\n1. Sample preparation and characterization\nHigh-quality epitaxial NFO thin films were deposited using pulsed laser deposition followed\nby in-situ Pd deposition by DC sputtering. For NFO film deposition we used a laser fluence of \u00181\nJ/cm2in an oxygen environment with a background pressure of 1.3 Pa. The temperature of the\nsubstrates was kept constant at 700\u000eC during film growth. We used three different (001)-oriented\nspinel substrates, namely MAO, MGO, and CGO. The MAO substrates were purchased commer-\ncially (CrysTec GmbH), while the MGO and CGO substrates were prepared from high quality\nsingle crystals, which were grown at the Leibniz Institute for Crystal Growth [33] and then cut\nand polished by CrysTec GmbH, Berlin, Germany. We investigated films with thicknesses ranging\nfrom 60 nm to 1 µm deposited on substrates with a size of 3 \u00025 mm2. For LSSE measurements,\nthe deposition of NFO film was followed by in-situ deposition of a 5 nm thick Pd layer by DC\nsputtering at 0.7 Pa Argon pressure and 20 W power.\nThe films were structurally characterized using a Philips X0Pert X-ray diffractometer. High\n4resolution STEM and high-angle annular dark field imaging (along the [001] direction) were car-\nried out on some of the samples in an aberration-corrected Nion UltraSTEMTM200 microscope\noperating at 200 kV . Two different imaging modes were used, the high-angle annular dark field\n(HAADF) and the low-angle annular dark field (LAADF) imaging modes. The HAADF imaging\nmode gives rise to the so-called Z-contrast, and it was acquired using an annular detector with a\nhigh inner collection angle [34]. On the other hand, the LAADF imaging mode is achieved us-\ning an annular detector with a smaller inner collection angle, which allows collection of electrons\nscattered by the strained regions giving rise to different angular distributions of the annular dark\nfield signal, thus causing extra contrast [35].\nThe films were magnetically characterized using vibrating sample magnetometry (VSM) in a\nPPMSrDynaCoolTMsystem (Quantum Design). Room temperature broadband ferromagnetic\nresonance (FMR) measurements were performed using a coplanar waveguide to determine the\neffective Gilbert damping parameter of two films deposited on MGO and CGO substrates. The\nFMR measurements were carried out in the in-plane geometry, i.e. with the quasi-static magnetic\nfield applied in the plane of the film.\n2. Measurement setup for spin Seebeck effect\n-100001000-40-2002040-\n100001000-200002000VLSSE (µV)N\nFO/CGO(b)N\nFO/MGO(c)H\n (Oe)NFO/MAO(d)1st meas2\nnd measE\nlectrical insulatorF\nilm + PtS\nubstrateT\nhermo coupleCu-block at T + ΔTC\nu-block at T(a)\nFIG. 1. (a) A schematic of the measurement setup for temperature gradient method. Panel (b), (c), and (d)\nshow the reproducibility of the VISHE signal using SiC spacer. The measurements were done on 330 nm\nthick NFO films deposited on CGO, MGO, and MAO. Black lines show the results of first measurement,\nwhile the red lines show a repeat measurement after remounting the same sample.\nWe used two methods to normalize the VISHE signal, namely by heat flux and by thermal gradi-\n5ent. For the heat flux setup in Bielefeld, we used a calibrated Peltier element clamped between the\nsample and one of the copper blocks to detect the heat flux as described in Ref. [19, 36–38]. The\nheat flux method developed by Sola et al. helps to improve the reproducibility when determined\nLSSE coefficients are compared between different setups as well as when remounting samples\nin the same setup [37, 38]. In the thermal gradient setups in Alabama and Bielefeld, only the\nsample was sandwiched between two copper blocks (Fig. 1(a)). The Cu-blocks were retained in\ngood thermal contact with Peltier elements for cooling and heating. A thermally conducting and\nelectrically insulating 250 µm thick SiC spacer was used between the top Pd layer and the upper\ncopper block. For a comparison of different spacers, see Fig. S2 in SI. For all measurements the\nspacing between the voltage probes ( w) was kept constant, w\u00194.8 mm. The temperature of the\nlower block was fixed at a base temperature T(room temperature, if not stated otherwise), while\nthe temperature of the upper block was varied ( T+DT) to obtain the desired temperature dif-\nference across the sample. A K-type thermocouple was used to measure the temperature at each\nCu-block. For angular-dependent measurements the sample was rotated in-plane with a manual\nstage. A helium-based closed cycle refrigerator was used to carry out the low-temperature mea-\nsurements. To check the reproducibility of the voltage signal in our setup, we have remeasured\nthe same sample repeatedly after remounting, but the voltage signal remains unaffected within the\nerror limit as shown in Fig. 1(b), 1(c) and 1(d). The primary source of error in our measurements\nis the distance between the electrical contact ( \u00184%). Since the voltage signal remains essentially\nunchanged after repeated measurements, we can compare results from the same setup using the\ntemperature difference method in addition to the heat flux technique. We used the temperature\ngradient method for the LSSE measurements of magnetic field and temperature variations. For a\nquantitative comparison of substrate effects in LSSE, we used the heat flux method.\nIII. RESULTS AND DISCUSSION\nA. Structural characterization of NFO films on different substrates\nAll three (001)-oriented substrates, namely CGO, MGO, and MAO, impose a compressive\nstrain on the NFO film, and hence the lattice parameter elongates in the out-of-plane direction. It\ncan be seen in Fig. 2(a) that the film peak position shifts to lower values of 2 q(2qbulk=43:33\u000e)\nwith increasing lattice mismatch. Omega scans in Fig. 2(b) indicate that epitaxial quality of the\n6-1.0-0.50.00.51.00.00.30.60.91.20.00.51.04\n243444546Intensity (arb. units) \n NFO/CGO \n NFO/MGO \n NFO/MAO∗∗2\nθ (deg)(a)∗ ( c)( b) \nNFO/CGO \nNFO/MGO \nNFO/MAOIntensity (arb. units)Δ\nω (deg)FWHM~0\n.05o0\n.04o0\n.68oNFO/CGON\nFO/MGON\nFO/MAORt\nfilm (µm)FIG. 2. (a) Standard q-2qdiffraction patterns around the (004) reflections of the substrates (*) and films\n(#), respectively. (b) The full width at half maximum (FWHM) of omega scans of NFO films grown on the\ndifferent substrates. (c) Variation of the strain parameter Rwith the thickness of NFO films deposited on\nthe different substrates. The films on MAO substrate are closer to being relaxed.\nfilms on MGO and CGO are significantly better than the film on MAO. We also performed off-\naxis XRD scans on few films deposited on three different substrates and calculated the strain in\nthe films which can be quantified by the parameter R= (af\u0000as)=(ab\u0000as), with a f, ab, and a sas\nin-plane lattice parameters of the NFO thin film (measured), NFO bulk (literature), and substrate\n(single crystal), respectively. Therefore, R=1 for a fully relaxed film and R=0 for a fully strained\nfilm. As shown in Fig. 2(c), the films on CGO and MGO substrates have significantly lower values\nof R than those on MAO and are not fully relaxed with even the thickest films remaining strained.\nWe also used X-ray reflectivity technique to determine the Pd layer thickness, which is essentially\nthe same (\u00185:0\u00060:4 nm) for all the samples.\nLow magnification STEM images of the films grown on CGO, MGO and MAO substrates\nare shown in Fig. 3(a), 3(b) and 3(c), respectively. Two different imaging modes are used, the\nhigh-angle annular dark field (HAADF) and the low-angle annular dark field (LAADF) imaging\nmodes. While the films grown on CGO and MGO exhibit sharp interfaces and are essentially\nfree of APBs and other defects, the film grown on MAO presents many structural defects. These\ndefects are clearly seen using the LAADF imaging mode, as shown in the lower panel of Fig. 3(c).\nThe bright contrast of this image stems from crystal defects, mainly APBs, with a crystallographic\ntranslation of 1/4a [001]. In high resolution STEM Z-contrast images they appear with a clear\ndistinct contrast, as highlighted in Figs. 3(d) and 3(e). These defects also appear as a superstruc-\nture in fast Fourier transform (FFT) patterns, as shown in the FFT of an NFO film grown on MAO\n7FIG. 3. (a) and (b) show low magnification HAADF Z-contrast images of NFO films ( \u001860 nm) grown on\nCGO and MGO substrates, respectively. The inset shows a characteristic FFT pattern from the Z-contrast\nimage of (b). (c) Upper and lower panels show low magnification Z-contrast and LAADF images of NFO\nfilms grown on MAO, respectively. The inset shows an FFT of the NFO film grown on MAO substrate. The\nyellow circles highlight the extra reflections arising due to the APBs. High resolution Z-contrast images of\nan APB within the bulk of the film (d) and close to the interface (e) of the NFO films grown on MAO. APBs\nare highlighted in yellow.\nsubstrate (inset Fig. 3(c)). The extra reflections marked with yellow circles in the FFT are due\nto the presence of APBs, and are absent in the FFT patterns of NFO films grown on CGO and\nMGO substrates (inset Fig. 3(b)). The LAADF image also shows that the defects are unevenly\ndistributed, as the density of APBs decreases near the surface of the film. Our previous studies\nhave established that even relatively thick NFO films (100-450 nm) grown on CGO and MGO sub-\nstrates remain essentially fully strained while those on MAO are partially relaxed with formation\nof misfit dislocations [29]. This is consistent with the X-ray diffraction results. The films on MAO\n8also show presence of threading dislocation and dark diffused contrast areas, likely from A-site\ncation vacancies [27, 29]. The APBs and other structural defects are known to cause a reduction\nof saturation magnetization and increase in the FMR linewidth of the thin films compared to their\nbulk values [39]. However, their effect on the spin transport properties and especially on ISHE\nremain unknown.\nB. Spin Seebeck effect measurements of NFO films on different substrates\n/s40/s99/s41\n/s40/s100/s41 /s40/s101/s41/s32/s78/s70/s79/s47/s67/s71/s79\n/s32/s78/s70/s79/s47/s77/s71/s79\n/s32/s78/s70/s79/s47/s77/s65 /s79\n/s32/s32/s83 /s105 /s110/s101/s32/s102/s105 /s116/s115\n/s32/s32/s86\n/s115/s97/s116/s32/s40 /s109 /s86/s41\n/s113 /s32/s40/s100/s101/s103/s41/s40/s98/s41\n/s40/s102/s41\n/s32 /s32 /s86\n/s115/s97/s116/s32/s40/s84/s41/s47 /s86\n/s115/s97/s116/s32/s40/s50/s57/s48/s32/s75/s41\n/s84 /s32/s40/s75/s41\nFIG. 4. (a) A schematic of the LSSE measurement geometry. A temperature gradient is created along\nthe ˆz-direction; the magnetic field is applied in the sample plane with an angle qwith respect to the ˆ x-\ndirection, and the voltage is measured in the same plane. (b) and (c) Results of COMSOL Multiphysicsr\nsimulation for the generation of the temperature gradient across the sample and the heating components.\nThe film thickness is \u0018330 nm. (d) LSSE measurements for Pd/NFO/MGO (001) with voltage contacts\nlocated along the ˆ x-direction and the external magnetic field applied in-plane at various angles with respect\nto the voltage contacts. A complete angular dependence of the saturation voltage for all the three films (330\nnm) is plotted in panel (e); the dotted lines are sine function fits. (f) Variation of normalized voltage signal\n(Vsat(T)=Vsat(290K)) with various base temperature ( T) for 330 nm thick films on different substrates. The\nsolid line is fit to the NFO/CGO data using the Eq. 1 described in the text.\n9In Fig. 4(a), we show a schematic of the measurement geometry for LSSE. The temperature\ngradient across the film and the substrate has been simulated using the heat transfer module and\nfinite element method available in COMSOL Multiphysicsr. The simulation for a 330 nm NFO\nfilm on MGO substrate is shown in Fig. 4(b) and 4(c). The temperature gradient ( DTf) is in the\nrange of tens of mK/µm when a temperature difference of \u001820 K is applied across the Cu-blocks.\nFig. 4(b) shows the cross-sectional view of temperature distribution across the stack. For clarity,\nthe temperature profile across the film is shown in the zoomed cross-section image (Fig. 4(c)).\nFurther details are provided in the Supplementary Information (SI) section I. We find that the\ntemperature difference across the film scales with the temperature difference across the Cu-blocks\nand is essentially independent of the choice of the substrate (MAO, MGO and CGO) because of\ntheir similar thermal characteristics (see Table I in SI).\nIn our geometry we are sensitive to the ˆ x-component of EISHE (with VISHE =EISHE\u0001w,wis\nthe distance between voltage probes), and according to Eq. 1 we are sensitive to the ˆ y-component\nof\u001band thusM. The background signal is subtracted from data presented here (for raw data\nplease see Fig. S3 in SI). In Fig. 4(d), we display the result for a 330 nm thick NFO/MGO film\nwith angular variation from 0\u000eto 90\u000ebetween the voltage contacts and the magnetic field. We\nobserve that upon reversing the direction of DTz, the voltage signal is also reversed, which is a\ncharacteristic behavior of VISHE induced by LSSE (see Fig. S4 in SI). To obtain the maximum\nLSSE voltage the external magnetic field is applied along the ˆ y-direction ( q= 90\u000e) to saturate\nthe magnetization aligned along this direction. This leads to a maximum Vsatof about\u001827\u0016V .\nAfter magnetic field reversal the magnetization direction is changed into the opposite direction\nandVsatof\u0018-27\u0016V is obtained. During the magnetic field reversal process (Fig. 4(d)), VLSSE acts\nin correspondence with the magnetization and correlates well with the VSM measurement (see\nFig. S5 in SI). When qis reduced, Vsatdecreases and follows the cross product of Eq. 1, which\nis evident from Fig. 4(e). During the magnetic field reversal process, the magnetization rotates\ntowards one of the magnetic easy axes aligned along 45\u000ein [011] directions [40]. For angles\nq>45\u000eupon reducing the magnetic field, the projection of the magnetization onto the ˆ y-direction\nalso decreases which results in a decrease of VLSSE. For q=45\u000e,VLSSE signal shows the maximum\nsquareness while the magnetization lies along one of the magnetic easy axes. For angles q<45\u000e,\nVLSSE signal increases when the magnetic field is decreased due to the increase of the projection\nof the magnetization in the ˆ y-direction. Across H= 0 Oe,Mlies along one of the magnetic easy\naxes and results in nearly the same remanent voltage signal for all angles q(see Fig. S6 in SI). For\n10q\u001430\u000ewe observe a slight difference around H= 0 Oe, which can be attributed due to the multi\ndomain formation during the reversal process [41]. We have additionally performed magnetic and\nLSSE measurements on an NFO film grown on (011)-oriented MGO substrate (see Section III in\nSI or the results in Ref. [42]).\nThe temperature dependence (from 30 K to 300 K) of normalized LSSE voltage for 330 nm\nthick films is shown in Fig. 4(f), with the DT across the stack being fixed at 20 K. This obser-\nvation is similar to the results for CVD deposited NFO films on MAO substrate [12]. In some\nprevious reports the temperature dependence of the ISHE signal has been discussed for magnetic\ninsulator/normal metal hybrid structures [43–45]. A T3=2variation at low temperatures has been\ntheoretically proposed, [43, 44] while a (Tc\u0000T)3(Tcis the Curie temperature) dependence at\nhigher temperatures has been experimentally observed for Pt/YIG [45]. We combined these two\ntemperature regimes and fitted our data (Fig. 4(f)) with VLSSEµT3=2(Tc\u0000T)3. This relationship\nfits well with our observation in the measured temperature range. From the fits, the Tcis found to\nbe in the range 700 K – 800 K, which is close to NFO bulk value ( \u0018850 K) [46].\nIn Figure 5(a), we plot the magnetic field variation of the SSE voltage of 600 nm thick films on\nMGO (circle), CGO (square), and MAO (triangle) obtained using the heat flux method. Here the\nVLSSE signal of the film on MGO is larger as compared to CGO. On the other hand, SSE voltage\nacross the films on MAO substrate remains lowest in both the measurement techniques which is\nevident from Fig. 4(e) and Fig. 5(a). In Fig. 5(b), we show the variation of normalized saturation\nelectric field ( Ec) generated in the Pd-layer as a function of the lattice mismatch with the three\nsubstrates. We observe a weak SSE signal for films grown on MAO substrate and larger SSE\nresponse for NFO films on MGO and CGO. Overall, it is noted that irrespective of the thickness\nof the films, MAO substrate shows the lowest LSSE signal. This signifies the importance of lattice\nmismatch in enhancing theEc\nFqsignal. In conjunction with the STEM results we conclude that\nAPBs and other structural defects present in the films are one of the reasons associated with the\nchange in the LSSE signal. The values ofEc\nFqfor Pd/NFO/MGO are in a similar range ( \u001830\nnm/A) as recently reported for Pt/YIG/GGG thin film heterostructure ( \u001840 nm/A) [48]. Here,\nthe effect of the lower spin Hall angle of Pd [30–32] is probably compensated by a larger SSE\nin the NFO. If directly compared to sputter-deposited Pt/NFO bilayers ( \u0018100 nm/A) [19], the\neffect of less efficient spin-to-charge conversion in Pd becomes obvious. However, complete SSE\nthickness dependencies are quite rare in the literature, especially when normalized to the heat flux,\nand should be investigated in future studies. Finally, since the spin Seebeck resistivityEc\nFqis only\n110.00 .51 .00.00.10.20.3-\n20000 2 000-202(c) \nH (Oe)(a)VLSSE (µV) NFO/CGO \nNFO/MGO \nNFO/MAO \n fitEc/Φq (´10-4 mV W-1m)t\nfilm (µm)01 2 3 0.10.2(b) \n \n Lattice mismatch (%) ~60 nm \n~600 nmEc/Φq (´10-4 mV W-1m)-50005 00 \nH (Oe)FIG. 5. (a) Magnetic field dependence of spin voltage signal generated at constant heat flux ( \u001824 kW/m2)\nfor three Pd/NFO films (600 nm) on different substrates. (b) Influence of lattice mismatch of NFO film with\nCGO, MGO, and MAO substrates for two different film thicknesses. (c) Thickness dependence for films\non different substrates (symbols), while solid lines are fit to the equation ELSSEµ1\u0000exp(\u0000tfilm=x),xis\nthe magnon propagation length [47]. All the measurements were performed using heat flux method at room\ntemperature.\nan effective SSE coefficient that still includes the heat conductivity of the NFO, any thickness-\ndependent change of the NFO heat conductivity can affect the thickness-dependence of the heat-\nflux-normalized SSE voltages. The study of this dependence will be part of future work.\nOur measurements show an increase in normalized saturation electric field ( E) generated across\nthe Pd-layer by the heat flux ( Fq) with increasing film thickness (Fig. 5(c)), which can be explained\nbased on characteristic magnon propagation length ( x), i.e. the number of magnons reaching the\nPd/ferrimagnetic interface increases with thickness [47–49] and contributes to the voltage signal.\nThe value of xdeduced from the fits is in the range of 400 to 700 nm. This is lower than the\nrecently reported value obtained from nonlocal magnon spin transport measurements in sputter\ndeposited NFO films ( \u00183 µm) [26, 50]. Such discrepancy between local LSSE and nonlocal\n12magnon spin transport results has also been observed for YIG [28, 47] and can be explained by the\ndifferent nature of the experiments. While magnons with different propagation lengths can reach\nthe Pd interface in the local experiment, the magnons with small diffusion length cannot make it to\nthe Pd detector in the nonlocal geometry. Upon further increasing the film thickness, we observe\nan increase/saturation in the voltage signal. Significant scatter is observed in the data points for\nthe films on CGO. This might be due to differences in the quality of CGO substrates which is also\nreflected in FMR measurements, where we find scatter in the FMR linewidth (see Fig. S8 in SI).\nC. FMR measurements of NFO films on different substrates\n0246810120102030407\n.58.08.50\n10203040020406080(\nb)P\nd/NFO/MGOP\nd/NFO/CGOf (GHz)H\nres (kOe)(a)F\nMR signalH\nres (kOe)Δ\nH (Oe)f\n (GHz)\nFIG. 6. Broadband FMR measurement results of 330 nm thick NFO films with Pd top layer deposited on\nCGO and MGO substrates. (a) Microwave frequency vs. resonance field data (symbols) are fitted to Kittel’s\nequation (solid lines). The inset shows a typical FMR spectra of the film at 30 GHz frequency. (b) The\ndependence of FMR line width signal with resonance frequency (solid data points) and solid lines are fits to\ncalculate the effective Gilbert damping and inhomogeneous linewidth broadening.\nIn addition to the LSSE measurements, we compared the dynamical properties of NFO\nfilms (330 nm) deposited on CGO and MGO substrates by FMR in the field along the in-\nplane hard axis geometry. From the measurements, we have estimated the effective magneti-\nzation ( Meff) and gyromagnetic ratio ( g0) from fitting the frequency ( f) versus resonance field\n(Hres) data (Fig. 6(a)) to the Kittel equation in the in-plane configuration using equation f=\ng0p\n(Hres+H4)\u0001(Hres+H4+4pMeff)with H4being the four-fold in-plane anisotropy. The FMR\n13linewidth ( DH) vs. frequency ( f) data is then used to calculate the effective Gilbert damping pa-\nrameter ( aeff) and inhomogeneous linewidth broadening ( DH0) from DH=DH0+2aeffp\n3g0f[51, 52].\nLinewidth vs. frequency data is shown in Fig. 6(b) for 330 nm thick NFO films on MGO and CGO\nsubstrates with Pd top layer. The estimated value of the aeffof the NFO/MGO and NFO/CGO\nthin films without Pd top layer are determined to be (22 \u00060.9)\u000210\u00004and (1.3\u00060.9)\u000210\u00004,\nrespectively (see Fig. S9 in SI). After Pd deposition we find an increase in the damping constant\nand the effective Gilbert damping parameter. The values derived from the fitting of the data in\nFig. 6(b) are (2.9\u00060.1)\u000210\u00003and (2.3\u00060.1)\u000210\u00003for the Pd/NFO/MGO and Pd/NFO/CGO\nfilms, respectively. The difference in the Gilbert damping parameter of the two films capped\nwith and without Pd can be directly related to the spin current density in the two films which can\nexplain the significant differences in the SSE voltage for the two films [53]. It should be noted\nthat the value of aefffor NFO/CGO ( (1.3 \u00060.9)\u000210\u00004) is comparable to the best reported value\nof YIG/GGG thin films ( \u00187.35\u000210\u00005) [17], suggesting that NFO/CGO is a promising candidate\nfor spin caloritronics and spin transport in general.\nIV . CONCLUSIONS\nIn summary, thin films of NFO exhibit improved structural, interfacial and dynamical proper-\nties when grown on lattice-matched substrates. The results clearly show that higher LSSE signal\nis obtained for the most closely lattice-matched substrates (MGO, CGO). We find that the thinner\nfilms on the CGO substrate provide larger LSSE voltage signal as compared to the other het-\nerostructures and this is consistent with the lower value of the effective Gilbert damping of these\nfilms. COMSOL Multiphysicsrsimulation indicates that the temperature gradient across the film\nis in the range of tens of mK/µm. The measurements using the heat flux method also affirm the im-\nportance of lattice matching to enhance spin generated voltage signal that also correlates with the\nFMR results. Apart from this, LSSE measurements provide a qualitative method to study in-plane\nmagnetic anisotropies by varying the angle between the external magnetic field and the direction\nof the contacts for the detection of the ISHE voltage. Improved quality NFO thin films exhibit\ndamping parameter comparable to that of YIG/GGG, which makes them attractive for spintronics\nas well as microwave applications. Further improvement of the LSSE efficiency of NFO could be\nreached by choosing substrates with even less lattice mismatch compared to MGO and CGO.\n14ACKNOWLEDGMENTS\nThe work at The University of Alabama was supported by NSF ECCS Grant No. 1509875 and\nNSF CAREER Award No. 0952929. The work at ORNL (AYB) was supported by the Materials\nScience and Engineering Division of the Office of Science of the US DOE. RM was supported by\na startup funding from Washington University. JG was supported by the Ramón y Cajal program\n(RyC-2012-11709). 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Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA \n3. Research Center for Functional Materials, National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan \n4. International Center for Materials Nanoarchitectonics, Nationa l Institute for Materials Science, 1-1 \nNamiki, Tsukuba 305 -0044, Japan \n \n \n \n \n \n \n \n \n \n \n \n*wu.2314@osu.edu \n \n2 \n Abstract: \nEngineering magnetic anisotropy in a ferro - or ferrimagnetic (FM) thin film is crucial in \nspintronic device. One way to modify the magnetic anisotropy is through the surface of the FM \nthin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced \nby interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y 3Fe5O12 \n(YIG), and the low -symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, \nWTe 2. At the same time, we also observed an enhancement in Gilbert damping in the WTe 2 \ncovered YIG area. Both the magnitude of interface -induced PMA and the Gilb ert damping \nenhancement have no observable WTe 2 thickness dependence down to single quadruple -layer, \nindicating that the interfacial interaction plays a critical role. The ability of WTe 2 to enhance the \nPMA in FM thin film, combined with its previously rep orted capability to generate out -of-plane \ndamping like spin torque, makes it desirable for magnetic memory applications. \n \nKey words: perpendicular magnetic anisotropy, magnetic resonance force microscope, transition metal \ndichalcogenides, ferrimagnetic i nsulator \n \n3 \n Perpendicular magnetic anisotropy (PMA) in a ferromagnetic thin film is of great interest in \nspintronics research and application s. Ferromagnetic nano -element s with PMA overcome their shape \nanisotropy , greatly ease the memory cell size reduction and improves memory retention . These \nexceptional properties, improving the performance of magnetic devices , make PMA highly desirable for \nmagnetic memory application s. PMA becomes even more important in the recent development of solid \nstate magnetic random -access memory (MRAM) since it allows MRAM to have lower switching current \nand faster switching speed compare d to in-plane magnetized materials 1, 2. \nMagnetic storage devices generally rely on metallic magnetic material s due to their robust \nelectrical response . Interfacial magnetic anisotropy plays a critical role in generat ing PMA in metallic \nferromagnet s. When interfacing with a nonmagnetic material (NM), electron orbital angular momentum \nof the magnetic ions at the ferromagnet surface will be modified, in some cases enabling strong covalent \nbonding, resulting in distinct magnetic properties compare d to the single layer 3-6. However, spintronics \ndevices made of metallic magnetic materials are inherently energy consumptive due to resistive losses. \nRecently, complex oxide ferro - or ferrimagnet insulator s (FMI) have attracted substantial interest due to \ntheir ability to transport spin excitation s with low dissipation 7. Inducing PMA in FMIs naturally \nbecomes an important topic both for scientific and technologic al reasons . Several successful route s to \nachiev ing PMA in FMIs ha ve been reported using bulk intrinsic anisotropy 8 or lattice strain 9-12. But in \nmost experiments, the sign of the resulting interfacial anisotropy in FMI/NM heterostructures is such as \nto enhance the easy-plane anisotropy 13-15. Only one recent experiment has shown the possibility of \ngenerating interfacial PMA, and this was attributed to topological surface states 16. Nevertheless, t hese \nresults demonstrate the possibility of controlling magnetic anisotropy through interfac ial interaction s in \n \n4 \n FMI/NM heterostructures . Here, we report a study on YIG/WTe 2/hBN heterostructures, which shows \nthat when interfacing with a low symmetry nonmagnetic van der Waals material , WTe 2, an additional \ninterfac e-induced PMA (iPMA) term emerges in the magnetic anisotropy of the YIG thin film . The \nabsence of topological surface states at room temperature in WTe 2 17, 18 forces us to seek an explanation \nfor our observation of enhanced PMA that is distinct from that proposed for top ological insulator/YIG \nbilayers 16. We therefore turn to an analysis of the broken symmetries in WTe 2. We point out that low \nsymmetry WTe 2 has recently shown the capability of generating both in -plane and out -of-plane spin \npolarization in charge -spin conversion experiments 19-22. It also enables field-free switching of PMA \nmagnet ic material , which ease s the application of PMA material s in MRAM application s 23-25. \nFerrimagnetic insulator YIG is of significant research interest in spintronics due to its \nexceptionally low Gilbert damping 26, which describes the relaxation rate of magnetization precession . \nAnd 1T’-WTe 2 is a semi -metallic transition metal dichalcogenide (TMD) layered material with strong \nSOC 27, 28. The crystal structure of 1T’-WTe 2 lacks twofold rotational symmetry about the c -axis (Fig. \n1a). The only symmetry in the WTe 2 crystal lattice ab plane is the mirror symmetry about the bc plane \n29. This u nique symmetry breaking allows out-of-plane damping -like torque to be generated 30, 31, \nenabling efficient switching of the out-of-plane magnetization of the adjacent magnetic material 24. \nA 20nm thick YIG thin film used in our experiment is epitaxially grown on (111) -oriented \nGd3Ga5O12 (GGG) substrate by off -axis sputtering 32. WTe 2 flakes are then mechanically exfoliated \nfrom a flux-grown crystal, and dry transferred on to the clean top YIG surface without touching any \nother substances. This whole process is carried out in an Ar -filled glove box with <0.1 ppm of H 2O and \n \n5 \n O2 to protect the flakes from degradation and ensure the clean liness of the YIG/WTe 2 interface. We \nemploy hexagonal boron nitride ( hBN ) encapsulation to protect the WTe 2 flakes from oxidation after \nbeing removed from the glove box. We make two samples and focus on the data taken from sample 1 in \nthe main text. The raw data taken from sample 2 can be found in Supporting Information Fig. S 2. \n \nFig. 1 Crystal structure of WTe 2 and sample schematic . a) Crystal lattice structure of WTe 2 viewed \nfrom the top along the c-axis and looking from the side along the a-axis. The black dashed box in the \nside view indicates a monolayer of WTe 2. b) Schematic of the ferromagnetic resonance force \nmicroscope. RF excitation is generated by a stripline underneath the sample , where the hBN \nencapsulation is not shown . The region of localized mode is shown as a yellow dot adjacent to the WTe 2 \nflake, and the probe magnetic moment is shown as a yellow arrow on the particle. The cantilever \noscillation is detected by a fiber laser interferometer. \nFigure 2a shows an optical image of the sample 1. Due to the small lateral size of the exfoliate d \nWTe 2 and hBN flake s having length scale s of 10 μm, we use a home -built ferromagnetic resonance \nforce microscope (FMRFM) to measure the local ferromagnetic resonance (FMR) signal. FMRFM is a \nsensitive technique to detect the local magnetic properties with high spatial and spectral resolution 33. In \nour FMRFM, the external magnetic field 𝐻⃗⃗ ext is aligned perpendicular to the sample plane. The \ncantilever tip holds a high coercivity SmCo 5 magnetic particle , whose moment is magnetized in the \n \n \n6 \n direction opposite to 𝐻⃗⃗ ext to create a magnetic field well . The field well supports a set of localized \nstanding spin wave modes (LMs). During the measurement, we excite spin precession uniformly by a \nstripline underneath the sample at a fixed RF frequency (2 GHz) and sweep the magnetic field. The \nresonance of each LM generate s a stray field, whic h can then be detected by the SmCo 5 magnetic \nparticle attached on the cantilever through their magnetic dipole -dipole interaction (Fig. 1b). During the \nmeasurement, we keep the probe -to-sample separation around 4 μm. The operation of FMRFM is \ndescribed in detail in Ref s. 34-36. For reference, w e separate a region of YIG that does not contain \nWTe 2/hBN heterostructures and measur e its Gilbert damping using broadband FMR. To eliminate two -\nmagnon scattering, w e perform broadband FMR in the out -of-plane field geometry. The FMR linewidth \nas a function of frequency measured on bare YIG (sample 1) shows a linear dependence (Fig. 2b), from \nwhich we can extract the Gilbert damping of bare YIG 𝛼YIG=1.05×10−3. We also confirm that the \nWTe 2 used in the experiment is indeed the 1T’ phase through polarized Raman measurements. The \npolarization angle dependence of the Raman peak at 212 cm-1 (spectrum is shown in Fig. S4) exhibits \nminimum intensity when the excitation laser polarization is along the crystallographic a axis of WTe 2 37 \nas shown by the polar plot in Fig. 2c and Raman intensity plot in Fig. 2d . \nWe find t he position of the YIG/WTe 2/hBN heterostructure with the assist ance of magnetic \nalignment markers (Fig. 2a) . Figure 2e shows t wo raw FMRFM scans taken in the region of YIG/hBN \nand YIG/WTe 2/hBN , indicated by the blue and the red dot in Fig. 2a, respectively , which reveals the \nchange in FMRFM spectra at two different location s. Here we focus on the 𝑛=1 LM because it has the \nmode radius of around 1 μm and gives the highest spatial resolution. Higher order modes have \nincreasing mode radius and therefore, detect less local ized magnetic properties. This is the reason why \n \n7 \n the quasi -uniform mode at ~ 3325 Oe does not show obvious change in resonance field or signal \namplitude. We further take a line scan across the edge of WTe 2 flake (Fig. 2 f) to resolve the spatial \nevolution of FMRFM spectra . The line scan in Fig. 2f (along the dashed line shown in Fig. 2a) shows \nthree main features : first, the magnitude of the LM resonance signal is reduced in the YIG/WTe 2/hBN \nregion compare d to the YIG/hBN region; second, the LM resonance field for all LMs is decreased by \n~40 Oe in the YIG/WTe 2/hBN region; third , the LMs show complex splitting and crossing when the \nprobe is close to the boundary (−5 μm<𝑋<10 μm). \n \nFig. 2 FMRFM and Raman measurement data . a) An optical micrograph of the YIG/WTe 2/hBN \nheterostructure under study . WTe 2 crystal a and b axis are labeled. b) Broadband FMR measurement of \nthe frequency -dependent linewidth of the YIG thin film. The measurement is done on the same piece of \nYIG used to make sample shown in Fig. 1b. c) Polar plot of the 21 2 cm-1 peak Raman intensity. Angle \ndenote s the relative angle between the measurement laser polarization and the WTe 2 a axis. d) 2D \nintensity plot showing Raman peak intensities versus polarization angle . e) FMRFM spectra, one over \nthe YIG/hBN region (blue line) and the second over the YIG/WTe 2/hBN region (red line); these \nlocations are indicated by the blue and red dot s in Fig. 2a respectively . f) Color plot of field -dependence \n \n \n8 \n FMRFM scans as a function of position along the trace indicated by the black dashed line in Fig. 2a. A \nconstant background is subtracted to show only the signal from the several LM resonance s. \nIn the following, we will explain the origin of the three observed effects using spin pumping and \nmagnetic anisotropy. The first effect , i.e. signal reduction in the YIG/WTe 2/hBN area relative to the \nYIG/hBN area, is the result of enhanced relaxation due to spin pumping from YIG to WTe 2 38. The 𝑛=\n1 LM resonance signal amplitude ∆𝐴 is inversely proportional to the square of Gilbert damping , 𝛼2. We \ndetermine the Gilbert damping constant 𝛼 for YIG/WTe 2/hBN using 𝛼YIG/WTe2/hBN=\n𝛼YIG/hBN×√∆𝐴YIG/hBN∆𝐴YIG/WTe2/hBN ⁄ (see Ref. 39), where 𝛼YIG/hBN is assumed to be the same as \n𝛼YIG=1.05×10−3 due to the low SOC and insulating character of hBN . The second effect is the \ndecrease of 𝑛=1 LM resonance field 𝐻r,1 by ~40 Oe . And the third effect is splitting and crossing of \ncomplex modes in the region −5 μm<𝑋<10 μm. The second and the third effects are due to an \nabrupt change of uniaxial anisotropy across the boundary separating the YIG/WTe 2/hBN and YIG/hBN \nregions 15. Here, the uniaxial anisotropy refers to the magnetic free energy depends on the angle between \nmagnetization and sample normal ℱu=−𝐾u𝒎z2, where 𝒎z is the component of magnetization unit \nvector in the direction normal to sample plane and 𝐾u is the uniaxial anisotropy constant specific to \nsample and depends on the total interaction in the sample . When 𝐾u is positive, ℱu is called to be of \nPMA type, on the other hand, if 𝐾u is negative, ℱu is called to be of easy -plane type. This uniaxial \nanisotropy will lead to an effective uniaxial magnetic field 𝑯u=−𝜕ℱu𝜕𝑴⁄ , where 𝑴 is the \nmagnetization . And therefore, a change in 𝐾u can modify the resonance field in a FMR measurement . In \nFMRFM spatial mapping, a n abrupt change in 𝐾u spatially could disturb the LM and lead to mode \nsplitting and crossing as described in Ref. 15. Moreover, i n striking contrast to the previously studied \n \n9 \n YIG/Au interface 15, which result s in a 32 Oe increase of 𝐻r,1 due to the enhanced easy -plane \nanisotropy, the observed decrease of 𝐻r,1 indicates that the WTe 2 overlayer induces an iPMA in YIG. \nWe note that the magnitude of the shift in 𝐻r,1 is comparable to the easy -plane anisotropy induced by a \nheavy metal 15, 40 or the iPMA generated by topological surface state 16 on garnet ferrimagnetic material . \nIn order to probe the global effect of a WTe 2 overlayer on YIG, we spatial ly map 𝐻r,1 using the \n𝑛=1 LM. Figure 3a present s an optical image of WTe 2 flakes on a Si/SiO 2 (285nm ) substrate , where \ndifferent c olors of WTe 2 flakes indicat e different WTe 2 thickness es. Figure 3b and 3 c show spatial maps \nof magnetic properties in the region enclosed by the black dashed rectangle in Fig. 3a . We acquire the \nmaps using the procedure described in Ref. 39, i.e., simultaneously measuring spatial variation of the \nmagnetic anisotropy and Gilbert damping using the 𝑛=1 LM resonance field 𝐻r,1 and signal amplitude \n∆𝐴. The entire WTe 2-covered area show s uniformly lower ed 𝐻r,1 and increased Gilbert damping relative \nto the area without WTe 2. In Fig. 3c, despite the not great signal to noise ratio in damping imaging, the re \nis a clear Gilbert damping enhancement in WTe 2-covered area . The averaged Gilbert damping of YIG in \nWTe 2-covered area is 𝛼̅YIG/WTe2/hBN≈1.30×10−3, about 24% higher than 𝛼YIG. We note that due to \nthe slight relative tilting of the scan plane and the sample plane, there is a color shift in Fig. 3b that \nmight conceal the contrast difference in different WTe 2 thickness region . Therefore, to study the WTe 2 \nthickness dependence, we will show fine line scans across edge s of flakes having different WTe 2 \nthickness es. \n \n10 \n \nFig. 3 Two -dimensional FMRFM scan resolving the spatial variation of magnetic anisotropy and \nGilbert damping. a ) Optical micrograph showing the color contrast of different thickness WTe 2 flake s \n(ranging from 4.7 nm to 44.8 nm) on Si/SiO 2(300 nm). Black dashed box outlines the FMRFM scanned \narea for 2D mapping. b) 2D map of the 𝑛=1 LM resonance field . The d ashed line s labeled 1 -4 \ncorrespond to the four line-scans shown in Fig. S1a-S1d. c) 2D mapping of the Gilbert damping \nextracted from the 𝑛=1 LM resonance peak amplitude. \n \n \n \n11 \n Next, we want to understand what gives rise to the PMA in WTe 2/YIG. We rule out the effect \ninduced by a modification of the gyromagnetic ratio by showing the resonance field shift across the \nWTe 2 edge does not depend on RF excitation frequency (See Fig. S3). We also exclude a strain induced \neffect given the absence of an epitaxial relation and the weakness of the van der Waals interaction \nbetween YIG and WTe 2. We further note that we can ignore the role of topological surface states 16 in \nour analysis; they are not relevant for our room temperature experiment since WTe 2 is a topological \nWeyl semimetal only below 100 K 17, 18. \nWe show how an analysis based on symmetry and the nature of the interfacial SOC , generalizing \nthe theory in Ref. 41, gives insight into the PMA observed in our experiment. This will also help us \nunderstand why the easy-axis anisotropy we observe in WTe 2/YIG is so different from the results of \nRef. 13, 15 on YIG interfaces with a dozen different metallic and semiconducting materials, all of which \nexhibit interface -induced easy-plane anisotropy, as is predicted by theory 41. \nYIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via \nantiferromagnetic (AFM) superexchange interactions. We focus on how interfacial SOC impacts AFM \nsuperexchange in YIG and show that it leads to a very specific form of the mag netic anisotropy that is \ngoverned by the direction of the effective B-field (see Supporting Information for a details). \nBefore turning to WTe 2/YIG, it is useful to first consider the simpler case when the only broken \nsymmetry is the mirror plane defined b y the interface. The abrupt change in lattice potential then results \nin an effective electric field that points normal to the interface, which in turn leads to an effective \nmagnetic field in the rest frame of the electron that couples to its spin. Since t he E-field points normal to \n \n12 \n the interfacial plane in which the electron moves, the resulting B-field arising from SOC lies within the \ninterfacial plane. As we show in the SI, this leads to a SOC -induced correction to AFM superexchange \nthat necessarily lead s to an easy-plane anisotropy. \nIn the case of WTe 2/YIG, however, when there are additional broken symmetries. Not only does \nthe interface break inversion symmetry , but the crystal structure of WTe 2 itself breaks in -plane inversion \nsymmetry . The electric field is now no longer normal to the interface, and the effective B-field arising \nfrom SOC necessarily has an out -of-plane component, as shown in Fig S5b in SI. Thus, we see why the \nlower symmetry of WTe 2/YIG can naturally result in an easy-axis or perpendicular magnetic anisotropy \n(PMA); see Supporting Information for details. \nWe note that the lack of two -fold rotational symmetry in the ab plane in WTe 2 that plays a \ncritical role in our understanding of PMA in WTe 2/YIG, has also been pointed out be crucial for the out -\nof-plane damping -like torque in WTe 2/Permalloy30. We note, however, that the out -of-plane damping -\nlike torque necessarily involves current flow in WTe 2, while the PMA is an equilibrium property of the \nsystem independent of current flow. \nWe further demonstrate the interfacial origin of the observed effect by studying the influence of \nWTe 2 thickness. We show four line-scans , labeled in Fig. 3b, across the edges of WTe 2 with different \nthickness es, ranging from 4. 7 nm to 44.8 nm . From these four line -scans, we extract the 𝑛=1 LM \nresonance field 𝐻r,1 and the 𝑛=1 LM resonance signal amplitude ∆𝐴. Figures S1a-d in the Supporting \nInformation show the evolution of 𝐻r,1 and ∆𝐴 along the traces labeled correspondingly . The thickness \nof WTe 2 at each measurement location is later measured using atomic force microscop y. From these \n \n13 \n line-scans , we choose the region s where the probe is far away from the edge of WTe 2 so that the \nmagnetic propert ies are uniform, to obtain spatial average s of 𝐻r,1 and ∆𝐴, which are denoted \n𝐻̅r,1,YIG/hBN and ∆𝐴̅̅̅̅YIG/hBN in the YIG/hBN region , and 𝐻̅r,1,YIG/WTe2/hBN and ∆𝐴̅̅̅̅YIG/WTe2/hBN in the \nYIG/WTe 2/hBN region , respectively . We further extract the 𝑛=1 LM resonance field difference \nbetween two regions using ∆𝐻r,1=𝐻̅r,1,YIG/hBN−𝐻̅r,1,YIG/WTe2/hBN, as well as the Gilbert damping \ndifference using ∆𝛼=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ −1) as a function of the WTe 2 \nthickness . We note that the hBN overlayer does not change the Gilbert damping in YIG . The \nsummarized results containing the data from both sample 1 and sample 2 are shown in Fig s. 4a and 4 b. \nRaw data from sample 2 can be found in Supporting Information Fig. S 2. The thinnest WTe 2 acquired in \nthe experiment is 3.2nm from sample 2, which is approximately the thickness of a quadruple -layer \nWTe 2. \nFigures 4a and 4 b indicate that both ∆𝐻r,1 and ∆𝛼 have almost no WTe 2 thickness dependence . \nThere is a small sample -to-sample variation possibly due to different YIG/WTe 2 interfacial quality . The \nchange of 𝑛=1 LM resonance field, ∆𝐻r,1, is as large as ~38 Oe even when the WTe 2 thickness \napproaches the quadruple -layer thickness . This indicates that the modification of magnetic anisotropy is \ndue to the YIG/WTe 2 interfac ial interaction , with no bulk contribution. For the increase of Gilbert \ndamping ∆𝛼, no obvious thickness dependence is observed when comparing the data from the same \nsample. In sample 2, the Gilbert damping enhancement due to the quadruple -layer WTe 2 has almost the \nsame value as the 50 nm thick WTe 2 flake, indicating that no thickness dependence of spin pumping can \nbe resolved from our measurement. There are two possible interpretations of these results . First, if the \n \n14 \n spin current injected into WTe 2 is mainly relaxed due to spin relaxation in the bulk, then the \nexperimental result is a demonstration of ultra -short spin diffusion length along the c axis38, smaller or \ncomparable to the thinnest WTe 2 flake (3.2 nm), employed in this experiment . It is much smaller than \nthe 8nm spin diffusio n length in the in-plane direction measured using inverse spin Hall effect 22. Note \nthat due to the chang e in mo bility and the metal -insulator transition in few layer WTe 2 when its \nthickness reduces 42, the spin diffusion length approximated here could be inaccurate . Alternatively , it is \npossible that the spin relaxation is primarily due to the interfacial SOC induced by inversion symmetry \nbreaking at the interface and in the WTe 2 crystal lattice. In this case, the Gilbert damping enhancement \nwill have no WTe 2 thickness dependence. \n \n15 \n \nFig. 4 WTe 2 thickness dependence of resonance field and damping enhancement . a) 𝐻r,1 in the \nYIG/hBN and YIG/WTe 2/hBN regions are averaged respectively to get 𝐻̅r,1,YIG and 𝐻̅r,1,YIG/WTe2, and \n∆𝐻r,1=𝐻̅r,1,YIG−𝐻̅r,1,YIG/WTe2. b) ∆𝛼 as a function of WTe 2 thickness , and ∆𝛼=𝛼YIG/WTe2−𝛼YIG \nwhere 𝛼YIG is the Gilbert damping of bare YIG measured using broadband FMR for each sample , and \n𝛼YIG/WTe2=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ ). \nIn conclusion, we have shown that the YIG/WTe 2 interface plays a critical role in both interfacial \nmagnetic anisotropy and spin relaxation , making WTe 2 a promising material in magnetic memory \n \n \n16 \n application s. Combining the iPMA created by WTe 2 with the out-of-plane spin orbit torque generated by \nflowing a charge current along the a axis of WTe 2, one can possibly achieve field -free switching of a \nPMA magnetic cell for magnetic memory application s. It will improv e the scalability , reduc e the power \nconsumption and increas e operation speed of magnetic solid -state devices . Our result reveals new \npossibilities in selecting materials and designing spintronic devices. For example, one can consider other \nmaterials with low lattice symmetry and strong SOC to induce larger PMA type interfacial ani sotropy in \nFMIs. To achieve a fully PMA material, one could utilize thinner FMIs to magnify the role of iPMA. \nMoreover, interfacial SOC also plays an important role in generat ing topologically protected magnetic \ntextures in the FMIs 43. These findings will motivate further research to reveal the fundamental physics \narising at the interface between FMIs and nonmagnetic materials. \n \nData availability: \nThe data generated by the present study are available from the corresponding author on request. \nSupporting Information: \nA description of raw data on WTe 2 thickness dependence, a FMRFM measurement on a second sample, \na FMRFM measurement at different RF frequency, a description of polarized Raman measurement \nresult, and a detailed illustration of impact of broken mirror reflection symmetries on the magnetic \nanisotropy. \nAckno wledgements: \n \n17 \n This work was primarily supported by the Center for Emergent Materials: an NSF MRSEC under award \nnumber DMR -2011876 (GW, NV , YC, SG, FY , MR and PCH) . KW and TT acknowledge support from \nthe Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) \nand JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233). DW, GC, CNL, and MB \nare supported by NSF under award DMR -2004801. We gr atefully acknowledge N. Trivedi for insightful \ndiscussions. Fabrication and some characterization were performed in the Ohio State University \nNanoSystems Laboratory. \n \n \n18 \n Methods: \nSample fabrication \nOur YIG/WTe 2/hBN heterostructure was prepared by means of dry transfer and stacking 44. hBN crystals \nwere mechanically exfoliated under ambient conditions onto SiO 2/Si substrates (285 nm thick SiO 2). 20-\n40 nm thick hBN flakes were identified under an optical microscope and used for the capping lay er for \nthe stack. The hBN was picked up using a polymer -based dry transfer technique and then moved into an \nAr-filled glove box with oxygen and water level below 0.1 ppm. Flux -grown WTe 2 crystals 45 were \nexfoliated inside the glove box and flakes with different thicknesses were optically identified and \nquickly picked up with the capping hBN layer then transferred to the YIG substrate. Finally, we removed \nthe fully encapsulated sample from the glove b ox and performed the e -beam lithography and \nmetallization (Ni/Au) step for alignment in our ferromagnetic resonance force microscope (FMRFM). \n \nPolarized Raman measurement \nPolarized Raman spectra from the WTe 2 sample were collected using 633 nm excitation w avelength in \nan inVia Renishaw Raman microscope. The sample was loaded onto the microscope stage and \npositioned in such a way that the long edge of the flake was aligned parallel to the laser polarization ( θ = \n0°). In this configuration, the incident illu mination is polarized vertically coming out of the laser and is \naligned with the long axis of the WTe 2 flake. The polarization of the incident laser was rotated from 0 to \n360° by 10° increments using a polarization rotator, while an analyzer was set to onl y allow vertically \npolarized light to enter the spectrometer. Raman spectra were collected at each polarization for 3 \nacquisitions with a 20 s time per acquisition. The laser power was set to 0.5 mW at the sample to avoid \nany damage by heating. Followin g spectral collection, the (baseline corrected) integrated intensities \nunder each peak were calculated to make the contour plots and polar plots in Fig. 2c and 2d. \n \nFMRFM measurement and signal fitting \nOur FMRFM perform s local ly measures FMR at room temper ature in vacuum. The cantilever has \nnatural frequency of ~18 KHz, spring constant of 0.2 N/m and Q factor of ~20000, resulting in force \ndetection sensitivity of 10-15 N/Hz1/2. The SmCo 5 magnetic particle attached on the cantilever has a \nmagnetic moment of ~4 nemu. When a LM is on resonance, the local reduction of magnetization in out -\nof-plane direction will generate a stray field, which will couple the altered magnetization to the magnet ic \ntip thus changing the cantilever oscillation amplitude and frequency. 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B 2015, 92, (4), 041104. \n \n \n \n23 \n Enhancing Perpendicular Magnetic Anisotropy in Garnet Ferrimagnet \nby Interfacing with Few-Layer WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nish chhal Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, \nGuixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun Ning Lau1, Fengyuan Yang1, \nMohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA \n3. Research Center for Functional Materials, National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan \n4. International Center for Materials Nanoarchitectonics , National Institute for Materials Science, 1-1 \nNamiki, Tsukuba 305 -0044, Japan \n*wu.2314@osu.edu \n \n \n24 \n FMRFM line-scan across edge of WTe 2 with different thickness \n \nFig. S1 a-d, FMRFM line-scans along the traces 1~4 indicated in Fig. 3b respectively. The gray shaded \narea in four figures are outlining the location of WTe2 flake . 𝐻r,1 and ∆𝐴 at each position are derived by \nfitting the 𝑛=1 LM to a Lorentzian line shape. The thickness of WTe2 flake at each location are \nmeasured by atomic force microscope. \n \n \n \n25 \n FMRFM measurement on Sample 2 \n \nFig. S 2 FMRFM measurement on sample 2. a, The optical picture of the YIG/WTe 2/hBN \nheterostructure. b, 2D mapping of the 𝑛=1 LM resonance field in the black dash line circled area. c, \n2D mapping of the Gilbert damping extracted from 𝑛=1 LM resonance peak amplitude in the black \ndash line circled area. d, FMRFM line scan along the trace indicated by the solid black line in Fig. S 2a. \nA constant background is subtracted to show only the signal from the LMs resonance. e, Fine scan \nzoomed in o n the quadruple layer WTe 2 stripe area \n \n \n \n26 \n FMRFMR measurement at 4 GHz \n \nFig. S 3 FMRFM measurement across WTe 2 edge at 4 GHz. FMRFM line scan is measured at 4 GHz \nacross the WTe 2 flake edge. The shift of the resonance field 𝐻r,1 is 36 Oe, similar to the 𝐻r,1 shift \nmeasured at 2 GHz. This result excludes the possibility that the resonance field shift arises from \nmodification of the gyromagnetic ratio. \n \n \n \n \n27 \n Polarized Raman measurement \n \nFig. S 4 Polarized Raman measurement. As shown by the red curve, the Raman spectrum taken on \nGGG/YIG/WTe 2/hBN heterostructure contains more peaks than WTe 2. The Raman spectrum taken in \nthe GGG/YIG/hBN area identifies the peaks arising from the substrate GGG/YIG or top hBN \nencapsulation layer. By subtracting the Raman spectrum in the GGG/YIG/hBN area, the Raman spectra \nfrom WTe 2 layer are extracted and plotted in Fig. 2d. The black dash line are the markers indicating the \nRaman peaks of WTe 2 \n \n \n28 \n Impact of broken mirror reflection symmetr ies on the magnetic anisotropy \nWe describe theo retical constraints on the interface -induced magnetic anisotropy in the WTe 2/YIG \nbilayer. We first show that symmetry arguments alone do not provide strong constraints on the anisotropy \ntensor, given that we are dealing with an interface between two crystalline materials at an arbitrary \norientation with respect to each other . We then present qualitative arguments, based on the interfacia l spin -\norbit coupling, that give insight into the magnetic anisotropy in WTe 2/YIG. This helps us understand why \nthe easy-axis anisotropy that we observe in WTe 2/YIG differ s from the results of Lee et al. [1] on YIG \ninterfaces with a dozen different metallic and semiconducting materials , all of which exhibit interface -\ninduced easy-plane anisotropy as predicted by theory [2]. \nOn general grounds, the anisotropy (free) energy can be written as \nℱ𝑎𝑛𝑖𝑠= ∑ 𝐾𝑎𝑏𝑎,𝑏 𝑚𝑎 𝑚𝑏, (S1) \nwhere a and b take on values x,y,z. We focus here on the leading term, quadratic in the magnetization, and \nignore higher order anisotropy terms like (mx4+my4+mz4) or (mx2my2+my2mz2+mz 2mx2). The form of \n 𝐾𝑎𝑏= 𝐾𝑏𝑎 is constrained by symmetry. Let us consider three cases , going from the most symmetric to \nthe least . \nCase I: The only broken symmetry is interfacial inversion (z → - z), which is relevant for the \nexperiments of Ref. [1]. The magnetization is an axial vector (or pseudovector) that transforms under \nrotation s like a vector but is unchanged under inversion . Thus (𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧)→ (𝑚𝑥 ,− 𝑚𝑦 ,−𝑚𝑧) under \nreflection in a mirror plane with normal 𝑥̂. Using reflection symmetry in mirror planes normal to 𝑥̂ and to \n \n29 \n 𝑦̂ , we can see that all off -diagonal components of 𝐾𝑎𝑏 vanish. Further, f our-fold rotational symmetry \nabout the 𝑧̂ axis shows that 𝐾𝑥𝑥= 𝐾𝑦𝑦. Using mx2+my2+mz2=1, we write 𝐾𝑥𝑥(m𝐱2+m𝐲2) in terms \nof m𝑧2, and d efining 𝐾𝑢= (𝐾𝑥𝑥− 𝐾𝑧𝑧), we obtain \nℱ𝑎𝑛𝑖𝑠= − 𝐾𝑢 m𝑧2. (S2) \nThis symmetry analysis only constrains the form of the anisotropy energy, but not the sign of 𝐾𝑢. We will \ngive below a simple microscopic argument [2] that shows that 𝐾𝑢<0 (easy plane) for Case I. \nCase II: In addition to broken interfacial inversion (z → - z), let u s also break reflection symmetry \nin the plane normal to 𝑥̂. This would be the case if the crystalline axes of WTe 2 were aligned with YIG. \nThis also breaks four-fold rotational symmetry about 𝑧̂, so that 𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. However, we can still use \nreflection symmetry in the plane normal to 𝑦̂ to conclude that 𝐾𝑥𝑦= 𝐾𝑦𝑧=0. Thus we find that \n K = (𝐾����𝑥0 𝐾𝑥𝑧\n0 𝐾𝑦𝑦0\n 𝐾𝑥𝑧0 𝐾𝑧𝑧) (S3) \nCase III: When the crystalline axes of WTe 2 are not aligned with YIG, which is the experimentally \nrelevant case, all mirror reflection and rotation symmetries are broken. Then there are no symmetr y \nconstrain ts on 𝐾𝑎𝑏 and all six components of this symmetric tensor are in general non -zero. \nLet us now see how, despite the lack of general symmetry -based constrai nts, we can still get some \nqualitative insight about the form of the anisotropy from simple microscopic considerations informed by \nsymmetry. YIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via \n \n30 \n antiferromagnetic (AFM) superexch ange interactions. We thus focus on how interfacial spin -orbit \ncoupling (SOC) impacts AFM superexchange. \nThe broken symmetry at the interface leads to an electric field ℇ=−𝛁𝑉(𝒓), whose direction will \nbe discussed in detail below for three cases. This in turn produces a magnetic field in the rest frame of the \nelectron which underlies SOC. As the electron moves along 𝐫̂ij from site i to j, it experiences an SOC field \nin the direction 𝐝̂ij which is determined by ℇ ×𝐫̂ij . The SOC Hamiltonian is thus given by \n−𝑖𝜆∑𝑐𝐢𝛼†(𝐝̂ij∙𝝈𝛼𝛽)𝑐𝐣𝛃 𝛼𝛽 . Including the effect of this term in addition to the usual hopping t and Hubbard \nU in the standard strong coupling expansion calculation leads to the Hamiltonian \n ℋex=J∑𝐒i∙𝐒j <𝐢,𝐣>+D∑𝐝̂ij∙𝐒i×𝐒j <𝐢,𝐣> +K0∑(𝐝̂ij∙𝐒i)(𝐝̂ij∙𝐒j). <𝐢,𝐣> (S4) \nHere the spin 𝐒i at site i is coupled to its neighbors via the AFM superexchange 𝐽 ~𝑡2\n𝑈 and the \nDzyaloshinskii -Moriya interaction (DMI) 𝐷 ~𝑡𝜆\n𝑈. The K0 term will be the focus of our attention belo w as \nit leads to magnetic anisotropy. We note that the general form of ℋex is in fact substantially independent \n[2] of the microscopic mechanism and very similar results are obtained not only for superexchange but \nalso for Zener double exchange and RKKY interactions. \nCase I: Let us again return to the simplest case with broken interfacial inversion (z → - z). This \nleads to an electric field ℇ=−𝛁𝑉(𝒓) along ẑ , the normal to the interface. The SOC magnetic field \ndirection is then given by 𝐝̂ij= ẑ ×𝐫̂ij ; see Figure S4(a). This is the well -known Rashba SOC at \ninterfaces . We note in passing that 𝐝̂ij is antis ymmetric under the interchange of i and j, and thus leads to \na DMI term where 𝐒i×𝐒j is also antisymmetric. \n \n31 \n \n \nFig. S 5 Symmetry based selection of magneto -crystalline anisotropy. Interfacial SOC originates from \nan effective Electric field ℇ=−𝛁𝑉(𝒓) whose direction is determined by the broken mirror planes in the \nsystem. This electric field leads to spin-orbit coupling (SOC), with the 𝐝̂ij= ℇ̂ ×𝐫̂ij, the direction of the \nSOC magnetic field. Note that the direction of the electron hop 𝐫̂ij lies in the xy plane of the interface. As \nshown in the text 𝐝̂ij controls the interface -induced magnetic anisotropy. (a) When only surface inversion \nis broken, 𝐝𝐢𝐣 is constrained to lie in the int erface and interfacial SOC leads to easy -plane anisotropy. (b) \nIf there are other broken mirror planes, the 𝐝𝐢𝐣 must lie outside the interfacial plane. This can lead to a \nperpendicular magnetic anisotropy in systems like YIG/WTe 2 bilayers. \n \nWe see that in Case I, 𝐝̂ij lies in the plane of the interface, and the third term in eq. (S 4) then takes \nthe form K0∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦) 𝐢 for a square lattice . To make the connection with magnetic anisotropy, \nwe look at a continuum approximation with a s lowly varying magnetization 𝐦(𝐫). We make a Taylor \nexpan sion of 𝐒r in terms of its value at 𝒓, denoted by 𝐦(𝐫), and its spatial derivatives . The exchange and \nDMI terms involve gradients of 𝐦(𝐫), but we focus here on local terms that do not involve derivatives to \n \n \n32 \n understand the magnetic anisotropy . The leading term is + K0(m𝐱2+m𝐲2) which can be rewritten as \n– K0 m𝑧2 using the fact that mx2+my2+mz2=1 at each 𝒓. Thus, we may identify K0 with the anisotropy \nK𝑢 defined in eq. (S2). \nThe microscopic analysis leads to the result K0= − 𝜆2\n𝑈 < 0 and this explains the easy-plane \nanisotropy arising Rashba SOC at the interface . The easy-plane nature of the anisotropy is in fact a general \nfeature of various microscopic models as emphasized in Ref. [2]. We note however that these author s use d \nthe opposite sign convention for anisotropies from the one we use here . The easy plane vs. easy -axis \ncharacter is , of course, independent of sign conventions. The FMR experiments of Ref. [1] have seen the \ninterface -induced easy-plane anisotropy predicted by the theory in a YIG interfaces with several metallic \nand semiconducting materials . \nThe key difference between the YIG/WTe 2 bilayer studied here and systems studied earlier [1] is \nthat WTe 2 has a broken mirror plane (the ac plane ) as shown in Fig. 1(a) of the paper . We now look at the \neffect of this lower symmetry on the microscopic analysis. \nCase II: Let us break reflection symmetry in the plane normal to 𝑥̂ in addition to broken interfacial \ninversion. We choose x̂ parallel to the b axis, ŷ parallel to a, and ẑ parallel to c. Reflection symmetry in \nthe ŷ mirror plane constrains the electric field ℇ =−𝛁𝑉(𝒓) to lie in the xz plane, at an angle 𝜃 from the \nz-axis as shown in Fig. S 5(b). Thus \n𝐝ij=(sin𝜃𝑥̂+cos𝜃𝑧̂)×𝐫̂ij (S5) \n \n33 \n where 𝐫̂ij is a vector in the interface (xy plane ) and 0≤𝜃≤𝜋. Using eq. (S5), we may rewrite the last \nterm in the Hamiltonian (S4) as \n K0 sin2𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑧)\n𝐢+K0 cos2𝜃∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦)\n𝐢\n−K0sin𝜃cos𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑥+S𝐢𝑥𝑆𝐢+𝑦𝑧)\n𝐢 \nAs before, we make a continuum approximation with a smoothly varying 𝐦(𝐫) and focus only on the \nlocal terms, without gradients, to obtain the magnetic anisotropy . We find that the leading order \ncontribution to anisotropy is −K0cos2𝜃m𝑧2+K0sin2𝜃 mzmx. This analysis correctly captures the non -\nzero K𝑥𝑧 expected on general grounds; see eq. (S3). We did not include here , for simplicity, the effects of \nbroken four -fold rotation that would have led to 𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. \nCase III: When we lose all mirror symmetries, the case relevant to the YIG/WTe 2 experiment, the \nelectric field ℇ =−𝛁𝑉(𝒓) will point in a general direction specified by 0≤𝜃≤𝜋 and 0≤𝜑≤2𝜋, and \nthere will be no symmetry constraints on the anisotropy tensor 𝐾𝑎𝑏. \n Let us conclude by highlighting the key qualitative difference between Case I on the one hand and \nCases II and III on the other . In Case I, the only broken symmetry is interfaci al inversion (z → - z). Then \nsymmetry constrains the 𝐝̂ij, the direction of the SOC B-field, to lie in the plane of the interface and this \nleads to easy -plane anisotropy as described above. In Cases II and III, there are other additional broken \nmirror planes, and this leads to the 𝐝̂ij vector being pulled out of the plane of the interface. This \nimmediately leads to the possibility of an easy -axis like character to the anisotropy, although in the general \ncase one has a non -trivial anisotropy tensor 𝐾𝑎𝑏. \n \n34 \n Reference \n[1] Lee, A. J.; Ahmed, A. S.; McCullian, B. A.; Guo, S. D.; Zhu, M. L.; Yu, S. S.; Woodward, P. M.; \nHwang, J.; Hammel, P. C.; Yang, F. Y . Interfacial Rashba -Effect-Induced Anisotropy in Nonmagnetic -\nMaterial -Ferrimagnetic -Insulator Bilayers. Phys. Rev. Lett. 2020, 124, (25), 257202. \n[2] Banerjee, S.; Rowland, J.; Erten, O.; Randeria, M. Enhanced Stability of Skyrmions in Two -\nDimensional Chiral Magnets with Rashba Spin -Orbit Coupling. Physical Review X 2014, 4, (3), 031045. \n \n " }, { "title": "1902.02978v1.Magnetic_Multipoles_in_a_Ruthenate_Ca3Ru2O7.pdf", "content": " \nMagnetic Multipoles in a Ruthenate Ca 3Ru 2O7 \n \nS. W. Lovesey 1,2, D. D. Khalyavin 1, and G. van der Laan 2 \n1 ISIS Facility, STFC, Didcot, Oxfordshire OX11 0QX, UK \n2 Diamond Light Source, Didcot, Oxfordshire OX11 0DE, UK \n \nAbstract Compulsory Dirac multipoles in the bilayer perovsk ite Ca 3Ru 2O7 are absent in \npublished analyses of experimental data. In a first step at correcting knowledge of the magnetic \nstructure, we have analysed existing Bragg diffract ion patterns gathered on samples held well \nbelow the Néel temperature at which A-type antiferr omagnetic order of axial dipoles \nspontaneously develops. Patterns were gathered with neutrons, and linearly polarized x-rays \ntuned in energy to a ruthenium atomic resonance. Ne utron diffraction data contains solid \nevidence of Dirac dipoles (anapoles or toroidal mom ents). No such conclusion is reached with \nexisting x-ray diffraction data, which instead is a mbiguous on the question. To address this \nshortcoming by future experiments, we calculated ad ditional diffraction patterns. Chiral order \nof Dirac multipoles is allowed by magnetic space-gr oup P Cna2 1, and it can be exposed in Bragg \ndiffraction using circularly polarized x-rays. Like wise, a similar experiment can expose a chiral \norder of axial dipoles. A magnetic field applied pa rallel to the b-axis creates a ferrimagnetic \nstructure in which bulk magnetization arises from f ield-induced nonequivalent Ru sites \n(magnetic space-group Pm 'c'21). \n \nI. INTRODUCTION \n \n The laminar perovskite Ca 3Ru 2O7 contains one ruthenium-oxygen bilayer per formula \nunit and crystallizes in an orthorhombic space grou p with a unit cell containing four formula \nunits [1, 2]. While the ruthenate is structurally s imilar to a bilayer cuprate, e.g., YBCO \n(YBa 2Cu 3O6 + x), not all the strange physics of underdoped cuprat es is present [3-6]. It is a low \ncarrier-concentration material on the verge of a me tal-insulator transition, with a colossal \nmagneto-resistance [7, 8]. At a Néel temperature T N ≈ 56 K the material is metallic. On \nreaching 48 K the resistivity along the c-axis abru ptly increases and continues to increase with \ncooling. Resistivity in the a-b plane is not change d so dramatically and becomes only weakly \ndependent on temperature below 48 K, with evidence of a quasi-two-dimensional metallic \nground state below 30 K [2, 9]. \n \n Lattice constants of Ca 3Ru 2O7 jump at the first-order metal-to-nonmetal phase \ntransition at 48 K without a change of the polar sp ace-group symmetry Bb2 1m illustrated in \nFig. 1 [1]. Magnetic reflections present below 48 K in neutron and resonant x-ray Bragg \ndiffraction patterns are consistent with A-type ant iferromagnetic order of axial dipoles and a \npropagation vector along the c-axis [1, 10 11]. The RuO 2 bilayers depicted in Fig. 1 (after \nreference [1]) are ferromagnetically ordered in the a-b plane and are antiferromagnetically \nstacked normal to the plane. \n Dirac multipoles are a compulsory component of a m agnetic structure when a centre of \ninversion symmetry is absent in sites occupied by m agnetic ions. To date, these multipoles \nhave been omitted from all experimental studies of Ca 3Ru 2O7, e.g., Zhu et al . and Sokolov et \nal . [12, 13]. We address this shortcoming with an analysis of published neutron and resonant \nx-ray Bragg diffraction patterns [10, 11], taking a ccount of Dirac multipoles and axial (parity-\neven) multipoles [14]. For Dirac dipoles, also call ed anapoles or toroidal dipoles, a persuasive \ncase can be extracted from a neutron diffraction pa ttern at hand [10]. Resonant x-ray Bragg \ndiffraction data are inconclusive, however. \n \n By way of a potential remedy, we propose addition al experiments that can vastly \nimprove the knowledge of multipoles in the ruthenat e in question. Specifically, we predict that \na handed setting will reveal chiral order among the primary and secondary axial dipoles. \nLikewise, chiral order exists within Dirac multipol es of even rank, e.g., the monopole and \nquadrupoles. Circular polarization in the primary x -ray beam, and collinear Bragg and \npropagation vectors, furnishes a suitable handed se tting. A magnetic field applied parallel to \nthe b-axis creates a ferrimagnetic structure in whi ch bulk magnetization arises from field-\ninduced nonequivalent Ru sites. \n \n The magnetic structure of Ca 3Ru 2O7 precludes a direct observation of Dirac multipoles \nby neutron diffraction accomplished for SmAl 2 that presents a diamond-type structure [15]. Set \nagainst controversial evidence for Dirac multipoles in the pseudo-gap phase of YBCO - the \nvery existence of magnetic Bragg spots in neutron d iffraction patterns is an ongoing debate \n[16-19] - our evidence in favour of anapoles in Ca 3Ru 2O7 is compelling. \n \nII. CRYSTAL AND MAGNETIC STRUCTURES \n \n The parent structure of Ca 3Ru 2O7 is orthorhombic Bb2 1m (#36, polar crystal-class mm2 \n(C 2v ), cell lengths a ≈ 5.3677, b ≈ 5.5356, c ≈ 19.5219 Å at 8 K [1]) that is a B-face centred \ncell with glide symmetry along the b-axis, and the structure is more distorted than the \ncompanion double-layer ruthenate Sr 3Ru 2O7 (Bbcb, #68). Below 48 K, ruthenium axial dipoles \nalign principally along the b-axis derived from mag netic space-group P Cna2 1 (#33.154 [20], \ncrystal class mm21 '). Crystal and magnetic structures are displayed in Fig. 1. Ruthenium ions \noccupy sites 8b with no symmetry and coordinates (1 /4, 0.4008, 1/4). Note that general \ncoordinates (x, y, z) are not fixed by symmetry in a polar crystal, and our use of x = z = 1/4 is \nan approximation, albeit an excellent approximation [1]. A basis (1, 0, 0), (0, 0, 1), (0, −1, 0) \nwith respect to the Bb2 1m paramagnetic space-group defines orthonormal prin cipal axes ( ξ, η, \nζ), with Miller indices h = H o, k = L o, l = − Ko. Magnetic Bragg spots are indexed by Miller \nindices ( h + k) = (H o + Lo) odd, a consequence of anti-translation in the spa ce-group, and (H o \n+ Ko) even. Looking ahead, Section IV. C is devoted to the influence of a magnetic field that \ncreates ferrimagnetic order. With the field applied along the b-axis, purely magnetic reflections \noccur for H o = 0, K o odd and all L o. \n Electronic degrees of freedom present in the grou nd state of a Ru ion are encapsulated \nin multipoles UKQ with rank K and projections Q that obey − K ≤ Q ≤ K. Angular brackets \n... denote the expectation value of the enclosed sphe rical tensor operator, defined for neutron \nand x-ray diffraction in references [21, 22]. The o perator UKQ is time-odd, of course, and its \nparity is labelled σπ = + 1 ( − 1) axial (polar). Hermitian operators yield multip oles that obey \nUKQ* = ( −1) Q UK−Q, and UKQ = UKQ′ + iUKQ′′ is our convention for real and imaginary \nparts. Multipoles are evaluated in principal axes, e.g., the dipole U1η is parallel to the crystal \nc-axis ≡ propagation vector (Cartesian and spherical compon ents of a vector R are related by \nRξ = (R −1 − R+1)/ √2, R η = i(R −1 + R+1)/ √2 and R ζ = R 0). \n Multipoles that are both parity-odd ( σπ = − 1) and time-odd are referred to as Dirac \nmultipoles. Dirac dipoles (anapoles) observed in ma gnetic neutron diffraction are simple \nproducts of spin S or orbital angular momentum L with the electronic position operator n [21, \n22]. For these anapoles we use ΩΩ ΩΩS = S ⤫⤫ ⤫⤫ n and ΩΩ ΩΩL = L ⤫⤫ ⤫⤫ n − n ⤫⤫ ⤫⤫ L, and it is noted that \noperators S and n commute whereas L and n do not commute. A monopole S ·· ·· n is allowed \nin resonant x-ray diffraction [23] but it is forbid den in neutron diffraction [22], while L ·· ·· n = \n0. \n Unit-cell structure factors for Bragg diffraction are derived from ΨKQ = {∑ exp(i d ·· ·· k) \nUKQd}, where k is the Bragg wavevector. Sites 8b, with positions d in a cell, used by Ru 4+ \nions are asymmetric. Symmetry operations in P Cna2 1 lead to the central result, \n ΨKQ = exp[i π(h + l)/2] [exp(i ϕ) + (−1) h + l + Q exp( −iϕ)] (1) \n \n × [UKQ − σπ (−1) h + l + K UK−Q], \n \nwith a spatial phase angle ϕ = (2 πyLo) and y ≈ 0.4008 [1]. Site symmetry adds nothing to \nsymmetry displayed by ΨKQ, in contrast to many materials where site symmetry is hugely \ninfluential [24]. The leading phase factor in (1) f or ΨKQ can be set aside in the calculation of \nBragg intensities, for they depend on the absolute value of the unit-cell structure factor derived \nfrom ΨKQ. \n \n The energy levels the free Ru 4+ 4d4 ion in intermediate coupling are determined by the \n4d-4d Coulomb interaction (with atomic Slater integ rals F 2 = 9.214 eV and F 4 = 6.095 eV) and \nspin-orbit interaction (with parameter ζd = 160 meV = 1290 cm -1). The ground state is 5D (S \n= 2 and L = 2 with cancelling opposite moments) whi ch has a purity of 96%. The remaining \n4% is made up of other SL terms which are mixed in the ground state by the spin-orbit \ninteraction. According to the third Hund 's rule the lowest energy level has total angular \nmomentum J = 0, for which all magnetic multipoles a re zero. To this atomic scenario we add \nthe effect of the spin-orbit interaction and a spat ially symmetric ligand field, which is a useful, \nalbeit approximate, rendition of Ru sites that have no symmetry. \n \n In pure octahedral symmetry, the 5D splits into 5Eg + 5T1g states. In weak ligand field \nthe lowest energy state is the high-spin t 2g 3eg1 (5Eg) (see, e.g., [25]). Under spin-orbit interaction, the 5E splits into Γ1, Γ2, Γ3, Γ4, Γ5 levels (in Koster notation, or A 1, A 2, E, T 1, T 2 in \nSchönflies notation), where the Γ1 level has the lowest energy. A strong ligand field (typical \ncubic crystal-field parameter 10Dq ≈ 3 eV, appropriate for ions in the palladium group) results \nin a t 2g 4 (3T1g ) ground state, which is low spin (S = 1). Under sp in-orbit interaction, the 3T1 \nsplits into Γ1, Γ4, Γ3, Γ5 levels with multiplicities of 1, 3, 2, 3, respecti vely. These levels have \nan energy separation in the order of ζd. Thus in all cases the total angular momentum in t he \nground state has the total symmetric representation Γ1. \n \n However, the magnetic exchange interaction H ex mixes higher J levels into the ground \nstate if H ex is in the order of the spin-orbit interaction. Thi s effect is clearly seen for Cr d 4 5D0 \n(with ζd ≈ 30 meV), which has no dichroism in the absence of J mixing [26]. For Ru d 4, where \nthe spin-orbit interaction is much larger, the J mi xing due to H ex is much smaller but still \nobservable. J mixing does also occur due to hybridi zation with the neighbouring ligands, \nbecause the 4d bandwidth of several eV is much larg er than ζd. \n \nIII. NEUTRON DIFFRACTION \n \n Bao et al . [10] report Bragg intensities for a sample temper ature 3.5 K indexed by Miller \nindices h and l even. In consequence, the spatial phase-factor in ΨKQ is {i sin( ϕ)} for Q odd or \ncos( ϕ) for Q even. \n \n Dipoles (K = 1) possess either Q = 0 or Q = ±1. Thus, axial dipoles are aligned with \nthe ζ-axis (Q = 0) or the η-axis, according to the structure factor (1). Magne tic moments µζ = \n2S ζ + Lζ aligned with the ζ-axis form a primary dipole-order. Secondary dipole -order arises \nfrom µξ and µη. Use of general coordinates x = z = 1/4 for Ru sit es 8b eliminates the axial ξ-\ndipole from the structure factor. Axial dipoles alo ng η- and ζ-axes are 90 o out of phase in the \nstructure factor. In consequence, µη can only add intensity to Bragg spots with h and l different \nfrom zero, while there is no contribution to (0, k, 0) Bragg spots from µη. The unit-cell structure \nfactor confronted with Bragg spots contains quadrup oles (K = 2) and octupoles (K = 3) with \neven projections and a spatial phase cos( ϕ). Values for the axial multipoles are inferred fro m \nexperimental data [10]. \n \n Anapoles allowed in neutron diffraction include t = [3 ΩξS (h 1) − ΩξL (j 0)] \naccompanied by a spatial phase sin( ϕ). Radial integrals (h 1) and (j 0) in t are displayed in Fig. \n2a using a dimensionless variable w = 12 πaos, where ao is the Bohr radius and s = sin( θ)/ λ \n(atomic code due to R. D. Cowan [27]). Values of t are inferred from fitting our analytic \nexpressions (2) and (3) for intensities to observed intensities. An anapole t ' = { nξ (g 1)} is also \naccompanied by a spatial phase sin( ϕ) and its contribution to the unit-cell structure f actor is \n90 o out of phase with t. We have tested our analytic e xpressions against experimental data for \nboth t ≠ 0, t ' = 0 and, also, t = 0, t ' ≠ 0. In the latter case, inferred values of t ' are unrelated to \nthe radial integral (g 1) displayed in Fig. 2a. We achieve better success w ith t versus a linear \ncombination of (h 1) and (j 0) shown in Fig. 3. \n Bragg spot intensities confronted with data are ca lculated from, \n \n ℑ(t) = [{ Q ·· ·· Q*} − |κκ κκ ·· ·· Q|2], (2) \n \nwith the following amplitude Q derived from (1) and expressions recorded in refer ence [22], \n \n Q ξ ≈ κξ κζ cos( ϕ) j2 [− q − p + p'], \n \n Q η ≈ κη κζ cos( ϕ) j2 [q − p − p'] − sin( ϕ) t κζ, (3) \n \n Q ζ ≈ cos( ϕ) [ j0 µζ + j2{Lζ + (κξ2 − κη2) (q + p'/2) + (3 κζ2 − 1) (p/2)}] + sin( ϕ) t κη, \n \n κκ κκ ·· ·· Q ≈ κζ cos( ϕ) [ j0 µζ + j2{Lζ + (κξ2 − κη2) (3p '/2) + (5 κζ2 − 3) (p/2)}]. \n \nA standard dipole approximation to the neutron scat tering amplitude is obtained from (3) on \nsetting t = p = q = p ' = 0. Evidently, this approximation depends on { j0 µζ} and { j2 Lζ} \narising from Q ζ, with Q ξ = Q η = 0. We go beyond this simple expression for the sc attering \namplitude through addition of some other multipoles allowed by magnetic symmetry. \nAmplitudes (3) include anapoles, t, and parity-even quadrupoles and octupoles q = 2 √3 \u00072+2'' , \np = (3/2) √7 \u000730 and p ' = (3/2) √(70/3) \u00073+2'. Multipoles in question are actually denoted \nTKQ in references [21, 22] but here we choose to reser ve this particular notation for parity-\neven multipoles in x-ray diffraction encountered in Section IV. Moreover, a departure in \nnotation is warranted on the grounds that we make e xplicit radial integrals jn which in \nprevious work are factors in multipoles. Observe th at parity-even quadrupoles and anapoles do \nnot contribute to ( κκ κκ ·· ·· Q). \n \n Radial integrals jn with n = 0, 2, 4 & 6 calculated for Ru 4+ are shown in Fig. 2a (atomic \ncode due to R. D. Cowan [27]). Evidently, j4 and j6 provide small corrections to an amplitude \nbased on j0 and j2 with w < 6, and they are omitted from our analysis. This si mplification \ndirectly affects octupoles, which are proportional to a linear combination of j2 and j4. Fig. \n2b compares j0, j2 and j4 for atomic configurations 4d4 and 4d7. There are significant \ndifferences for the two configurations, particularl y with j0, that influence data analysis. Also \nincluded in Fig. 2b are results for Ru 1+ (4d7) derived from standard tabulations prepared by P. \nJ. Brown [28] that are satisfyingly almost indistin guishable from our results for the same atomic \nconfiguration. \n \n In fits of (2) and (3) to observed intensities we used Lζ = + 0.11, and µζ = 1.8 µB. \nThese results are consistent with an independent ob servation |Lζ|/Sζ ≈ 0.13 [29]. Spin and \norbital magnetic moments are parallel in the config uration (t 2g )4 appropriate for Ru 4+ in a strong \noctahedral ligand field [25]. As regard to the magn etic moment, Crawford et al ., [30] quote µ \n= 1.8(3) µB for Ru 4+ in Sr 4Ru 3O6, while Yoshida et al . [1] and Bao et al . [10] quote µζ = 1.59 ± .07 µB and µζ = 1.8(2) µB, respectively, for Ru 4+ in Ca 3Ru 2O7. Fits of ℑ(t) to observed \nintensities imply q ≈ 1.31, p ≈ 3.10, and p ' ≈ − 3.85, and these values are used in calculated \nvalues reported in Table I. Two values of t are obt ained for most Bragg spots, because intensity \n(2) is a product of amplitudes that contain t. Tabl e I lists experimental data [10], ℑ(t) , t and \nℑ(0) in ascending order of w. A goodness-of-fit R F = 1.2% with inclusion of anapoles slumps \nto 10 times this value when anapoles are omitted fr om calculated intensities. \n \n Our results for t in Table I versus w are displaye d in Fig. 3. In light of the excellent \nagreement between the observed cross-sections and ℑ(t) we attribute the evident scatter to \nlimitations in the data that might be reduced on re visiting the experiment. A linear combination \nof radial integrals associated with spin, (h 1), and orbital, (j 0), anapoles is fitted to 12 selected \nvalues of t. The fit reported in Fig. 3 yields an e stimate ΩξS /ΩξL ≈ − (1/12). \n \n The quality of our fit of ℑ(t) to intensities of Bragg spots R F = 1.2% is also achieved \nusing oppositely aligned spin and orbital magnetic moments. Specifically, Lζ = − 0.125 and \nµζ = 1.8 µB imply q ≈ 1.17, p ≈ 2.90, and p ' ≈ − 3.50, and these multipoles yield an identical R F \nwith changes to t that are entirely negligible. \n \nIV. RESONANT X-RAY DIFFRACTION \n \n Bragg diffraction of x-rays tuned to a ruthenium a tomic resonance is considered in this \nsection [21, 31]. First, a study of published data [11], followed by feasibility studies for future \nexperiments suggested by results from the foregoing analysis of neutron diffraction data. Unit-\ncell structure factors are derived from the electro nic structure factor (1) that respects all \nelements of symmetry in the magnetic space-group P Cna2 1. Results in Section IV. C \nincorporate a magnetic field applied along the b-ax is with ferrimagnetic order described by the \nspace group Pm 'c'21. Throughout, we use universal expressions for unit -cell structure factors \napplicable to an azimuthal-angle scan, in which the crystal is rotated about the Bragg \nwavevector [23, 32]. States of polarization are def ined in Fig. 4, and definitions of parity-even \nTKQ and Dirac GKQ multipoles comply with references [21, 23, 32, 33] . \n \nA. Analysis of published data \n \n An azimuthal-angle scan of intensity at the (H o, H o, 0) Bragg spot with H o = 1 and \nsample temperature 17 K is reported by Bohnenbuck et al . [11]. There is no intensity in the \nunrotated channel of polarization σ'σ with enhancement by an E1-E1 absorption event, bec ause \nBragg reflections of interest forbid charge-like (t ime-even) contributions. Magnetic \ncontributions occur in both π'π and the two rotated channels. Data displayed in Fi g. 5 are for \nthe rotated channel of polarization σ'π. \n \n With Miller indices ( h + l) = 0 and phase angle ϕ = 0, the electronic structure factor (1) \nreduces to, \n ΨKQ = [1 + (−1) Q] [UKQ − σπ (−1)K UK−Q]. (4) \n \nEvidently, ΨKQ can be different from zero for projections Q even. Magnetic Bragg spots are \nindexed by ( h + k) odd. Recall that we use general coordinates x = z = 1/4, which is an \napproximation albeit a very good one [1]. In the su bsequent calculation we add the further \nassumption that the Bragg wavevector (1, 1, 0) subt ends 45 o with the a-axis and the b-axis, \nwhich neglects a small difference between the cell lengths a and b (the angle between the a-\naxis and (1, 1, 0) is ≈ 44.12 o). The two approximations mentioned do not modify o ur principal \nfindings that rely on magnetic symmetry. \n \n Magnetic diffraction enhanced by an E1-E1 event ( σπ = +1) is determined by a dipole \nT1Q and only Q = 0 is allowed by the electronic struct ure factor (4). The corresponding unit-\ncell structure factor for diffraction by Ru ions is found to be, \n \n F(+)σ'π(1, 1) = − (iT1ζ/2) [cos( θ) cos( ψ) + sin( θ)]. (5) \n \nWe note in passing that F(+)π'π(1, 1) = { (iT1ζ/2) sin(2 θ) sin( ψ)}. \n \n The origin of the azimuthal angle ( ψ = 0) in (3) is such that a- and b-crystal axes spa n \nthe plane of scattering [11]. The Bragg angle θ ≈ 32.8 o for (1, 1, 0) and an x-ray energy 2.965 \nkeV for the L 2 absorption edge of Ru 4+ (sin( θ) ≈ 0.130 ( λ Ho) Å−1 with λ ≈ 12.40/E and photon \nwavelength λ and primary energy E in units of Å and keV, respec tively). Note that the dipole \nT1ζ is purely real while F(+)σ'π(1, 1) is purely imaginary. Intensity in the σ'π channel is \nproportional to I(1, 1) = |F(+)σ'π(1, 1)| 2, and it is confronted with available experimental data in \nFig. 5a. The proportionality factor for I(1, 1) con tains a dipole radial integrals 2p|r|4d or \n3p|r|4d about which we say more later in the context of re lative strengths of E1-E1, E1-M1 \nand E1-E2 diffraction amplitudes. \n \n In the E1-E2 parity-odd event, Dirac multipoles GKQ have ranks K = 1, 2 and 3. \nAnapoles (K = 1) are forbidden by (4), however, and three multipoles contribute to the \ndiffraction pattern. The unit-cell structure factor can be neatly expressed in terms of two \nfunctions of θ and ψ, namely, \n \n ν = [sin 2(θ) − {cos( θ) cos( ψ)} 2] and χ = [2 sin(2 θ) cos( ψ)], (6) \n \nand we find, \n \n F(−)σ'π(1, 2) = (1/4 √5) ℜ(1, 2) [G20 {ν + χ} + √(2/3) G2+2' {3 ν − χ} (7) \n \n + (2/ √3) G3+2'' χ {3 cos 2(ψ) − 2}]. \n Here, G2+2' and G3+2'' are real and imaginary parts of a quadrupole and a n octupole, \nrespectively. The parity-even and parity-odd struct ure factors (5) and (7) are seen to differ by \na 90 o phase. While the azimuthal-angle dependence of F (+)σ'π(1, 1) is simply one harmonic, \ncos( ψ), F(−)σ'π(1, 2) also includes harmonics cos(2 ψ) and cos(3 ψ) as a consequence of \nquadrupoles and an octupole. Intensities I(1, 1) an d I(1, 2) = |F(−)σ'π(1, 2)| 2 are symmetric around \nψ = 180 o. \n \n Data in Fig. 5a [11] are in conflict with our stru cture factors because they are not \nsymmetric about ψ = 180 o. However, intensities I(1, 1) and I(1, 2), derived from (5) and (7), \nrespectively, fit the data very well if the origin of the azimuthal-angle scan is offset, which is \nreported in Figs. 5b, c. For the moment, offsets in the azimuthal angle that create an accord \nbetween data and calculated intensities are attribu ted to experimental conditions (we thank Dr \nE. Schierle for clarification [11]). Revisiting res onant x-ray Bragg diffraction might add insight \nto the problem of asymmetry in diffraction at (H o, H o, 0). Feasibility studies of different Bragg \nspots are reported in the next sub-section with thi s problem in mind. The octupole in I(1, 2) is \na mismatch to experimental data and only the allowe d Dirac quadrupoles are used in the fit \nshown in Fig. 5c. \n \n To be meaningful, s tructure factors (5) and (7) should include strengths of E1-E1 and \nE1-E2 events, in particular radial integrals of the position variable taken between the core state \nand valence states. In our case, the ratio of E1-E2 to E1-E1 strengths is expressed by a \ndimensionless factor included in the result (7) for F(−)σ'π(1, 2). With the photon energy E tuned \nto an L-edge, 2p, \n ℜ(1, 2) = [( α E) ̸(2 ao R∞)] ( 2p|r 2|5p ̸2p|r|4d), (8) \n \nwhere α, ao and R ∞ are the fine structure constant, Bohr radius and t he Rydberg unit of energy, \nrespectively. A relativistic atomic code provides t he estimates ( 2p|r 2|5p ̸2p|r|4d ) = − 0.047 \nao while (2p|r 2|4f ̸2p|r|4d ) = − 0.031 ao [27]. \n \nB. Chiral order \n \n We report unit-cell structure factors for magnetic (0, 0, L o) Bragg spots, with odd L o. \nOur analysis of the corresponding Bragg spots in ne utron diffraction patterns, Section III, \nindicates that anapoles are significant, and it is reasonable to conjecture that higher-order Dirac \nmultipoles are likewise significant. The conjecture is supported by results from the simulation \nof electronic structure performed by Thöle and Spal din [34]. These authors include the \nmonopole in their study and it contributes to diffr action enhanced by an E1-M1 absorption \nevent, and witnessed in the structure factor (12). We continue with calculations of structure \nfactors for E1-E1, E1-M1 and E1-E2 events. \n \n The unit-cell structure factor for E1-E1 and (0, 0, L o) is, \n F(+)σ'π(1, 1) = (1 /√2) [ iT1ζ cos( ϕ) cos( θ) sin( ψ) − T1η sin( ϕ) sin( θ)], (9) \n \nwith F(+)π'σ(1, 1) = {F(+)σ'π(1, 1)}*. Recall that ϕ = (2 πyLo) with sin( ϕ) ≈ 0 for L o = 5, to a good \napproximation. The b-axis is normal to the plane of scattering for ψ = 0. Note in (9) that the \ncontribution using the secondary dipole T1η = T1c is independent of the azimuthal angle, as \nit should be when the Bragg and propagation (c-axis ) vectors are collinear. Intensity derived \nfrom (9) has a sin 2 (ψ) dependence. The Bragg angle satisfies sin( θ) ≈ 0.316 (L o/E) with E in \nunits of keV. Thus, only L o = 1 is accessible at M-edges (3p) and E ≈ 0.47 keV, while L o = 1, \n3, 5, 7 are accessible at L-edges and E ≈ 3.0 keV. \n \n The 90 o phase difference between the two axial dipoles in (9) indicates a chiral (handed) \norder that manifests itself with intensity that dep ends on circular polarization in the primary x-\nray beam. Intensity in the handed setting is propor tional to the imaginary part of interference \nbetween rotated and unrotated channels of polarizat ion [35]. Since F(+)σ'σ(1, 1) = 0 for magnetic \nreflections (0, 0, L o) the E1-E1 intensity of interest reduces to, \n \n Π(+) = P 2 Im.{[F(+)π'π(1, 1)]* F(+)π'σ(1, 1)} \n \n = − (P 2/2) sin(2 ϕ) sin 2(θ) cos( θ) cos( ψ) T1ζ T1η. (10) \n \nHere, P 2 is the Stokes parameter for primary circular polari zation [21, 35]. Evidently, intensity \nin the proposed experiment is an interference betwe en the primary and secondary axial dipoles. \n \n Our first parity-odd event is E1-M1. Dipoles M1 an d E1 in ℜ(1, 1), the analogue of (8), \nhave magnitudes µB and (e ao), respectively, where µB is the Bohr magneton. Using µB/(e ao) \n= α/2 leads to, \n \n ℜ(1, 1) = ( α ao λ|λ')/2p|r|4d. (11) \nIn this expression, λ|λ' is the overlap of orbitals in the M1 event that po ssess the same angular \nmomentum, because the magnetic moment operator is d iagonal in this basis [36]. We go on to \nfind, \n F(−)σ'π(1, 1) = − (i/√2) ℜ(1, 1) G1ξ sin(ϕ) sin(2 θ) sin( ψ) \n \n − (2/√3) ℜ(1, 1) cos( ϕ) [G00 sin 2(θ) + (1/2√2) G20 {1 + cos 2(θ) [2 − 3 sin 2 (ψ)]} \n \n + (√3/2) G2+2'{1 + [cos ( θ) sin( ψ)] 2}]. (12) \n \nand F (−)π'σ(1, 1) = {F (−)σ'π(1, 1)}*. The anapole G1ξ is depicted in Fig. 1, and its contribution \nin (12) is out of phase with the contribution from even-rank multipoles. The magnetic \nmonopole, G00, carries no dependence on the azimuthal angle, of course [37]. Comparing (9) and (12) we see that, dipoles T1ζ and G1ξ are accompanied by sin( ψ), with the anapole \ncontribution in (12) very small for L o = 5. \n \n Dirac multipoles possess a chiral configuration. I t is exposed in a handed setting \nprovided by circular polarization in the primary be am of x-rays, and parallel Bragg and \npropagation vectors. Structure factors for all four channels of polarization contribute to the E1-\nM1 intensity in question, unlike the corresponding calculation for E1-E1 that is significantly \nsimplified by the result F(+)σ'σ(1, 1) = 0. In the present case, E1-M1 intensity is given by [35], \n \n Π(−) = − (4 P 2/√3) ℜ2(1, 1) sin(2 ϕ) cos ( θ) cos( ψ) G2+1'' \n \n × [G00 sin 2(θ) + (1/2 √2) G20 {1 + cos 2(θ) [2 − 3 sin 2 (ψ)]} \n \n + (√3/2) G2+2'{1 + [cos ( θ) sin( ψ)] 2}]. (13) \n \nNotably, intensity does not depend on the anapole, whereas corresponding intensity Π(+) created \nin an E1-E1 event arises from dipoles alone. Howeve r, (10) and (13) are both proportional to \nsin(2 ϕ) cos( ψ). \n \n Octupoles in the E1-E2 structure factor are omitte d in subsequent calculations, \nprincipally on the grounds that they are at odds wi th data for (1, 1, 0) but also because the \ndominant contribution to the diffraction amplitude is likely given by an anapole G1ξ. \nIncluding all dipoles and quadrupoles allowed in (0 , 0, L o) reflections we find, \n \n F(−)σ'π(1, 2) ≈ ℜ(1, 2) {(2i/5) √3 [{G1ξ − (2/3) √10 G2+1'' } sin( ϕ) sin(2 θ) sin( ψ)] \n \n − (1/2) √(1/5) cos( ϕ) [ G20 ν + √(2/3) G2+2' {6 sin 2(θ) − ν − 2}]}, (14) \n \nwith ν defined in (6), and F (−)π'σ(1, 2) = {F (−)σ'π(1, 2)}*. The two components of (14), which \ndiffer by a 90 o phase, can be separated in an experiment by compar ing data for different Miller \nindices using sin( ϕ) ≈ 0 for L o = 5. \n \nC. Applied magnet field \n \n A magnetic field applied to Ca 3Ru 2O7 induces a (first-order) metamagnetic transition \nto a canted antiferromagnetic structure in which ax ial dipoles are partially polarized in the \ndirection of the field. In the low temperature phas e, T < 48 K, a canted structure occurs in a \nfield that exceeds ≈ 5.5 T parallel to the b-axis [10, 13]. In such a m agnetic field, a small \nmagneto-resistive effect occurs in the transition [ 10]. The metamagnetic transition no longer \noccurs at temperatures above the metal-to-nonmetal transition at T = 48 K. \n \n The precursor to the transition is a ferrimagnetic order described by the space group \nPm 'c'21 (#26.70) [20]. The crystal class m 'm'2 is asymmetric, polar and compatible with ferromagnetism. Bulk magnetization parallel to the applied field (b-axis) comes from two non-\nequivalent Ru sites that are allowed to have differ ent axial magnetic dipoles moments in the \nzero-field A-type antiferromagnetic order described by P Cna2 1. \n \n A basis (0, 0, 1), (1, 0, 0), (0, 1, 0) with respe ct to the Bb2 1m paramagnetic space-group \ndefines principal axes ( ξ, η, ζ) ≡ (c, a, b) not shown in Fig. 1, with Miller indices h = L o, k = \nHo, l = K o. The applied field is parallel to the ζ-axis. The electronic structure factor for Pm 'c'21 \nis, \n \n ΨKQ = exp(i2 πzl) {UKQ [exp(i ϕ) + (−1) l + Q exp( −iϕ)] (15) \n \n + σθ σπ (−1) l + K + Q UK−Q [exp(i ϕ') + (−1) l + Q exp( −iϕ')]}, \n \nwith ϕ = 2 π(x h + yk) and ϕ' = 2 π(x h − yk), and general coordinates (x, y, z). Bulk magnetiz ation \nis revealed by a non-zero value of ΨKQ(Pm 'c'21) evaluated for h = k = l = 0, K = 1 and σθ σπ = \n−1 (axial magnetism). For these conditions, the stru cture factor (15) is non-zero for Q = 0, i.e., \na bulk ferromagnetic moment parallel to the b-axis. \n \n Restrictions for purely magnetic Bragg spots are i dentified in (15) evaluated with Q = \n0, K even and σθ σπ = +1 appropriate to charge or nuclear scattering. With said conditions, ΨK0 \n= 0 for ( h, 0, l), l odd and no restrictions on h. (In the absence of a magnetic field, using space \ngroup P Cna2 1, the corresponding restrictions are ( h + l) = (H o + Ko) even with ( h + k) = (H o + \nLo) odd arising from anti-translation.) Unit-cell str ucture factors for purely magnetic diffraction \nare derived from, \n \n ΨKQ = exp(i2 πzl) [exp(i ϕ) − (−1) Q exp( −iϕ)] (16) \n \n × [UKQ − σθ σπ (−1) K + Q UK−Q], \n \nwhich is obtained by setting k = 0 and l odd in (15). Spatial phases are reduced to ϕ = ϕ' = \n(2 πxh) = (2 πxLo), while ( h, 0, l) ≡ (0, K o, L o) with K o odd and all L o. The leading phase factor \nin (16) can be set aside in the calculation of Brag g intensities. \n \n Ruthenium ions occupy non-equivalent sites 4c that possess no symmetry. The two sites \nhave general coordinates (x, y, z) ≈ (0.4008, 1/4, 3/4) and (0.4008 − 1/2, 3/4, 3/4). For purely \nmagnetic Bragg spots ϕ = (2 πxLo) with x ≈ 0.4008 [1], and ϕ is identical to the spatial angle \nused in all foregoing calculations; phase factors f or the two Ru sites are exp(i ϕ) and ( −1) Lo \nexp(i ϕ). \n \n With a photon energy E ≈ 3.0 keV for Ru L-edges and K o = 1 the Bragg condition can \nbe met with L o = 0, 1, 2, ... , 7, while K o > 1 is not achievable with the specified energy. T he \nE1-E1 unit-cell structure factor in the rotated cha nnel of polarization is, \n F(+)σ'π(1, 1) = − (iT1η/2) cos( ϕ) cos( θ) sin( ψ). (17) \n \nwhile F (+)σ'σ(1, 1) = 0. The axial dipole T1η = T1a is normal to the plane of scattering when \nthe azimuthal angle ψ = 0. It is notable that F(+)σ'π(1, 1), as well as F(+)π'π(1, 1), does not \nspecifically depend on the orientation of magnetic field, which is along the b-axis. Also, there \nis no chiral order in axial dipoles to be exposed b y circular polarization and Π(+) = 0. \n \n Parity-odd diffraction enhanced by an E1-M1 event does not contain a monopole. \nInstead, diffraction engages anapoles parallel ( G1ζ) and perpendicular ( G1ξ) to the magnetic \nfield, and two quadrupoles. For the rotated channel of polarization, \n \n F(−)σ'π(1, 1) = 2 ℜ(1, 1) cos( θ) sin( ψ) {cos( ϕ) [− (1/ √2) G1ξ sin(α) sin( θ) + A G2+1'' ] \n \n − i sin( ϕ) [(1/ √2) G1ζ cos( α) sin( θ) + B G2+2'' ]}, (18) \n \nwith factors A = [cos( α) cos( θ) cos( ψ)] and B = [sin( α) cos( θ) cos( ψ)]. The angle α arises from \nthe orientation of the b-axis (field direction) wit hin the plane of scattering at ψ = 0, and it finds \nno place in E1-E1 structure factors. Specifically, \n \n cos( α) = − Lo/√[(cK o/b) 2 + (Lo)2], (19) \n \nwith α = π/2 for L o = 0, which also means ϕ = 0. The Bragg wavevector (0, K o, 0) is parallel \nto the magnetic field and F(−)σ'π(1, 1) ∝ [G1ξ sin(2 θ) sin( ψ)], because sin( ϕ) = 0 and A = 0. In \nthe general case, F(−)σ'π(1, 1) contains two harmonics of the azimuthal angl e, namely, sin( ψ) \nand sin(2 ψ). \n \n Diffraction in the rotated channel of polarization enhanced by an E1-E2 absorption \nevent is described by a structure factor similar to (18) when octupoles are neglected as they are \nin (14). In this approximation, F(−)σ'π(1, 1) and F(−)σ'π(1, 2) have identical factors from the \nazimuthal angle. Indeed, F(−)σ'π(1, 2) can be derived from (18) with the substituti ons A → \n(1/3) √5 [A − 2 sin( α) sin( θ)], B → (1/3) √5 [B + 2 cos( α) sin( θ)] and ℜ(1, 1) → ℜ(1, 2), apart \nfrom unimportant numerical factors. The change to A means that F(−)σ'π(1, 2) depends on two \nmultipoles, G1ξ and G2+1'' , for (0, K o, 0). \n \nV. CONCLUSIONS \n \n In our study of antiferromagnetic Ca 3Ru 2O7 we have presented: \n- the magnetic space-group P Cna2 1 (crystal class mm21 ') that is appropriate for A-type \n antiferromagnetic order \n- compelling evidence derived from a neutron diffra ction pattern [10] for the existence of Dirac \n dipoles (anapoles) - radial integrals in neutron diffraction amplitude s for Ru 4+ (4d4) \n- inconclusive evidence for Dirac multipoles derive d from published diffraction pattern \n obtained with x-rays tuned to a ruthenium atomic r esonance [11] \n- chiral order of both axial dipoles, created by pr imary and secondary dipoles, and Dirac \n multipoles that can be exposed in the resonant dif fraction of circularly polarized x-rays \n- the magnetic space-group Pm 'c'21 (crystal class m 'm'2) that is appropriate for ferrimagnetic \n order induced by a magnetic field applied parallel to the crystal b-axis, together with \n feasibility studies of pertinent resonant x-ray di ffraction experiments. \n \n Our evidence for anapoles in the low-temperature m agnetic configuration is bolstered \nby their appearance in a simulation of the electron ic structure [34]. \n \nAcknowledgements. We are grateful to Dr S. Agrestini, Dr A. Bombardi , Professor J. W. \nLynn, Professor A. Michels, Dr E. Schierle and Dr D . A. Sokolov for useful advice and \nopinions. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n TABLE I. Observed neutron cross-sections σobs listed against increasing w. Sample \ntemperature = 3.5 K. Bragg spots are here labelled by orthorhombic Miller indices (H o, K o, L o). \nInferred values of anapoles, t, using Lζ = + 0.11 and µζ = 2S ζ + Lζ = 1.80. Calculated values \nof the cross-section ℑ(t) derived from (2) and (3) in the fourth column p ossess a goodness-of-\nfit R F = 1.2%, with R F defined in standard manner R F = {∑ | σobs − ℑ(t)| / ∑ σobs } with both \nsums over all 15 Bragg spots. Omission of anapoles in ℑ(0) yields a vastly inferior R F = 13.6%. \nFor L o = 5 no realistic value can be assigned to t becaus e the corresponding structural phase \nfactor sin( ϕ) ≈ 0.0. \n \n(Ho, K o, L o) w σobs ℑ(t) t ℑ(0) \n(0, 0, 1) 0.51 9.64(3) 9.64 −0.01 9.57 \n \n(0, 0, 3) 1.53 1.10(2) 1.09 +0.035, −0.98 0.95 \n \n(0, 0, 5) 2.56 5.91(6) 5.91 ∼ \n \n(0. 0, 7) 3.58 0.43(2) 0.43 +0.56, −0.09 0.23 \n \n(0, 2, 1) 3.65 0.070(7) 0.070 +0.39, −0.04 0.05 \n \n(2, 0, 1) 3.75 2.39(1) 2.41 +0.50 2.69 \n \n(0, 2, 3) 3.92 0.051(7) 0.047 +0.01, −0.20 0.038 \n \n(2, 0, 3) 4.02 0.216(5) 0.217 −0.05 0.26 \n \n(0, 2, 5) 4.43 0.44(2) 0.44 ∼ \n \n(2, 0, 5) 4.51 1.66(1) 1.43 ∼ \n \n(0, 0, 9) 4.60 1.34(5) 1.33 +0.60, −1.13 0.14 \n \n(0, 2, 7) 5.09 0.05(1) 0.05 +0.175, −0.05 0.017 \n \n(0, 0, 11) 5.63 0.63(4) 0.63 +0.47, −0.88 0.055 \n \n(0, 2, 9) 5.85 0.39(3) 0.38 +0.48, −0.44 0.0 \n \n(0, 2, 11) 6.69 0.27(3) 0.27 +0.14, −0.75 0.12 \n \n \n \n \n \n \n FIG. 1. Crystal and magnetic structures of Ca 3Ru 2O7 (cell lengths a ≈ 5.3677, b ≈ 5.5356, c ≈ \n19.5219 Å at 8 K). Anapoles parallel to the crystal a-axis a re depicted together with principal \naxial dipoles parallel to the crystal b-axis. Axes (ξ, η, ζ) ≡ (a, c, −b) for the magnetic structure \nPCna2 1 are displayed. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 2. (a) Radial integrals for Dirac multipoles t hat appear in the anapole t = [3 ΩξS (h 1) − \nΩξL (j 0)] using a dimensionless variable w = 12 πaos, where ao is the Bohr radius and s = \nsin( θ)/ λ. Atomic wavefunctions 4d4 - 5p1 (see text). Legend: ( ) (h 1), () [w × (j 0)] and ( ) \n[w × (g1) /10 ]. Note that (j 0) and (g 1) are proportional to 1/w as the wavevector approac hes \nzero, and t ' = { nξ (g 1)}. Radial integrals j0 (black), j2 (orange), j4 (brown) and j6 (black) \nfor axial multipoles using Ru 4+ (4d4). (b) j0, j2 and j4 using black continuous curves for the \nconfiguration 4d4, and red continuous 4d7. Green dashed are results for 4d7 obtained from a \nstandard tabulation (see text). \n \n \n \n \n \n \nFIG. 3. Values of the anapole t listed in Table I a re displayed as a function of w using black \nsquares and red dots. For most Bragg spots there ar e two values of t, one black and one red, \nwhile no value can be assigned to t at Bragg spots (Ho, K o, L o) with L o = 5. The solid curve is \na fit to a linear combination of radial integrals f or spin and orbital anapoles reported in Fig. 2a \nto 12 values of t denoted by black squares. \n \n \n \n \n \n \n \n \n \n \nFIG. 4. Primary ( σ, π) and secondary ( σ', π') states of polarization. Corresponding wavevectors \nq and q' subtend an angle 2 θ, and k = q − q'. Principal axes ( ξ, η, ζ) for a magnetic structure \nand depicted Cartesian co-ordinates (x, y, z) coinc ide in the nominal setting, and k for a \nparticular Bragg spot is aligned with − x. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 5. Green triangles: Measured intensity of the (1, 1, 0) Bragg spot in the σ'π channel of \npolarization as a function of azimuthal angle, ψ. (a) Symmetric and antisymmetric components \nof the data are displayed as red and blue spots, re spectively. Continuous green curve is intensity \nI(1, 1) = |F(+)σ'π(1, 1)| 2, derived from (5) for enhancement by an E1-E1 even t. Origin ψ = 0 at \nwhich a- and b-axes span the plane of scattering de fined in Fig. 4. (b) Continuous red curve is \nintensity I(1, 1) using an ψ-origin off-set by 9.54 o. (c) Continuous red curve is intensity I(1, 2) \nderived from (7) for an E1-E2 event. The ψ-origin is off-set by 12.39 o and allowed Dirac \noctupoles are set equal to zero in the calculation. \n \n \n \n \n \n \n \n \n( a ) \n( b ) \n( c ) Y - > Y +9.54 References. \n \n[1] Y. Yoshida, C. H. Lee, and S. Katano, Phys. Rev . B 72 , 054412 (2005). \n \n[2] Y. Yoshida et al ., Phys. Rev. B 69 , 220411(R) (2004). \n \n[3] M. R. Norman, D. Pines, and C. Kallin, Adv. Phys. 54 , 715 (2005). \n \n[4] T. M. Rice, K. –Y. Yang, and F. C. Zhang, Rep. Prog. Phys. 75 , 016502 (2012). \n \n[5] R. B. Laughlin, Phys. Rev. B 89 , 035134 (2014). \n \n[6] B . Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, an d J. Zaanen, Nature 518 , 179 \n (2015). \n[7] G. Cao et al ., Phys. Rev. B 67 , 060406(R) (2003). \n \n[8] X. N. Lin et al ., Phys. Rev. Lett. 95 , 017203 (2005). \n \n[9] G. Cao et al ., New J. Phys. 6, 159 (2004). \n \n[10] W. Bao, Z. Q. Mao, Z. Qu, and J. W. Lynn, Phys . Rev. Lett. 100 , 247203 (2008). \n \n[11] B. Bohnenbuck, et al ., Phys. Rev. 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B 80 , \n 140410(R) (2009). \n \n \n \n \n \n \n \n \n \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:(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": "2103.02449v1.THz_Field_induced_Spin_Dynamics_in_Ferrimagnetic_Iron_Garnets.pdf", "content": "arXiv:2103.02449v1 [cond-mat.mtrl-sci] 3 Mar 2021THz Field-induced Spin Dynamics in Ferrimagnetic Iron Garn ets\nT.G.H. Blank,1K.A. Grishunin,1E.A. Mashkovich,1M.V. Logunov,2A.K. Zvezdin,3and A.V. Kimel1\n1Radboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, The Netherlands.\n2Kotel’nikov Institute of Radioengineering and Electronic s, 125009 Moscow, Russia.\n3Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia.\n(Dated: March 4, 2021)\nTHz magnetization dynamics is excited in ferrimagnetic thu lium iron garnet with a picosecond,\nsingle-cycle magnetic field pulse and seen as a high-frequen cy modulation of the magneto-optical\nFaraday effect. Data analysis combined with numerical model ling and evaluation of the effective\nLagrangian allow us to conclude that the dynamics correspon ds to the exchange mode excited by\nZeeman interaction of the THz field with the antiferromagnet ically coupled spins. We argue that\nTHz-pump — IR-probe experiments on ferrimagnets offer a uniq ue tool for quantitative studies of\ndynamics and mechanisms to control antiferromagnetically coupled spins.\nAll magnetically ordered materials, depending on the\nalignment of spins, are divided into two primary classes:\nferro- and antiferromagnets. Ferromagnets are charac-\nterized by parallel alignment of spins which results in\nnet magnetic moment, while spins in antiferromagnets\nare aligned in a mutually antiparallel way with zero net\nmagnetization in the unperturbed state. Antiferromag-\nnets represent the largest, but the least explored class of\nmagnets with a potential to have a dramatic impact on\nspintronics and other magnetic technologies. In particu-\nlar, the higher frequency ( ∼THz) of spin resonances in\nantiferromagnetscan bring the clock-speed of spintronics\ndevices into the THz range [1–3].\nUnfortunately, proceedings in both fundamental re-\nsearch and the development of antiferromagnetic spin-\ntronics are considerably hindered by the lack of net mag-\nnetization in antiferromagnets, as even the discovery of\nantiferromagnetic order itself had to wait for the advent\nof neutron diffraction experiments in the late 1940s [4].\nThisiswhyapproachesandmechanismsallowingefficient\nexcitation of antiferromagnetic spins in the THz range\nbecame a subject of not only intense, but also challeng-\ning and intriguing research. In particular, recently it was\nsuggested that THz magnetic fields can excite antifer-\nromagnetically coupled spins with a significantly higher\nefficiency when accounting for the new, relativistic mech-\nanism of field derivative torque (rFDT) [5]. This torque\ncan reach strengths comparable with conventional the\nZeeman torque [6]. However, the lack of methods for\nquantitative detection of spins in antiferromagnets pre-\nventsthese claims fromexperimental verificationand can\neven lead to mistakes in interpretation of experimental\nresults [7].\nA substantial progress in understanding THz light-\nspin coupling can be achieved by studying ferrimagnets,\nwhich are a subclass of antiferromagnetshaving two non-\nequivalent magnetic sublattices. Within each sublattice\nthe spins are aligned ferromagnetically, while the inter-\nsublattice interaction is antiferromagnetic. The sublat-\ntice magnetizations can be different in size, and therefore\nthe net magnetization is not necessarily zero. The lat-\nter greatly simplifies experimental studies, but it doesnot ruin the presence of THz resonances called exchange\nmodes, as antiferromagnetic order is still present. In this\narticle, we demonstrate and explore the high-frequency\nresponse of antiferromagnetic spins in a ferrimagnet to\nTHz magnetic field. We experimentally reveal the ori-\nentation of the THz field which causes the largest de-\nviation of spins from their equilibrium. Using simula-\ntions we show that the oscillations correspond to the\nexchange mode of spin resonance. The applied experi-\nmental technique is shown to have a great potential to\nfacilitate quantitative conclusions. In particular, due to\nthe non-zero Faraday rotation in the unperturbed state\n(αF) and having the calibrated dynamic Faraday rota-\ntion (∆αF), the ratio ∆ αF/αFunambiguously defines\nspin deviations caused by the calibrated THz magnetic\nfield. The technique allows us to show that the conven-\ntional Zeeman torque does play in the spin-excitation the\ndominant role, while alternative mechanisms can essen-\ntially be neglected.\nThe garnet structure (crystallographic space group\nIa¯3d) of rare-earth iron garnets (REIGs) gives rise to un-\nusual magnetic properties [8, 9]. Three of five Fe3+-ions\nperformulaunit(R 3Fe5O12)formasublatticewithtetra-\nhedral symmetry and are antiferromagnetically coupled\nto the remaining two iron ions occupying sites of octahe-\ndral symmetry. The imbalance between these iron ions\nresultsin a net magnetic moment MFeto which the rare-\nearth site magnetization MRaligns anti-parallel. The re-\nsult is a three-sublattice ferrimagnet with net magnetiza-\ntionM=MR+MFe. The antiferromagnetic exchange\nbetween the iron sublattices is large compared to any\nother interactions experienced by the Fe3+spins, justify-\ning the approximation of treating it as a single sublattice\nwith magnetization MFe[10]. The RE-sublattice expe-\nriences the exchange-field generated by this iron magne-\ntization [8], while intra-sublattice exchange interaction is\nweak and can be ignored, resembling a paramagnet in\nthe exchange field.\nThe REIG structure studied in this work is a 19 µm\nfilm of Bi- and Ga- substituted thulium iron garnet\nTm3-xBixFe5-yGayO12(TmIG) with targeted composi-\ntionx= 1,y= 0.8. The ���lm was grown by liquid2\nphase epitaxy on a 500 µm thick (111)-oriented GGG\nsubstrate. The sample was doped with Bi3+to enhance\nmagneto-optical effects [11–13]. Previous research on\nfilms grown in this way show that the sample is char-\nacterized by an uniaxial out-of-plane type anisotropy, as\nthe thin-film shape anisotropy is shown to be overcome\nby stress-induced anisotropy from a lattice mismatch be-\ntween substrate and sample [14] together with a small\ncontribution of growth-induced anisotropy due to the\nsite preference of bismuth ions along the growth direc-\ntion [15, 16]. Consequentially, this gives an “easy-axis”\nalong the [111] crystallographic direction. The expecta-\ntions are confirmed by measurements of static magneto-\noptical Faraday rotation as a function of magnetic field\n(Supplemental Material [17]).\nIn the pump −probe experiment, we use optical pulses\nfrom a Ti:Sapphire amplifier with a central wavelength\nof 800 nm, 4 mJ energy per pulse, 100 fs pulse dura-\ntion and 1 kHz repetition rate. These pulses were em-\nployed to generate single-cycle THz pulses by a titled-\nfront optical rectification technique in a lithium niobate\ncrystal as described in Ref. [18] and written in detail\nin Ref. [19]. The generated THz beam was tightly fo-\ncused onto the sample [20] and spatially overlapped with\na low intensity optical probe beam that was chopped out\nbeforehand from the original beam. Varying time retar-\ndation between the THz pump and optical probe pulse,\ntime-resolved measurements were obtained by mapping\nprobe polarization changes induced by the THz pulse us-\ning a balanced photo-detector. The strength of the THz\nelectric field was calibrated using the Pockels effect in a\nthin (110)-cut GaP crystal and yields a maximum peak\nstrength of |ETHz| ≈1 MV/cm, implying a peak mag-\nnetic field of 0 .33 T. The THz pulse waveform and the\ncorresponding Fourier spectrum are shown in the Sup-\nplemental Material. Both the generated THz and optical\nprobe pulses are linearly polarized. The experimental ge-\nometry is schematically depicted in Fig. 1(a). The THz\nmagnetic field is initially along the x-axis, but this di-\nrection can be controlled by a set of wire-grid polarizers.\nNote that using this approach, a polarization rotation of\nπ/2 from the initial state always reduces the THz mag-\nnetic field at leastbyone half. Astatic externalmagnetic\nfieldµ0Hextof at most 250 mT was applied at an angle\nof∼10◦with the sample plane. Using static Faraday\nrotation αFwe see that such maximum field strength is\nsufficient to saturate magnetization in the garnet film.\nFigure 1(b) shows THz-induced ultrafast dynamics of\nthe probe polarization ∆ αFand how it depends on the\nTHz-pump polarization. By rotating the THz polariza-\ntion from HTHz/bardblM/bardbltoHTHz⊥M/bardbl, the symmetry\nof the high-frequency oscillations with respect to the po-\nlarity of the external magnetic field is altered. To reveal\nthe originofthesepeculiar THz-induced modulations, we\nperformed systematic studies as a function of pump and\nprobe polarizations, external magnetic field, THz field\nstrength and temperature.\nThe observed oscillations of the probe polarization ro-\nFIG.1. (a)Schematicoftheexperimentalsetup. Theillustr a-\ntion on the top-right shows the distribution of dodecahedra l\nTm3+and tetrahedral/octahedral Fe3+ions. Any magnetic\nmoment will tend to align along the [111] “easy-axis”. (b) Po -\nlarization rotation ∆ αFmeasured as a function of the delay\nτbetween THz pump and visible probe pulses. Depending\non the THz polarization, the mapped dynamics is either odd\n(HTHz/bardbly⊥M/bardbl) with the external magnetic field or even\n(HTHz/bardblx/bardblM/bardbl). The measurements were performed at\nT= 6 K.\ntation, obviously, are a result of a periodic modulation of\noptical anisotropy (birefringence) in the sample. A THz\npulse is able to induce such optical anisotropyby modify-\ning the dielectric permittivity tensor ǫij. If one neglects\ndissipation, which is a safe approximation for iron gar-\nnets at the wavelength of 800 nm [13, 21], the tensor is\nHermitian [22]. Such type of tensor ǫijcan be written as\nasumofthesymmetric(real) ǫ(s)\nij=ǫ(s)\njiandantisymmet-\nric (imaginary) ǫ(a)\nij=−ǫ(a)\njiparts. Measurements of the\nTHz-induced dynamics as a function of probe polariza-\ntion angle show no dependency (Supplemental Material\n[17]), indicating that the THz-induced modulations orig-\ninate from the anti-symmetric part of the dielectric ten-\nsor. It means that the polarization rotation ∆ αFmust\nbe assigned to the magneto-optical Faraday effect. In a\n[111] garnet crystal, this effect is a measure of the mag-3\nnetization along the z-axis [23, 24]:\nǫ(a)\nxy∼Mz. (1)\nWhenHTHz⊥M/bardbl, changing the external magnetic\nfield polarity from + Hextto−Hextflips the sign of the\nobserved dynamics (Fig. 1(b), red waveforms). More-\nover, by increasing the strength of the static magnetic\nfield we found that the amplitude of the oscillations and\nthe net magnetization saturate at the same field (Supple-\nmental Material [17]). This fact implies that the THz-\ninduced dynamics must by assigned to dynamics of the\nmagnetization M. Due to peculiarities of the detection\ntechnique (Eq. (1)), the measurements are sensitive to\nmodulations of the out-of-plane magnetization. Thus,\nif we compare the size of the amplitude of the oscilla-\ntions ∆αFwith the saturated static Faraday rotation\nαF(∼20◦at 800 nm), this allows us to quantitatively\nestimate the relative change of the magnetization along\nthe z-axis during the oscillations ∆ Mz/Mz∼0.012. For\nanothercase HTHz/bardblM/bardbl, the signalalsosaturatesin line\nwith the magnetization M, but the phase of the oscilla-\ntions is unaffected by the polarity of the external field\n(see Fig. 1(b)).\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s48/s48/s46/s53/s49/s49/s46/s53/s50/s50/s46/s53/s51/s51/s46/s53/s52/s52/s46/s53\n/s48 /s49/s48/s48 /s50/s48/s48/s49/s53/s48/s51/s48/s48/s52/s53/s48/s70/s111/s117/s114/s105/s101/s114/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s46/s117/s46/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s54/s75/s51/s54/s75/s55/s54/s75/s49/s48/s54/s75/s49/s51/s48/s75/s49/s54/s49/s75/s50/s48/s48/s75/s50/s51/s56/s75\n/s72\n/s84/s72/s122 /s77/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s84/s101 /s109 /s112/s101 /s114/s97/s116/s117/s114/s101 /s32/s40/s75/s41\nFIG. 2. Fourier spectrum of the THz-induced signal ( HTHz/bardbl\nM/bardbl) measured at various temperatures. Central frequencies\nof the peaks deduced from the fit are plotted as a function of\ntemperature in the inset. The dotted line denotes a fit with\nEq. (8) ( ω0= 400 GHz, TC= 314 K) and the bars denote\n±half-width-half-maximum of the fitted Lorentzians. The\nFFT spectrum for HTHz⊥Mis added to the Supplemental\nMaterial [17].\nFigure 2 shows the Fourier spectra of the THz-induced\nwaveforms ranging in the entire accessible temperature\nrange when HTHz/bardblM/bardbl. The inset summarizes the tem-\nperature dependence of the peak frequency, and this be-\nhaviour is in qualitative agreement with what could be\nexpected for an exchange mode in rare-earth iron gar-\nnets [25, 26]. In order to get a better insight into the\nTHz-induced magnetization dynamics, we modelled the/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48/s45/s49/s46/s50/s45/s48/s46/s54/s48/s48/s46/s54/s49/s46/s50\n/s84/s105/s109/s101/s32/s40/s112/s115/s41/s72\n/s84/s72/s122 /s32 /s32/s77 /s68 /s77\n/s70/s101/s32/s122/s32/s47/s32/s77\n/s70/s101/s32\n/s45/s48/s46/s51/s48/s48/s46/s51/s48/s46/s54/s77\n/s70/s101 /s32/s122/s120/s49/s48/s45/s50\n/s72\n/s84/s72/s122/s77 \n/s43/s72\n/s101 /s120 /s116\n/s45/s32/s72\n/s101 /s120 /s116/s124/s109\n/s48/s72\n/s84 /s72/s122/s124/s32/s61/s32/s51/s51/s51/s32/s109/s84\n/s124/s109\n/s48/s72\n/s84 /s72/s122/s124/s32/s61/s32/s49/s54/s55/s32/s109/s84/s124/s109\n/s48/s72\n/s101 /s120 /s116/s124/s32/s61/s32/s57/s48/s32/s109/s84\nFIG. 3. Dynamics in the z-component of iron MFemodeled\nby LLG equations.\nresponse with the help of the Landau-Lifshitz-Gilbert\n(LLG) equations [27]. The equations, in particular, ac-\ncount for rFDT derived by [5]:\ndMi\ndt=−γiMi×Beff\ni(t)+αi\nMiMi×/parenleftBigdMi\ndt+a3\ni\nµBdH\ndt/parenrightBig\n,\n(2)\nwherei= Fe, Tm. We use literature g-values for thulium\ngTm= 7/6 and iron gFe= 2 [28]. Based on the Ga-\ncontent, the sublatticemagnetizationofiron |MFe|= 4.2\n(µBper formula unit R 3Fe5O12) [28] is antiferromagnet-\nically coupled to the magnetization of thulium |MTm|=\n2. The latter is taken to match the effective g-factor\ngef≡(MFe−MTm)/((MFe/gFe)−MTm/gTm)≈6 mea-\nsured in this sample (Supplemental Material [17]). The\nvolumeof the unit cell a3\ni[29] per spin constitutes a small\nfactora3/µB∼10−5m/A. The effective magnetic fields\nBeff\ni≡ −δΦ/δMi(in T) are derived from the thermody-\nnamic potential Φ [27], containing exchange interaction\nand Zeeman coupling to the external field and THz mag-\nnetic field H(t) (in A/m). For the model we use a real-\nistic exchange constant Λ = −30 T/µB[28, 30, 31] and\nTHz magnetic field modelled by the Gaussian derivative\nfunction fitted to the experimental waveform (see Sup-\nplemental Material [17]). The initial state of the net\nmagnetization vector is taken along the external field,\nconsidering we saturated the magnetization experimen-\ntally. The numerical solution of these equations reveals\nthat the THz magnetic field induces dynamics of the\nN´ eel vector L≡MFe−MTmand the magnetization\nM≡MFe+MTm. The dynamics of MFe, which dom-\ninates the detected magneto-optical signal, is shown in\nFig. 3. The phenomenological Gilbert damping factors\nofαFe/MFe=αTm/MTm= 0.0015 have been taken\nto match the experimental observations. The simulation\ncontains a high-frequency magnetic resonance at around\n380 GHz, which we identify as the Kaplan-Kittel ex-\nchange mode since its frequency depends linearly on the\nexchange constant [32]. The dynamics of MFe,z(t) in4\nFig. 3 is in agreement with our experimental results in\nFig. 1(b). It has a larger amplitude and changes sign\nupon reversing MwhenHTHz⊥M/bardbl, while the sign is\nconserved if HTHz/bardblM/bardbl. The amplitude matches very\nwell to the experimental values even if the rFDT term\nis not taken into account. As proposed in Ref. [6] the\ncontribution of this term will be indeed small in cases\nof low damping α1,2<0.01. Altogether, the simulations\npoint out that the observedoscillationscorrespondto the\nexchange mode of spin resonance and show that Zeeman-\ntorque plays the dominant role in the excitation of this\nmode with THz magnetic field.\nThese conclusions can also be confirmed analytically\nusing Lagrangianmechanics and the effective Lagrangian\n(see Supplemental Material [17] for full derivation):\nLeff=M2\n2δ/bracketleftBigg/parenleftBigg/parenleftBig˙φ\nγ−H/parenrightBig\nsinθ+hycosθcosφ/parenrightBigg2\n+/parenleftBigg˙θ\nγ+hysinφ/parenrightBigg2/bracketrightBigg\n+m/parenleftBig\nH−˙φ\nγef/parenrightBig\ncosθ\n+mhysinθcosφ+KUsin2θsin2φ.(3)\nWhereM=MFe+MTm,m=MFe−MTm,δ≡\n−4ΛMFeMTm, 1/γ= (MFe/γFe+MTm/γTm)/(MFe+\nMTm),γef=gefµB//planckover2pi1,hx(t) andhy(t) arethe THz mag-\nnetic field componentsin the sample x−yplane asin Fig.\n1 andH(t)≡Hext+hx(t) the total field along the exter-\nnalfieldx-direction(here10◦inclinationangleofexternal\nmagnetic field is ignored). The polar angle θ∈[0,π] is\ndefined with respect to the external field x-axis. In this\ncoordinate system, the net magnetization vector can be\nexpressed as M=m(cosθ,sinθcosφ,sinθsinφ). Equa-\ntions of motion now follow from Euler-Lagrange equa-\ntions, taking into account a phenomenological damping\nterm through a Rayleigh function [33]. The results can\nbe linearized about the ground state angles θ0,φ0, found\nby minimization of the thermodynamic potential Φ for\nwhich we find φ0=π/2 andθ0depending on the ra-\ntio of external field to anisotropy. This has been done\nfor general θ0in the Supplemental Material [17], yield-\ning complex equations of motion. In the special case of\nzero external field, the spins lie along the easy axis of\nanisotropy ( θ0=π/2). Linearizing around the ground-\nstate angles θ=θ0+θl,φ=φ0+φlwithθl,φl≪1, the\nequations of motion then take the simple form:\n¨θl+αMγ\nχ⊥˙θl+2KUγ2\nχ⊥θl−mγ2\nγefχ⊥˙φl=−γ˙hy−mγ2hx\nχ⊥,\n(4)\n¨φl+αMγ\nχ⊥˙φl+2KUγ2\nχ⊥φl+mγ2\nγefχ⊥˙θl=γ˙hx−mγ2hy\nχ⊥.\n(5)\nHereχ⊥≡M2\nδis a constant inversely proportional to\nthe exchange constant. It is seen that the large THzfield derivative term γ˙hiappearsas the dominant driving\nforce, in accordance with our understanding how dynam-\nical THz fields may excite antiferromagnetic magnons in\nantiferromagnets (where m→0) by Zeeman interaction\n[34, 35]. Moreover, each equation of motion contains\na mutually orthogonal component of the field-derivative\n˙hx,y. Noting that HTHz⊥M/bardblleads to ˙hx= 0 and\nHTHz/bardblM/bardblto˙hy= 0, the symmetry with respect to ex-\nternal field ±Hext, as observed experimentally, can now\nbe explained (see Supplemental Material [17]).\nMoreover,consideringfree precession α→0,hx,y→0,\nthe absolute eigenfrequencies of the coupled set of equa-\ntions (4) - (5) are:\nωex=mγ2\nγefχ⊥≈ |Λ|(|γTm|MFe−|γFe|MTm),(6)\nωFM=γef2KU\nm≡γefHa. (7)\nEquation (6) corresponds to Kaplan-Kittel’s exchange\nresonance frequency [32] while Eq. (7) describes the con-\nventional ferromagnetic precession of the net magneti-\nzation in the anisotropy field Ha. Using Eq. (6) and\nBloch’s law for the spontaneous magnetization of iron\nwhileMTm(T)∼MFe(T), we fitted the temperature de-\npendence of the oscillations frequency shown in inset of\nFig. 2 using:\nωex(T)∼ω0/parenleftBig\n1−(T/TC)3\n2/parenrightBig\n. (8)\nwhereω0the exchange resonance frequency at zero\nKelvin. In reality MTmdrops faster with temperature\nthan the magnetization of iron, accounting for the slight\nrise of frequency at low temperatures. In general, the fit\nis another confirmation of the validity of our assumption\nthat the observedoscillations correspondto the exchange\nmode.\nIn conclusion, investigating the response of ferrimag-\nnets to THz fields and comparing the data with theoret-\nical predictions from numerical solutions of the Landau-\nLifshitz-Gilbert equations and analytical solutions de-\nrived from Euler-Lagrange equations of motion, we\nshowed that the THz field excites the exchange mode\nin the ensemble of antiferromagetically coupled spins.\nWe demonstrated that the Zeeman-torque plays a dom-\ninant role in the coupling of the THz-field to the spins.\nWhile quantitative studies of spin dynamics in compen-\nsated antiferromagnets seem to require complex mag-\nnetometry techniques, ferrimagnets facilitate an excel-\nlent playground to study dynamics of antiferromagneti-\ncally coupled spins. At last, we would like to point out\nthat previous measurements of ferrimagnetic resonance\n[26, 36] could only reveal an effective gyromagnetic ratio.\nUsing excitation of exchange mode with THz magnetic\nfield, magneto-optical detection via the Faraday effect\nand comparison of the observed amplitudes of magneti-\nzationdynamicswith theresultsofnumericalsimulations\nprovidesauniversaltechnique todirectly estimatethe in-\ndividual gyromagnetic ratio of the ions.5\nACKNOWLEDGMENTS\nThe authors thank S. Semin, Ch. Berkhout and P. Al-\nbers for technical support. The work was supported byde Nederlandse Organisatie voor Wetenschappelijk On-\nderzoek (NWO). M.V.L. acknowledges the support from\nthe Russian Foundation for Basic Research (Nos. 18-29-\n27020 and 18-52-16006).\n[1] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Antiferromagnetic spintronics,\nRev. Mod. Phys. 90, 015005 (2018).\n[2] P. Nˇ emec, M. Fiebig, T. 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Express 23, 28985 (2015).arXiv:2103.02449v1 [cond-mat.mtrl-sci] 3 Mar 2021SUPPLEMENTAL MATERIAL TO: THZ FIELD-INDUCED SPIN DYNAMICS IN\nFERRIMAGNETIC IRON GARNETS\n1 Static magneto-optical characterization of TmIG sample\nFigure 1: Measurements of the magneto-optical Faraday rotation usin g a continuous-wave helium-neon laser ( λ= 632.8 nm)\nwith both the external field and the light’s wave-vector perp endicular to the sample plane. A paramagnetic contribution\n∼0.56 deg/T from the cryostat glass windows has been subtracted from the raw data. The data exhibits a large rotation\nand demonstrates a weak easy-axis type of anisotropy with sm all coercive <6 mT and saturation <25 mT field. No\ncompensation point was observed above nitrogen temperatur es>77 K.\nFigure 2: Static polarization rotation measurements with light at th e wavelength of 800 nm in the experimental geometry\n(see Fig. 1(a) in article). The evolution of hysteresis loop form can be due to temperature dependent anisotropy constan ts.\nClearly, no magnetization compensation point is observed i n this temperature range. At T = 6 K, the saturated polarizati on\nrotation is ∼ ±1.65◦, and given the angle of the magnetic field of 10◦, this has been used to estimate the Faraday rotation\n(∼ ±10◦) when the magnetization is along the sample normal.\n1Supplemental Material\nFigure 3: Domain pattern seen by magneto-optical microscopy in trans mission at zero field. The typical ”labyrinth” type\ndomains grow with decreasing temperature, which indicates a growing role of easy axis anisotropy and thuliummagnetiza tion\n[1]. When applying an external magnetic field along the out-of- plane easy axis, the domains along this field expand and the\nsample will be uniformly magnetized for relatively small fie ld (see Suppl. Fig. 1)\n2Supplemental Material\n2 Experimental setup\nThe experimental setup regarding THz generation by optical rect ification in lithium niobate is described in detail\nin [2,3]. The THz path was purged with nitrogen to avoid water absorption lin es in the THz spectrum. A small\npart of the initial 800 nm beam is chopped out beforehand (ratio 1 : 1 00) and is brought to spectral and temporal\noverlap with the THz pump pulse. The focused spot size of the probe beam is considerably smaller than that of\nthe THz. The waveform of the THz pulse was mapped using electro-o ptical sampling in a 50 µm GaP [110] crystal\nseen in Supplemental Fig. 4.\n/s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s49/s45/s48/s46/s53/s48/s48/s46/s53/s49\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s69/s108/s101/s99/s116/s114/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s77/s86/s47/s99/s109/s41\n/s84/s105/s109/s101/s32/s40/s112/s115/s41\n/s70/s111/s117/s114/s105/s101/s114/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s46/s117/s46/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41\nFigure 4: THz waveform and corresponding Fourier spectrum measured b y EO sampling in GaP.\n3 Supplemental Results\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s45/s49/s53/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s80/s111/s108/s97/s114/s105/s122/s97/s116/s111/s105/s110/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s109/s100/s101/s103/s41\n/s84/s105/s109/s101/s32/s100/s101/s108/s97/s121/s32/s40/s112/s115/s41/s32/s48/s111/s32/s32/s32/s32/s32/s32\n/s32/s50/s48/s111\n/s32/s54/s48/s111/s32/s32/s32\n/s32/s56/s48/s111\n/s32/s49/s50/s48/s111\n/s32/s49/s54/s48/s111/s72\n/s84/s72/s122/s32 /s77\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s109/s100/s101/s103/s41\n/s84/s105/s109/s101/s32/s100/s101/s108/s97/s121/s32/s40/s112/s115/s41/s32/s48/s111\n/s32/s50/s48/s111\n/s32/s54/s48/s111\n/s32/s56/s48/s111\n/s32/s49/s50/s48/s111\n/s32/s49/s54/s48/s111/s72\n/s84/s72/s122/s32 /s32/s77\nFigure 5: THz induced polarization rotation waveforms for two orthog onal THz pump polarizations (two figures) and for\nseveral orientations of the probe polarization. The angle d epicted is the angle of the probe electric field with respect t o the\nexperimental x-axis (Fig. 1(a) of the article). This data implies that the T Hz induced signals are Faraday rotation (see\nmain text).\n3Supplemental Material\nFigure 6: Peak-to-peak amplitudes of THz induced waveforms as a funct ion of external magnetic field (a) and THz field\n(b) for two orthogonal THz pump polarizations. Bending of th e red dots at low THz fields is attributed to the facts that\nthe THz light is not perfectly linearly polarized and imperf ections of the wire grid polarizers.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s49/s53/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s45/s72\n/s101/s120 /s116/s45/s52/s53/s111\n/s32\n/s43/s57/s48/s111 \n/s32/s40/s112/s41/s32/s43/s52/s53/s111\n/s32/s48/s111\n/s32/s40/s115/s41/s70/s97/s114/s97/s100/s97/s121/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s109/s100/s101/s103/s41\n/s80/s114/s111/s98/s101/s32/s100/s101/s108/s97/s121/s32/s40/s112/s115/s41/s45/s57/s48/s111\n/s32/s40/s45/s112/s41\n/s43 /s72\n/s101/s120 /s116/s32/s97/s32/s61/s84/s32/s61/s32/s49/s53/s48/s75\nFigure 7: THz induced Faraday rotation waveforms obtained at several angles of the pump polarization angle αand applied\nexternal field of ±250 mT. Besides the previous figure, this is the only graph whe re a weaker THz electric field of 160 kV/cm\nhas been used. The result shows that measuring at α=±45◦, in between the fully symmetric α= 0◦and fully antisymmetric\northogonal α=±90◦, results in a mix of symmetry/antisymmetry. Moreover, the e ffects gradually become weaker towards\nα= 0◦(HTHz/bardblM/bardbl).\n4Supplemental Material\n/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49 /s49/s46/s50 /s49/s46/s52/s48/s49/s48/s50/s48/s51/s48/s52/s48/s70/s70/s84/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s46/s117/s46/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s50/s57/s51/s75\n/s50/s55/s48/s75\n/s50/s51/s56/s75\n/s50/s48/s48/s75\n/s49/s54/s49/s75\n/s49/s51/s48/s75\n/s49/s48/s54/s75\n/s55/s54/s75\n/s51/s54/s75\n/s54/s75\n/s72\n/s84 /s72/s122/s32 /s32/s77\n/s124/s124\nFigure 8: FFT spectra of THz induced Faraday rotation for HTHz⊥M/bardbl. The exchange-mode frequency at about 375\nGHz shows softening similar to the case HTHz/bardblM/bardblas is presented in article Fig. 2. At lower temperatures anot her\nhigh frequency (725 GHz) appears. It is known that crystal fie ld transition may appear in this region [ 4], but if it can be\nattributed to these transitions is yet unclear.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/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/s100/s101/s108/s97/s121/s32/s40/s112/s115/s41/s54/s75/s51/s54/s75/s55/s54/s75/s49/s48/s54/s75/s49/s51/s48/s75/s49/s54/s49/s75/s50/s48/s48/s75/s50/s51/s56/s75/s50/s55/s48/s75/s50/s57/s51/s75\n/s56/s55/s46/s49/s109 /s84/s56/s55/s46/s49/s109 /s84/s56/s55/s46/s49/s109 /s84/s56/s55/s46/s49/s109 /s84/s49/s48/s53/s46/s55/s109 /s84/s49/s48/s53/s46/s55/s109 /s84/s49/s50/s50/s46/s52/s109 /s84/s49/s51/s55/s46/s49/s109 /s84/s49/s51/s55/s46/s49/s109 /s84/s49/s51/s55/s46/s49/s109 /s84\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/s100/s101/s108/s97/s121/s32/s40/s112/s115/s41/s54/s75/s51/s54/s75/s55/s54/s75/s49/s48/s54/s75/s49/s51/s48/s75/s49/s54/s49/s75/s50/s48/s48/s75/s50/s51/s56/s75/s50/s55/s48/s75/s50/s57/s51/s75/s72\n/s84/s72/s122 /s77 /s72\n/s84/s72/s122 /s32 /s32/s77\nFigure 9: Experimental waveforms of THz induced Faraday rotation as a function of temperature. In both cases the same\nexternal fields (specified in the first figure) have been applie d to ensure saturation of static magnetization.\n5Supplemental Material\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s50/s52/s50/s54/s50/s56/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s69/s120/s116/s101/s114/s110/s97/s108/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41/s126/s55/s56/s46/s56/s32/s71/s72/s122/s47/s84\n/s126/s49/s49/s46/s56/s32/s71/s72/s122/s32/s97/s116/s32/s72\n/s101/s120 /s116/s61/s48\nFigure 10: Preliminary data of THz-induced ferromagnetic resonance, used to estimate the effective g-factorgeff≈6.\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s52/s48/s48/s45/s50/s48/s48/s48/s50/s48/s48/s52/s48/s48/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41\n/s84/s97/s114/s103/s101/s116/s32/s100/s101/s108/s97/s121/s32/s40/s112/s115/s41/s69/s120/s112/s46/s32/s119 /s97/s118 /s101/s102/s111/s114/s109\n/s70/s105/s116/s116/s101/s100/s32/s119 /s97/s118 /s101/s102/s111/s114/s109\nFigure 11: The dotted line shows the experimentally calibrated THz mag netic field pulse, which has been fitted using the\nGaussian derivative function G′(x) =−2A((x−d)/w)exp/bracketleftbig\n((x−d)/w)2/bracketrightbig\nwithA= 404 mT, d= 1.17 ps (variable, determines\narrival time of pulse) and w= 0.2223 ps pulse-width.\n6Supplemental Material\n/s48 /s49/s48 /s50/s48 /s51/s48/s45/s49/s45/s48/s46/s53/s48/s48/s46/s53/s49/s49/s46/s53/s50\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s73/s114/s111/s110/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40 /s109\n/s66/s41/s77\n/s70/s101/s32/s120 \n/s72\n/s84/s72/s122/s77 /s32\n/s72\n/s84/s72/s122/s32/s32/s32/s32/s77 /s120 /s49/s48/s45/s50\n/s48 /s49/s48 /s50/s48 /s51/s48/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s77\n/s70/s101/s32/s121 /s120 /s49/s48/s45/s50\n/s48 /s49/s48 /s50/s48 /s51/s48/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48\n/s43 /s72\n/s101/s120 /s116\n/s45/s32/s72\n/s101/s120 /s116\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s77\n/s70/s101/s32/s122/s120 /s49/s48/s45/s50\nFigure 12: Simulated dynamics of the iron magnetization MFeusing LLG equations and plotted separately for the x,y\nandzcomponents where zcoincides with the sample out-of-plane axis. It shows how th e symmetry with respect to external\nfield is exactly opposite when looking at the y-component, to which we are not experimentally sensitive.\n4 Equations of motion derived from Lagrangian formalism\nWe start from the following Lagrangian and Rayleigh dissipation funct ions, which are equivalent to the LLG\nequations for a two-sublattice ferrimagnet [ 5]:\nL=T−Φ\n=−MFe\nγFecosθFe∂φFe\n∂t−MR\nγRcosθR∂φR\n∂t−Φ (1)\nR=RFe+RR,RFe,R=αMFe,R\n2γFe,R/parenleftbig˙θ2\nFe,R+sin2θFe,R˙φ2\nFe,R/parenrightbig\n, (2)\nwhereθiandφithe polar and azimuthal angles of the iron (Fe) and rare-earth (R) sublattices in the experimental\ncoordinate system with the x-axis aligned to the external magnetic field Hext(see Fig. 13, here we ignore the 10◦\ninclination of the field for simplification).\nFigure 13: Coordinate system used for Lagrangian equation.\n7Supplemental Material\nThe thermodynamic potential used is:\nΦ =−(MFe+MR)·Hef−ΛMFe·MR−KFe(MFe·n)2\nM2\nFe−KR(MR·n)2\nM2\nR. (3)\nHeren= (0,0,1) is the directional vector of the easy axis of anisotropy, Λ <0 the intersublattice exchange\nconstant and KFe,R>0 the uniaxial anisotropy constants. The Euler-Lagrange equatio ns w.r.t. θi,φigive rise\nto four (coupled) equations of motion (two for each sublattice), a nd this is generally not easy and sometimes even\nimpossible to solve. Instead, in Ref. [ 5] an effective Lagrangian is obtained by assuming the canting of the t wo\nsublattices are equal and are assumed to be small. This approach ge nerally works at field well below the exchange\nfield (small canting), and it is valid here as the static measurements in dicate we are well below the spin-flop field.\nWe introduce the usual definitions of the magnetization M=MFe+MRand antiferromagnetic (N´ eel) vector\nL=MFe−MR. These two vectors are parameterized using a set of angles θ,ǫandφ,βdefined as:\nθFe=θ−ǫ, θR=π−θ−ǫ, (4)\nφFe=φ+β, φ R=π+φ−β. (5)\nIn the quasi-antiferromagnetic approximation [ 5], the canting angles are assumed to be small ǫ≪1,β≪1. In first\norder approximation, the MandLare then naturally defined as:\nM=m(cosθ,sinθcosφ,sinθsinφ) (6)\nL=M(cosθ,sinθcosφ,sinθsinφ) (7)\nwherem≡MFe−MRandM ≡MFe+MR. Substituting our new set of angles ( 4)-(5) into the Lagrangian ( 1)\nand expanding up to quadratic terms in the small variables ǫ,βgives for the kinetic energy part:\nL=−m\nγef˙φcosθ−M\nγsinθ/parenleftBig\n˙φǫ+β˙θ/parenrightBig\n−Φ (8)\nwhere we defined:\n1\nγef≡MFe/γFe−MR/γR\nMFe−MRand1\nγ≡MFe/γFe+MR/γR\nMFe+MR. (9)\nThepotentialenergyΦcanbe expandedsimilarly. Here, wemakethe simplificationthatboth sublatticesexperience\nthe same effective anisotropy KU≡(KFe+KR)/2 in which case the anisotropy terms can by replaced by a single\nterm−KU(l·n)2wherel=L/|L|. Furthermore, the effective field Hefin (3) consists of the static external field\nand the time-dependent THz magnetic field Hef=Hext+HTHz. The external field is chosen along the x-axis\nHext= (H0,0,0), while we assume the THz magnetic field lies in the x−yplaneHTHz≡(hx(t),hy(t),0) (see Fig.\n1 from the article). Writing δ≡ −4ΛMFeMRandH≡Hext+hx, the potential energy becomes after expanding\ninǫ,β:\nΦ =−mHcosθ−MHǫsinθ−mhysinθcosφ+Mhyβsinθsinφ (10)\n+Mhyǫcosθcosφ−mhycosθsinφ ǫ·β+δ\n2/parenleftbig\nǫ2+β2sin2θ/parenrightbig\n−KUsin2θsin2φ.\nWe will ignore the term containing ǫ·βas it is very small, from the quadratic terms only the ones proportion al\nto the exchange constant ∼δsurvive. We exclude the variables ǫ,βby solving the Euler-Lagrange equations\nd\ndt∂L\n∂˙ǫ−∂L\n∂ǫ=−∂R\n∂˙ǫ≈0 andd\ndt∂L\n∂˙β−∂L\n∂β=−∂R\n∂˙β≈0, giving:\nǫ=M\nδsinθ/parenleftBigg\nH−˙φ\nγ/parenrightBigg\n−Mhy\nδcosθcosφ, (11)\nβsinθ=−M\nδ/parenleftBigg˙θ\nγ+hysinφ/parenrightBigg\n. (12)\n8Supplemental Material\nSubstituting in ( 8)-(10) and rearranging terms yields the effective Lagrangian from the ar ticle:\nLeff=M2\n2δ/bracketleftBigg/parenleftBigg/parenleftBig˙φ\nγ−H/parenrightBig\nsinθ+hycosθcosφ/parenrightBigg2\n+/parenleftBigg˙θ\nγ+hysinφ/parenrightBigg2/bracketrightBigg\n+m/parenleftBig\nH−˙φ\nγef/parenrightBig\ncosθ\n+mhysinθcosφ+KUsin2θsin2φ.(13)\nThe equations of motion are now determined by Euler-Lagrange equ ations:\nd\ndt/parenleftBig∂Leff\n∂˙θ/parenrightBig\n−∂Leff\n∂θ+∂R\n∂˙θ= 0, (14)\nd\ndt/parenleftBig∂Leff\n∂˙φ/parenrightBig\n−∂Leff\n∂φ+∂R\n∂˙φ= 0. (15)\nWe solve these equations and linearize them around the equilibrium (gr ound-state) equilibrium values θ0andφ0,\nwhich are found by minimizing ( 3) yielding φ0=π/2 andθ0depending on the ratio of external field to anisotropy\n(i.e. when Hext= 0 we have θ0=π/2 whileθ0= 0 when Hext≫Hanis). Linearizing around these values, i.e.\nθ=θ0+θlandφ=φ0+φlwithθl,φl≪1, the first equation ( 14) gives:\n¨θl+αMγ\nχ⊥˙θl+/parenleftBig\n−γ2H2cos2θ0+mγ2H\nχ⊥cosθ0−2KUγ2\nχ⊥cos2θ0/parenrightBig\nθl+/parenleftBig\nγHsin2θ0−mγ2\nγefχ⊥sinθ0/parenrightBig\n˙φl\n+/parenleftBig\n−γ2Hhycos2θ0+mγ2hy\nχ⊥cosθ0/parenrightBig\nφl=−γ˙hy+γ2H2\n2sin2θ0−mγ2H\nχ⊥sinθ0+γ2KU\nχ⊥sin2θ0.(16)\nwhere we introduced the notation χ⊥≡M2\nδ. Similarly for the second Euler-Lagrange equation ( 15):\n¨φl+αMγ\nχ⊥˙φl+φl/parenleftBig\n−γ˙hycotθ0+γ2h2\ny+2KUγ2\nχ⊥/parenrightBig\n+˙θl/parenleftBig\n−2γHcotθ0+mγ2\nγefχ⊥sinθ0/parenrightBig\n+θl/parenleftBig\n−2γ˙hxcotθ0+γ2Hhy(1−cot2θ0)+γ2mhy\nχ⊥cosθ0\nsin2θ0/parenrightBig\n=γ˙hx+γ2Hhycotθ0−mγ2hy\nχ⊥1\nsinθ0.(17)\nThese equations can be drastically simplified by noting that1\nχ⊥is proportional the the exchange constant and is\ntherefore relatively large. Also the field derivative term γ˙hiis strong, while terms proportional to ∼γhx,ywithin\nbrackets are driving terms proportional to the response and thu s negligible. Equations ( 16)-(17) are then given in\napproximation:\n¨θl+αMγ\nχ⊥˙θl+/parenleftBig\n−γ2H2cos2θ0+mγ2H\nχ⊥cosθ0−2KUγ2\nχ⊥cos2θ0/parenrightBig\nθl+/parenleftBig\nγHsin2θ0−mγ2\nγefχ⊥sinθ0/parenrightBig\n˙φl\n=−γ˙hy−mγ2H\nχ⊥sinθ0+γ2KU\nχ⊥sin2θ0,(18)\n¨φl+αMγ\nχ⊥˙φl+2KUγ2\nχ⊥φl+/parenleftBig\n−2γHcotθ0+mγ2\nγefχ⊥sinθ0/parenrightBig\n˙θl=γ˙hx+γ2Hhycotθ0−mγ2hy\nχ⊥1\nsinθ0.(19)\nThe large field derivatives of the THz field ˙hx,yappear as a dominant driving force in these equations of motion.\nInterestingly, only the y-component ˙hyappears in the equation of motion for θl, which we use here to understand\nthe qualitative difference in dependencies on THz pump polarization as suming the field-derivative driving force is\ndominant.\nIn the experiment we saturate the magnetization with the externa l field at a small angle θ0≈0 (thusθ0≈π\nfor−Hext). Given that φ0=π/2, we have that the modulations in the magnetization z-component are Mz(t) =\nMsinφsinθ∼ ±θlfor±Hextexternalfield. Theexperimentrevealsweareonlysensitiveto Mz(t), sothedetectable\nFaraday rotation modulations should also be proportional ∼ ±θl(t). When HTHz⊥M,˙hx=hx/negationslash= 0 and we have\na strong non-zero driving force in ( 18), it explains why we see immediate strong oscillations in Mz. Because the\n9Supplemental Material\ndriving term has the same sign for both external field polarities ±Hext, the forced oscillations must be sensitive to\nthe polarity of the external magnetic field ¨Mz(t= 0) =d2\ndt2sin/parenleftbig\nθ0+θl(t)/parenrightbig/vextendsingle/vextendsingle/vextendsingle\nt=0∼ ±¨θl(t= 0)∼ ∓γ˙hy(asθ0= 0, π\nfor±Hext) i.e. this is an H-odd effect. After the THz pulse has left the sample, the system of equations resembles\nthose for a harmonic oscillator in 2D, meaning the subsequent free o scillations will have opposite phases in the\ncases of opposite polarities of the external magnetic field.\nMeanwhile by a similar argument, it is clear why a strong response is abs ent inMzwhenHTHz/bardblM/bardbl(˙hy=\nhy= 0) as in this case only the equation of motion for the in-plane dynamic sφl(t) (Eq. (17)), to which we are not\nsensitive, has an initial non-zero driving force γ˙hxwhileθl(t) does not. Detectable oscillations in θl(t) are instead\nonly driven by cross-terms like −mγ2\nγefχ⊥sinθ0˙φl(Eq. (18)). Here it is important that the ground state θ0is not\nexactly equal to 0 and π, i.e. sin( θ0) =±ρfor±Hextwithρ >0 small constant (due to experimental canting of\nexternal field, otherwise no dynamics in this case is observed as was also seen in the experiment and simulations).\nThus for opposite externalfield polarities, the driving force in θlhas opposite sign ∓mγ\nχ⊥ρ˙φl, contraryto the previous\ncase. This means that subsequent oscillations are now expected to be even with Hext, in accordance with what was\nseen experimentally. Because these field-even oscillations are a sec ondary result from primary in-plane oscillations\nφ(t), it also explains why the observed effects are relatively weak when HTHz/bardblM/bardblcompared to HTHz⊥M.\nThe eigenfrequencies in the article have been found by solving the co upled set of equations:\n¨θl+2KUγ2\nχ⊥θl−mγ2\nγefχ⊥˙φl= 0, (20)\n¨φl+2KUγ2\nχ⊥φl+mγ2\nγefχ⊥˙θl= 0. (21)\nAssuming θl,φl∼exp(iωt) the frequencies can be solved by the equation\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−ω2+ω2\nK−iωωex\niωωex−ω2+ω2\nK/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0, (22)\nwhereω2\nK= 2KUγ2/χ⊥andωex=mγ2/γefχ⊥in the case of weak anisotropy ωex≫ωK, and we obtain:\nω=±ωex\n2±/radicalbigg\nω2ex\n4+ω2\nK≈ ±ωex\n2±(ωex\n2+ω2\nK/ωex). (23)\nThus we obtain two approximate absolute frequencies:\nω1=ω2\nK/ωex=γef2KU\nm, (24)\nω2≈ωex=mγ2\nγefχ⊥≈ |Λ|(|γR|MFe−|γFe|MR) (25)\nwhere in the last approximation we used that MFeMR≈(MFe+MR)2/4 and (MFe/γFe+MR/γR)2≈(MFe+\nMR)2/γFeγRto recover the approximate Kaplan-Kittel expression of exchang e resonance.\nReferences\n[1] A. Kalashnikova, V. Pavlov, A. Kimel, A. Kirilyuk, T. Rasing, and R. P isarev, “Magneto-optical study of\nholmium iron garnet Ho 3Fe5O12,”Low Temperature Physics , vol. 38, 09 2012.\n[2] E. A. Mashkovich, K. A. Grishunin, R. V. Mikhaylovskiy, A. K. Zvez din, R. V. Pisarev, M. B. Strugatsky,\nP. C. M. Christianen, T. Rasing, and A. V. Kimel, “Terahertz optoma gnetism: Nonlinear THz excitation of\nGHz spin waves in antiferromagnetic FeBO 3,”Phys. Rev. Lett. , vol. 123, p. 157202, Oct 2019.\n[3] M. Sajadi, M. Wolf, and T. Kampfrath, “Terahertz-field-induce d optical birefringence in common window and\nsubstrate materials,” Opt. Express , vol. 23, pp. 28985–28992, Nov 2015.\n10Supplemental Material\n[4] A. J. Sievers and M. Tinkham, “Far infrared spectra of rare-ea rth iron garnets,” Phys. Rev. , vol. 129, pp. 1995–\n2004, Mar 1963.\n[5] M. D. Davydova, K. A. Zvezdin, A. V. Kimel, and A. K. Zvezdin, “Ult rafast spin dynamics in ferrimagnets\nwith compensation point,” Journal of Physics: Condensed Matter , vol. 32, p. 01LT01, oct 2019.\n11" }, { "title": "1904.11977v1.Blueprint_for_deterministic_all_optical_switching_of_magnetization.pdf", "content": "1\nBlueprint for deterministic all-optical switching of \nmagnetization \nC. S. Davies1,2,*, T. Janssen1,2, J. H. Mentink2, A. Tsukamoto3, A. V. Kimel2, \nA. F. G. van der Meer1,2, A. Stupakiewicz4, and A. Kirilyuk1,2,† \n1 FELIX Laboratory, Radboud University, 7c Toernooiveld, 6525 ED Nijmegen, The \nNetherlands \n2 Radboud University, Institute for Molecule s and Materials, 135 Heyendaalseweg, 6525 AJ \nNijmegen, The Netherlands \n3 College of Science and T echnology, Nihon University, 7-24-1 Funabashi, Chiba 274-8501, \nJapan \n4 Laboratory of Magnetism, Faculty of Physics, University of Bialystok, 1L Ciolkowskiego, \n15-245 Bialystok, Poland \nAbstract \nWe resolve a significant contr oversy about how to understand and engineer single-shot all-\noptical switching of magnetization in ferrima gnets using femto- or picosecond-long heat \npulses. By realistically mode lling a generic ferrimagnet as two coupled macrospins, we \ncomprehensively show that the net magnetizati on can be reversed vi a different pathways, \nusing a heat pulse with durati on spanning all relevant timescal es within the non-adiabatic \nlimit. This conceptual understanding is fully validated by experiments studying the material \nand optical limits at which the switching proce ss in GdFeCo alloys loses its reliability. Our \ninterpretation and results constitute a bl ueprint for understanding how deterministic all-\noptical switching can be achieved in alternat ive ferrimagnets using short thermal pulses. \n\n* Corresponding author: C.Davies@science.ru.nl \n† Corresponding author: A.Kirilyuk@science.ru.nl 2\nThe societal thirst for smaller, faster and more energy-efficient ha rd-disk drive technology \nstimulates intense research devoted to finding and understanding magnetization switching \nprocesses. The industrially-favored approach uses just a writing magnetic field, but the \nsuperparamagnetic effect1 and the associated recording trilemma impedes further \nimprovements in this direction. In 2007, the discovery of all-optical switching (AOS)2, in \nwhich ultrashort optical pulses reverse magnetization without as sisting magnetic fields, gave \nbirth to the identification and study of a whole family of AOS-related effects3,4,5 displayed by \ndifferent materials and enabled by tailored optical pulses. To date, however, the only \nmaterials that have been known to display ultr afast single-shot AOS are amorphous alloys of \nGdFeCo3 and multilayered stacks of Pt/Gd/Co6. In these materials, a single optical pulse will \ntoggle the magnetization deterministically i. e. irrespective of it s initial polarity. \nSince the first discovery of AOS in GdFeCo, many reports ha ve aimed to elucidate its \nunderlying mechanisms, but due to the undeni able complexity of ultrafast magnetism7, many \nconflicting results and interpretations have emer ged. It was initially thought that the inverse \nFaraday Effect drove helicity -dependent AOS in GdFeCo3, but later careful and quantitative \nanalysis of the wavelength-dependent fluence re quirements irrefutably revealed that magnetic \ncircular dichroism was responsible8, in combination with deterministic AOS. It was also \nassumed3,9-11 that a sub-picosecond optical pulse was a compulsory prerequisite for \ndeterministic AOS, but experimental reports ha ve shown that even 15-picosecond-long pulses \ncan suffice in certain cases12,13. \nWhile it is clear that a femtosecond-long laser pulse generates a strongly non-\nequilibrium state in GdFeCo with a fully-demagnetized FeCo-sublattice3,9,10-11, it is also clear \nthat laser-pulses with duration longer than the electron-lattice interaction e-l (2 ps) cannot \ninduce dramatic overheating of the free electrons12,13. Such overheating is crucial for ultrafast \ndemagnetization as the spin-lattice relaxation ra te is proportional to the effective electron \ntemperature14,15. The community of ultrafast magnetis m therefore found it counter-intuitive \nthat deterministic AOS could be achieved using laser and current pulses with > 10 ps. These \nexperimental observations even led to statements13 about the insolvency of the mechanism \nvia a strongly non-equilibrium state. Such st atements, however, overlook the fact that the \nthree-temperature model7,12-13,16 does not always adequately represent a ferrimagnet. Two \nvery different magnetic sublattices are be tter represented not by one, but by two \ninterconnected reservoirs, where the characteristic time of interaction Gd-FeCo between the \nspin-reservoirs of Gd and FeCo is defined by the inter-sublattice exchange interaction. 3\nBecause of this, fast change of the magnetizati on of one of the sublattices is possible at the \ncost of the other another, and does not require a spike in the electronic temperature. The \noverarching criterion for deterministic AOS lies in the condition that the heating induces a strongly non-equilibrium state. If this is satisfi ed, even relatively slow heating of the system \ntriggering purely exchange-driven dynamics can ach ieve reversal, provided that (i) there is \nmore angular momentum in the Gd sublattice than in the FeCo one, and (ii) the spin-lattice \nthermalization time is slower than \nGd-FeCo . This leads to the observable transient \nferromagnetic state, whereby the magnetization of FeCo crosses zero while Gd is still \ndemagnetizing, which is a compulsory pr erequisite for deterministic AOS. \nIn this letter, we present a conceptual unde rstanding of deterministic AOS derived for \na generic ferrimagnet of composition A100-xBx, using laser pulses with duration covering all \nrelevant time scales. The magnetization dynamics of AB, which underpin the switching \nprocess, can be described using a master/slave relationship, with A being the “master” and B \nserving as the “slave”. Two di stinct pathways allow for deterministic AOS, either with \nangular momentum flowing from both sublattices to the exte rnal environmen t or between A \nand B themselves. The direction of the flow is dictated by the combination of the relative \nconcentrations of A and B and the temporal properties of the excitation. To validate our \nconceptual understanding, we use a phenomenol ogical mean field theory describing the \nsublattice-resolved longitudina l magnetization dynamics of A100-xBx, taking in to account both \nthe temporal profile of a therma l load and the alloy composition. To provide ultimate proof of \nour interpretation, we experimenta lly study the material and optical parameters that enable or \ndisable deterministic AOS in Gdx(FeCo)100-x alloys. Specifically, we identify a critical pulse-\nduration threshold that defines the determ inistic character of AOS, and increases \nmonotonically with the concentration x of slave gadolinium. Photons in a very wide spectral \nrange, from the visible to mid-in frared, are also shown to be equally capable of triggering \ndeterministic AOS. Our conceptual interpreta tion explains both our measurements and a \nwealth of other experimental and numerical findings that have , until now, not been unified \nwithin a common framework of understanding. Moreover, we believe our understanding may \nbe expanded to experimentally predict the gene ral conditions that will enable deterministic \nAOS in different materials. \n The master/slave relationship intrinsic to our considered generic ferrimagnet AB \nderives from the fact that, in isolation, sublattices A and B are ferromagnetic and \nparamagnetic respectively. Nevertheless, the inte rsublattice exchange coup ling gives rise to a 4\ncommon Curie temperature in equilibrium, a nd also the existence of two degenerate \nequilibrium states, with A and B having antiparallel magnetiza tion. These two states are \nindicated by green dots in the sublattice-re solved phase diagram of angular momentum S \nshown in Fig. 1, and trajectories connect ing the two correspond to deterministic AOS \npathways17. Under equilibrium conditions, it is im possible for AOS to occur without an \nassistive magnetic field. Adiabatic heating of the ferrimagnet, i.e. > s-l where s-l is the \nspin-lattice thermalizat ion time, results in SB decreasing more rapidly than SA (inset of \nFig. 1), and ends with the complete destruc tion of magnetization. Th is scenario corresponds \nto the dashed trajectory shown in Fig. 1. \n \nFig. 1 Conceptual phase map showing the different pa thways for thermally-induced relaxation and \nrecovery of the constituent-resolved angular momentum S of the ferrimagnet A100-xBx. The \nthick green dots indicate positions of equilibrium, and by varying x, these states are \ntranslated across the map. Excitation of the ferrimagnet by thermal pulses of varying \nduration lead to different trajectories. Shown in the inset is the adiabatic thermal \ndependence of the angular momentum. \n When the ferrimagnet is instead heated under non-equilibrium conditions, the \nmagnetization can relax via two distinct mechanisms. The first involves inter-sublattice \nexchange coupling (with a characteristic timescale A-B) whereby the angular momentum of \nthe master sublattice grows at the expense of the slave’s. If the dynamics are driven purely by \nexchange coupling, the to tal angular momentum of AB is conserved and so Bt At S S . As \none therefore reduces from above to below s-l, the solid trajectory shown in Fig. 1 becomes \nincreasingly linear along the fi gure diagonal (solid curve). Provi ded that (i) there is more \n5\nangular momentum in slave- B than in master -A, and (ii) A-B < s-l, relatively slow heating of \nthe system (> 10 ps) can still satisfy the obs ervable condition for deterministic AOS (that SA \ncrosses zero while SB is demagnetizing). Upon forming the transient ferromagnetic state, \ncontinuous exchange of angular momentum l eads to the slave switching its magnetization \npolarity, as dictated by master A, and so deterministic AOS is successfully achieved. \n Upon further reduction of towards the timescale of electron-lattice thermalization \n(2 ps in GdFeCo)12,16, temperature-induced dissipation of SA and SB to the external \nenvironment overwhelms the exchange coupli ng, and the sublattices essentially relax \nindependently. Furthermore, if A has a smaller spin than B, A will demagnetize faster9, \nresulting in a reasonably-horizon tal dotted trajectory as indicated in Fig. 1 (in GdFeCo, this \ngradient is approximately 4:1)9. SB is now even larger when SA crosses zero, and so the \nalready-cooling system enables the intersub lattice exchange coupling and subsequent \nmagnetization recovery to comple te the switching process. \nBy varying the all oy concentration of A100-xBx, the initial and final equilibrium states \n(green dots in Fig. 1) will shift. Increasing y shifts the initial and final equilibrium states of \nAB up and down respectively in Fig 1, allowing a steeper trajectory to join the two states. \nPhysically, the slave has more angular momentum available to transfer to the master, \nenabling a longer pulse (still satis fying the non-adiabatic condition A-B < s-l) achieve \ndeterministic AOS. Conversely, reducing y will disable the possibili ty for deterministic AOS \nto proceed via exchange coupling only, if | SB| < |SA|. However, a shorter pulse generating a \nmore horizontal trajectory in Fig. 1 would still suffice. \nTo numerically test our conceptual unders tanding summarized in Fig. 1, we have \nexpanded upon the phenomenological mean-f ield model of relaxation dynamics of a \nferrimagnet developed by Mentink et al10,18. In this model, the longitudinal dynamics of the \nSA and SB is governed by the interplay between the in ter-sublattice exchange and spin-lattice \nrelaxation of individual sublattices. The coupl ed equations of motion characterizing the \ntemperature-dependent angular momentum of each sublattice (which are treated as a pair of \nmacrospins) are \nA A A B eAH H HdtdS , (1)\nBB A B eBH H HdtdS , (2)6\nwhere A and B characterize the flow of angular mome ntum from the indicated sublattice to \nthe external environment (of temperature T), e characterizes the in ter-sublattice exchange, \nand H represents the effective field acting on the subscripted sublattice. A full description of \nthe mean-field model is supplied19 in Supplemental Note 1. \n \nFig. 2 (a) Calculated time-resolved dynamics of the angular momentum S of master Fe (red line) \nand slave Gd (blue line) in the ferrimagnet Gd26Fe74 triggered using different pulse durations \n as indicated. (b) Corresponding phase map of the sublattice-resolved angular momentum \ntrajectories of the ferrimagnet Gd26Fe74 obtained using different . Shown in the inset is a \nzoomed section. (c)-(d) Same as in panels (a)-(b) except = 2 ps and the alloy concentration \nof the ferrimagnet GdxFe100-x is varied. \nUsing Eqs. (1)-(2), we calculated the s ublattice-resolved magnetization dynamics of \nAB with different alloy concentrations in re sponse to thermal pulses of varying duration. \nMaterial parameters typical of the ferrimagnetic alloy Gdx(FeCo)100-x were adopted, taking \nthe transition metal component as a single s ublattice and using concentration-independent \nmaterial properties (thus restricting the independent parameters to just e, A and B). The \n7\nfull-width half-duration of the temporally-Gaussian pulse enters the model through a time-\ndependent temperature that captures the spirit of the two-temperature model. \nFigure 2 (a) shows the results of the calculations for the alloy Gd26Fe74, obtained with \nvarying pulse duration . With = 100 fs (solid curves), we successfully achieve \ndeterministic AOS via different demagnetization rates and the clear formation of a transient \nferromagnetic state. Generally, increasing leads to increasingly comparable demagnetizing \nrates of Gd and Fe. Stretching the pulse dura tion to 5 ps (dashed curves) still enables \ndeterministic AOS, but SGd and SFe almost completely quench simultaneously. In practice, \nthermal fluctuations may dominate at th is point, and the switching would lose its \ndeterministic character. Upon stretching the pul se duration even further (Supplemental Note \n2)20, the polarity of the transient fe rromagnetic state undergoes reversal21 i.e. the \nmagnetization of the slave switches before that of the master. This is consistent with both the \nexperimental and numerical results reported in Refs. [21]-[23], and we observe in this case \nthat AOS always fails. \nBy recasting the time-resolved trajectories of SGd and SFe as functions of each other, \nwe gain a numerically-supported insight of how the pulse duration controls the process of \ndeterministic AOS. Figure 2 (b) shows20 that by increasing , the AOS trajectory initially \nbecomes more linear, and then curves below the figure diagonal, refl ecting the increasing \ndominance of the inter-sublattic e exchange coupling. By repeating the same calculations for \nGdxFe100-x alloys with varying x and fixed pulse duration = 2 ps (Fig. 2 (c)), the initial \nferrimagnetic state in the plane SFe-SGd is shifted upwards (Fig. 2 (d)). This permits a steeper \ngradient of the AOS trajectory where SFe crosses zero while SGd is still demagnetizing. \nphysically allowing a ferrimagnet with more sl ave constituents to be deterministically \nswitched using a longer pulse. \nTo obtain ultimate experimental evidence of our interpretation, we performed a set of \nexperiments exposing 6 GdFeCo al loys with different sublattice concentrations to single laser \npulses of varying duration. The samples were all of elemental composition Gdx(FeCo)100-x, \nwith 22 ≤ x ≤ 27, and all possessed out-of-plane magne to-crystalline anisotropy. Specific \ndetails of the samples are supplied24 in Supplemental Note 3. The laser pulses had a photon \nenergy of 1.55 eV (central wavele ngth 800 nm) and a duration that could be adjusted between \n60 fs and 6.0 ps and resolved with an accuracy of < 100 fs. The effect of the optical pulse on \nthe sample magnetization at room temperat ure was monitored using a magneto-optical \nmicroscope sensitive to the out-of-plane compon ent of magnetization via the Faraday effect. 8\nThe insets of Fig. 3 show typical magneto -optical images recorded for the alloy \nGd23(FeCo)77 after exposure to a single laser pulse of duration = 1.4 ps (bottom-right inset) \nand = 1.5 ps (top-left inset). Deterministic AOS is clearly observed in the former, whereas \nthe latter displays a random spatial distribution of magnetic domains i.e. demagnetization. \nFurther measurements showed that pulse durati ons below and above 1.4 ps always result in \ndeterministic AOS and demagnetization resp ectively, and thus we conclude that \nGd23(FeCo)77 possesses a critical threshold c = 1.4 ps whereby deterministic AOS is enabled \nif < c but is disabled if > c. \nFig. 3 The critical pulse-duration threshold c is plotted (red circles) as a function of alloy \ncomposition for Gdx(FeCo)100-x, measured using pulses of photon energy 1.55 eV. \nDeterministic AOS is achieved if < c, but disabled if > c. Experimentally we could \nonly realize ≤ 6 ps, and so can only conclude that c > 6 ps for x 26. Also shown are \nthe calculated values of c for the alloy GdxFe100-x (blue squares). Insets: Typical \nbackground-corrected magneto-optical imag es, of side length 100 µm, obtained for \nGd23(FeCo)77 showing deterministic AOS (bottom-right panel, = 1.4 ps) and \ndemagnetization (top-left inset, = 1.5 ps). The contrast in the images is proportional to \nthe out-of-plane component of magnetization. \nWe repeated the measurements shown in the insets of Fig. 3 for each Gdx(FeCo)100-x \nalloy, and presented in Fig. 3 are the corresponding thresholds c as a function of x. Clearly, \nas the percentage of th e slave gadolinium in Gdx(FeCo)100-x increases, the pulse duration still \ncapable of enabling deterministic AOS increases monotonically. When x 26, we were \nunable to identify the threshold which exceeded 6.0 ps (a limit imposed by our regenerative \namplifier). However, in Ref. [13], c = 15 ps for x = 27.5, which is in good agreement with \nthe implications of our results. Using the calcula tions, we obtain the same linear trend, taking \nin to account that thermal fluctuat ions disable deterministic AOS if SFe and SGd cross zero \nalmost simultaneously25. These findings are clearly in excellent agreement with our \n9\nconceptual understanding, dem onstrating the deep physical insight one can obtain by \nconsidering AOS trajectories across the SA-SB plane. \nA fundamental assumption of our model lies in our use of the concept of \n“temperature”. Temperature can be asso ciated with equilibrium phenomena only26, but it is \nroutinely used in descriptions of non-equilibrium magnetization dynamics3,9,10-11,16. An \noptical excitation of high photon energy 1.55 eV stimulates a multitude of intra- and inter-\nband electronic excitations, causing the temperat ure of the spins to become poorly defined13. \nThe importance of these high-energy excitations in the effectiveness of the demagnetization process was also a subject of recent theoretical debate\n27-28. As an efficient and fast \ndemagnetization is an essential prerogative fo r switching in our model, we can provide a \ndirect experimental answer to this problem by considerably reducing th e photon energy of the \noptical excitation. We therefore use pulses in the mid-infrared spectral range at FELIX (Free \nElectron Lasers for Infrared eXperiments)29-30. A single optical pulse, with photon energy \nranging between E = 70 meV and E = 230 meV, is focused to a spot of diameter 100 µm31 on \nthe surface of the GdFeCo samples. The durati on of the pulse is controlled through cavity \ndesynchronization32, allowing the latter to be varied between 400 fs and at least 6.5 ps33. \nFigure 4 shows the experimentally-measured state map for Gdx(FeCo)100-x with \nx = 24, while the state maps for x = 25 and x = 26 are provided34 in Supplemental Note 6. In \nthese maps, we summarize how the dete rministic character of AOS in Gdx(FeCo)100-x alloys \ndepends on the photon energy and pulse-dura tion, obtained through analyzing magneto-\noptical images recorded af ter exposing the material to consecutive optical pulses34. For all the \nstudied compositions of Gdx(FeCo)100-x, we generally observed that the photon energy, \ndespite being adjusted by a f actor of more than 20 (betwe en 70 meV and 1.55 eV), always \nenabled deterministic AOS provided the pulse duration was sufficiently low. This result \nvalidates both the microscopic picture of u ltrafast demagnetization advanced by Schellekens \net al in Ref. [28] and the invocation of temper ature in our model. Mo reover, these results \nconfirm that relatively gentle heating of the free electrons in GdFeCo is sufficient to achieve the necessary strongly non-equilibrium st ate required for deterministic AOS. 10\n \nFig. 4 State map recorded for Gd24(FeCo)76 indicating how the switching process depends on the \nphoton energy and pulse duration. Points indi cated with a blue circle or red triangle \ncorrespond to observations of deterministic AOS or demagnetization respectively, whereas \ngreen triangles correspond to observations of both effects arising from jitter in the pulse duration. \n In summary, we have revealed a new conceptual understanding of the mechanism \nunderpinning deterministic AOS. We base our description on there being a master/slave \nrelationship between the constitu ents of a generic ferrimagnet AB, where A (the master) is \nferromagnetic and B (the slave) is paramagnetic in isolation. Deterministic AOS can be \nachieved through two distinct pathways, e ither by angular momentum flowing from A and B \nto the external bath or through angular mome ntum being transferred from the slave to the \nmaster. The choice of which pathway is follo wed depends solely on the pulse duration \nrelative to the timescales of the spin-lattice an d inter-sublattice exchange interactions, and \nincreasing the concentration of slaves in AB also increases the pulse duration that can still \nenable deterministic AOS. We use a phenomenol ogical mean field approach to validate our \nunderstanding, and provide ultimate proof by studying how the critical pulse-duration \nthreshold (above/below which deterministic AOS is disabled/enabled) evolves as a \nconcentration of the slave in GdFeCo alloys. Moreover, by demonstrating that mid-infrared \noptical pulses are capable of re alizing deterministic AOS, we experimentally show that the \n11\nthree-temperature model offers a valid descri ption of magnetization dynamics, provided that \nsuitable discrimination is made between the spin-reservoirs of A and B. We believe our \nconceptual understanding resolves many cont roversies surrounding deterministic AOS, and \ncould be deployed to understand how deterministic AOS can be achieved in a larger class of \nmaterials. \nAcknowledgements \nThe authors thank S. Semin, T. Toonen and all technical staff at FELI X for technical support. \nThis research has received funding from the European Union’s Horizon 2020 research and \ninnovation program under FET-Open Grant Agre ement No. 713481 (SPICE), de Nederlandse \nOrganisatie voor Wetenschappelijk Onderzoe k (NWO), the project TEAM/2017-4/40 of the \nfoundation for Polish Science, and the Grant-in -Aid for Scientific Research on Innovative \nArea, “Nano Spin Conversion Science” (Grant No. 26103004). \n\n1 C. P. Bean and J. D. Livingston, “Superparamagnetism.” J. Appl. Phys. 30, S120 (1959). \n2 C. D. Stanciu et al. “All-Optical Magnetic Recording with Circularly Polarized Light.” Phys. Rev. \nLett. 99, 047601 (2007). \n3 T. A. Ostler et al. “Ultrafast heating as a sufficient stimulus for magnetization reversal in a \nferrimagnet.” Nature Comms. 3, 666 (2012). \n4 C.-H. Lambert et al. “All-optical control of ferromagnetic thin films and nanostructures.” Science \n345, 1337 (2014). \n5 S. Mangin et al. “Engineered materials for all-optical helicity-dependent ma gnetic switching.” \nNature Mat. 13, 286 (2014). \n6 M. L. M. Lalieu, M. J. G. Peeters, S. R. R. Haenen, R. Lavrijsen and B. Koopmans, “Deterministic \nall-optical switching of synthetic ferrima gnets using single femtosecond laser pulses.” Phys. Rev. \nB 96, 220411 (2017). \n7 E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, “Ultrafast Spin Dynamics in \nFerromagnetic Nickel.” Phys. Rev. Lett. 76, 4250 (1996). \n8 A. R. Khorshand et al. “Role of Magnetic Circular Dichrois m in All-Optical Magnetic Recording.” \nPhys. Rev. Lett. 108, 127205 (2012). \n9 I. Radu et al. “Transient ferromagnetic-like state mediating ultrafast reversal of \nantiferromagnetically coupled spins.” Nature 472, 205 (2011). \n10 J. H. Mentink et al. “Ultrafast Spin Dynamics in Multisublattice Magnets.” Phys. Rev. Lett. 108, \n057202 (2012). \n11 S. Wienholdt, D. Hinzke, K. Carva, P. M. Oppe neer and U. Nowak, “Orbital-resolved spin model \nfor thermal magnetization switching in rare-earth-based ferrimagnets.” Phys. Rev. B 88, 020406 \n(2013). \n12 D. Steil, S. Alebrand, A. Hassdenteufel, M. Cinchetti and M. Aeschlimann, “All-optical \nmagnetization recording by tailoring optical excitation parameters.” Phys. Rev. B 84, 224408 \n(2011). \n12\n \n13 J. Gorchon, R. B. Wilson, Y. Yang, A. Pattabi, J. Y. Chen, L. He, J. P. Wang, M. Li and J. Bokor, \n“Role of electron and phonon temperatures in th e helicity-independent all-optical switching of \nGdFeCo.” Phys. Rev. B 94, 184406 (2016). \n14 O. Eriksson et al. “Atomistic Spin Dynamics: Foundations and Applications.” Oxford University \nPress, New York (2017). \n15 E. Iacocca et al. “Spin-current-mediated rapid magnon loca lisation and coalescence after ultrafast \noptical pumping of ferrimagnetic alloys.” Nature Comms. 10, 1756 (2019). \n16 B. Koopmans et al. “Explaining the paradoxical diversity of ultrafast laser-induced \ndemagnetization.” Nature Mater. 9, 259 (2010). \n17 Note that we refer interchangeably to magnetization M and angular momentum S, but these \nquantities are related via S M where γ is the gyromagnetic ratio. \n18 J. H. Mentink, “Magnetism on the timescale of the exchange interaction: explanations and \npredictions” Ph.D. thesis , Radboud University Nijmegen (2012). \n19 See Supplemental Note 1 at [url inserted by publis her] for a full description of the model used, and \nthe thermal and material paramete rs adopted in our calculations. \n20 See Supplemental Note 2 at [url inserted by pub lisher] for a brief discussion of the opposite polarity \nof the transient ferromagnetic state, and the underlying time-resolved calculations used to \nconstruct Fig. 2 (b). \n21 C. E. Graves et al. “Nanoscale spin reversal by non-local angular momentum transfer following \nultrafast laser excitation in ferrimagnetic GdFeCo.” Nat. Materials 12, 293 (2013). \n22 U. Atxitia, J. Barker, R. W. Chantrell, and O. Chubykalo-Fesenko, “Controlli ng the polarity of the \ntransient ferromagneticlike state in ferrimagnets.” Phys. Rev. B 89, 224421 (2014). \n23 R. Chimata et al. “All-thermal switching of amorphous Gd-Fe alloys: Analysis of structural \nproperties and magnetization dynamics.” Phys. Rev. B 92, 094411 (2015). \n24 See Supplemental Note 3 at [url inserted by publi sher] for details of the specific GdFeCo alloys \nstudied. \n25 See Supplemental Note 4 at [url inserted by publisher] for calculated phase maps obtained using \ndifferent threshold values of angular momentum, us ed to take in to account thermal fluctuations \nthat dominate (and obstruct deterministic AOS) when SFe and SGd almost simultaneously cross \nzero. \n26 A. Puglisi, A. Sarracino and A. Vulpiani, “Tempe rature in and out of equilibrium: A review of \nconcepts, tools and attempts.” Phys. Rep. 709, 1 (2017). \n27 K. Carva, M. Battiato and P. M. Oppeneer, “ Ab Initio Investigation of the Elliott-Yafet Electron-\nPhonon Mechanism in Laser-Induced Ultrafast Demagnetization.” Phys. Rev. Lett. 107, 207201 \n(2011). \n28 A. J. Schellekens and B. Koopmans, “Com paring Ultrafast Demagnetization Rates Between \nCompeting Models for Finite Temperature Magnetism.” Phys. Rev. Lett. 110, 217204 (2013). \n29 D. Oepts, A. F. G. van der Meer and P. W. va n Amersfoort, “The free-electron-laser user facility \nFELIX.” Infrared physics & technology 36, 297 (1995). \n30 G. M. H. Knippels and A. F. G. van der Meer, “FEL diagnostics and user control.” Nucl. Instrum. \nMethods Phys. Res. 144, 32 (1998). \n31 J. M. Liu, “Simple technique for measur ements of pulsed Gaussian-beam spot sizes .” Opt. Lett. 7, \n196 (1982). \n13\n \n32 R. J. Bakker, D. A. Jaroszynski, A. F. G. van der Meer, D. Oepts and P. W. van Amersfoort, “Short-\npulse effects in a free-electron laser.” IEEE J. Quantum Electron. 30, 1635 (1994). \n33 See Supplemental Note 5 at [url inserted by publisher] for a brief description of how we calculate \nthe duration of the mid infra-red pulses. \n34 See Supplemental Note 6 at [url inserted by pub lisher] for the state maps measured for the GdFeCo \nsamples with different concentrations of gado linium, and exemplary images showing the three \nprocesses that are triggered by the pulses (deterministic AOS, demagnetization, and a mixture of the latter two). " }, { "title": "1806.06334v1.Skyrmion_Formation_Induced_by_Antiferromagnetic_enhanced_Interfacial_Dzyaloshinskii_Moriya_Interaction.pdf", "content": "Skyrmion Formation Induced by Antiferromagnetic -enhanced Interfacial Dzyaloshinskii Moriya \nInteraction \nAuthor : Marco Chung Ting Ma1, Yunkun Xie2, Howard Sheng3, S. Joseph Poon1, and Avik Ghosh2 \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 22 030 USA \nAbstract \nNeél skyrmions originate from interfacial Dzyaloshinskii Moriya interaction (DMI) . Recent \nstudies have explored using ferromagnet to host Neél skyrmions for device applications. \nHowever, challenges remain to reduce the size of skyrmion to near 10 nm. Amorphous rare -\nearth -transitional -metal ferrimagnet s are attractive alternative material s to obtain ultrasmall \nskyrmion s at room temperature . Their intrinsic perpendicular magnetic anisotropy and tunable \nmagne tization provides a favorable environment for skyrmion stability. In this work , we employ \natomistic stochastic Landau -Liftshitz -Gilbert (LLG) algorithm to investigate skyrmion s in GdFe \nwithin the interfacial DMI model . Despite the rapid decay of DMI away from the interface, small \nskyrmions of near 10 nm are found in thick ~ 5 nm amorphous GdFe film at 300K . We have also \nconsidered three scenarios for the sign of DMI between Gd -Fe pa ir. It is revealed that \nantiferromagnetic coupling in the ferrimagnet plays an important role in enhanc ing the effect of \ninterfacial DMI and to stabilize skyrmio n. These results show that ferrimagnets and \nantiferromagnets with intrinsic antiferromagnetic couplings are appealing materials to host \nsmall skyrmions at room temperature , which is crucial to improve density and energy efficiency \nin skyrmion based devices. \nIntroduction \nMagnetic skyrmions are t opologically protected spin textures. Their potentials in advancing \nmemory density and efficiency have drawn extensive investigation s in recent years1-24. In \nmagnetic materials, skyrmions are stabilized th rough Dzyaloshi nskii Moriya interaction (DMI)25-26. \nDMI is generated by either intrinsic or interfacial effect. Intrinsic DMI arises in non-\ncentrosymmetric crystal , such as B20 alloys , where Bloch skyrmions are found to exist in MnSi \nand FeGe at low temperature13-14. Interfacial DMI originates from interf acial layer with strong \nspin-orbit coupling . Multilayer stacks, such as Ir/Fe/Co/Pt and Pt/Co/Ta , are found to host ~ 50 \nnm Neél skyrmions at room temperature15-16. Several challenges remain in developing skyrmion \nbased memory and logic devices. For example, further reduction in skyrmion sizes is needed to \noptimize skyrmion based devices . However, the stability of small skyrmion at room temperature \nbecomes a problem . Thicker magnetic layers are required to increase stability17-18. For \nferromagnet /heavy metal multilayer stacks, increase in thickness of magnetic layer can lead to \nthe loss of interfacial anisotropy and the reduction of the strength of DMI52-55. Both are critical \nfor skyrmion formations. Moreover, skyrmions Hall effect can provide great challenges on \nmoving skyrmions in electronics devices19-22. To overcome these challenges, one need s to \nexplore more materials. \nAmorphous rare -earth-transitional -metal ( RE-TM) ferrimagnet is one of the pote ntial material s \nto overcome these challenges. Several properties of RE -TM alloys provide favorable environment to host small skyrmion s at room temperature. Their Intrinsic perpendicular \nmagnetic anisotropy (PMA) 27-30 gives a crucial advantage in stabilizing small skyrmion by \nallowing the use of thicker films (~ 5 nm) . However, the effectiveness of interfacial DMI \ndecreases significantly away from the interface52-55. Besides PMA, the magnetization of RE -TM \nalloys vanishes at the compensation temperature31. With near zero magnetization, the skyrmion \nHall effect is vastly reduced . Another advantage of RE -TM alloys is the access to ultrafast \nswitching32-39. Recently, all-optical switching helicity -dependent has been demonstrated in RE-\nTM alloys using a circularly polarized laser32-35. This gives an additional tool to control spins in \nfuture devices . RE-TM alloys have begun to draw interest in the field of skyrmions research . \nLarge skyrmions of ~ 150 nm have been observed in Pt/GdFeCo/MgO23, and skyrmion bound \npairs are found in Gd/Fe multilayers24. Further tuning is needed to reduce the size of skyrmion in \nRE-TM alloys . To guide experiments , numerical model has served as a n important tool , \nespecially for complex systems such as RE-TM alloys34,40 -44. Several methods, such as atomistic \nLandau -Liftshitz -Gilbert (LLG) algorithm34,40 -43 and micromagnetic Landau -Lifshitz -Bloch (LLB) \nalgorithm44, has been employed to provide deeper understanding of magnetic properties in RE-\nTM alloys . \nIn this study , atomistic LLG algorithm34,40 -43 is employed to study properties of skyrmions in GdFe \nwith interfacial DMI . Although the sign of DM I at ferromagnet s/heavy metal interface is well \nstudied44-51, the sign of DMI involved ferrimagnet remains complex . Here, we consider three \nscenario s for the DMI between Gd and Fe (D Gd-Fe). First, the influence of DMI between \nantiferromagnetic pair is excluded by setting it to zero (D Gd-Fe = 0) . Second, DMI between \nantiferromagnetic pair is set to the same sign as DMI between ferromagnetic pair, where DGd-Fe > \n0. Finally, the case of D Gd-Fe < 0 is considered. Furthermore , to incorporate DMI being an \ninterfacial effect, an exponential decay DMI is utilized. Simulation results find that near 10 nm \nskyrmions remain robust in ~ 5 nm GdFe at room temperature . This demonstrates that \ninterfacial DMI remain s prominent in thicker ferrimagnet samples, which is critical in stabilizing \nsmall skyrmions at room temperat ure. \nSimulation Model \nThe classical atomistic Hamiltonian H in Eq. (1) is employed to investigate magnetic textures in \namorphous ferrimagnets . \n \n ∑ \n \n ∑ ( )\n ( ̂) \n \nWhere are the normalized spin at site i, j respectively, are the atomic moment at \nsite i, j respectively . Atomic moment is absorbed into the following constant, is the \nexchange interaction, is the DMI interaction and is the anisotropy. \nand is the external field and demagnetization field respectively. \nOnly nearest neighbor interactions are considered in exchange and DMI interactions. Periodic \nboundary condition is enforced in x and y direction. To find the ground state, spins are evolved \nunder the stochastic Landau -Lifshitz -Gilbert (LLG) Equation as sho wn in Eq. ( 2), and the constant \nparameters used in the simulation are listed in Table 1 . \n \n ( ) \n [ ( )] \nWhere is the gyromagnetic ratio , is the Gilbert damping constant, is the effective \nfield, is the Gaussian white noise term for thermal fluctuation and is the saturation \nmagnetization. \nTo incorporate the amorphous short range order, a n amorphous structure of a 1. 6 nm x 1. 6 nm \nx 1.6 nm box containing 250 atoms is generated from ab initio molecular dynamic s calculations \nby Sheng et al.56. Fig. 1 shows a plot of RE and TM atoms in the amorphous structure. Replicas of \nthis box (32 x 32 x 1) are place d next to each other to expand the simulated sample to 50.7 nm x \n50.7 nm x 1. 6 nm and 256000 atoms. For a 4.8 nm thick sample, replicas of the box are also \nplaced in z -direction, and the total number of atoms is 768000. \nResults and Discussion \nWith ferromagnetic DMI (D Gd-Gd and D Fe-Fe) remains positive, th ree scenarios of antiferromagnetic \nDMI (D Gd-Fe) are considered. Samples of 50.7 nm x 50.7 nm x 1.6 nm are simulated using \natomistic LLG equation from Eq. (2) at 0 K. Fig. 2 shows the equilibrium spin configurations at 0 K. \nFor DGd-Fe = 0 and DGd-Fe < 0, skyrmion ’s radius increase s as DMI increases and be come s stripe at \nlarge DMI value, which behaves similar to a ferromagnet17-18. On the other hand, with DGd-Fe > 0, \nsame sign as DGd-Gd and D Fe-Fe, the trend of skyrmion sizes is somewhat different. At s mall DMI \nvalue, skyrmion’s radius increases as DMI increases. However, at large DMI value, skyrmion’s \nradius decreases as DMI increases, which is different from what observed in DGd-Fe = 0 and DGd-Fe \n< 0, and in a ferromagnet. For a given DMI value, the ra dius of skyrmion is also different for the \nthree scenarios , where smallest skyrmions are found with DGd-Fe > 0, and the largest skyrmions \nare found with DGd-Fe < 0. \nTo understand the intriguing behavior of skyrmion’ s size in ferri magnet, in -plane spin \nconfigurations and the chirality of skyrmion ’s wall are investigated. Fig. 3 summarizes the \nchirality of skyrmion wall at 0 K. With DGd-Gd, DFe-Fe > 0 and DGd-Fe = 0, for Fe sublattice, the spins in \nthe skyrmion’s wall are turning cou nter -clockwise. For Gd sublattice, the spins in the skyrmion’s \nwall are also turning counter -clockwise. This can be explained by the dominance of exchange \ninteraction in the system . Antiferromagnetic coupling s between Gd and Fe align the spins of Gd \nand Fe in nearly antiparallel direction, with small canting due to presence of DMI. Identical \nbehavior is observed with DGd-Gd, DFe-Fe > 0 and DGd-Fe < 0, where spins in both Gd and Fe \nsublattice are turning counter -clockwise across the skyrmion’s wall. With DGd-Gd, DFe-Fe > 0 and \nDGd-Fe > 0, the chirality of skyrmion ’s wall is opposite to what observed in DGd-Fe = 0 and DGd-Fe < 0. \nThe spins in both Gd and Fe sublattice are turning clockwise across the skyrmion’s wall. \nIn order t o determine the reason behind the change in chirality, the total DMI energies of each \nnearest neighbor pair are computed using equilibrium configurations at 0 K. Table 2 summarizes \nthe sign of total DMI energies of different nearest neighbor pair. With DGd-Gd, DFe-Fe > 0 and DGd-Fe \n= 0, the total DMI energy between Gd and Gd pair E DMI(Gd-Gd) and Fe and Fe pair E DMI(Fe-Fe) is \nnegative, and the total DMI energy between Gd and Fe pair E DMI(Gd-Fe) is zero. This means that \nwith DGd-Gd, DFe-Fe > 0, it is energetically favorable for spins to turn counterclockwise across \nskyrmion’s wall. EDMI(Gd-Fe) is zero because DGd-Fe is set to zero. With DGd-Gd, DFe-Fe > 0 and DGd-Fe > \n0, E DMI(Gd-Gd) and E DMI(Fe-Fe) is positive, while E DMI(Gd-Fe) is negative. This implies that it is \nenergetically favorable for Gd -Fe pair to turn clockwise across skyrmion’ s wall, but it is \nenergetically unfavorable for Gd -Gd and Fe -Fe pair to do so . This means that in a ferrimagnet, if the DMI of ferromagnetic pair and antiferr omagnetic pair has the same sign, cancellation of DMI \noccurs because it is preferable for ferromagnetic pair to turn in opposite direction of \nantiferromagnetic pair. With DGd-Gd, DFe-Fe > 0 and DGd-Fe < 0, all three terms E DMI(Gd-Gd), E DMI(Fe-\nFe) and E DMI(Gd-Fe) is negative, so turning counterclockwise is energy favorable for both \nferromagnetic pair and antiferromagnetic pair in a ferrimagnet . These differences in sign of total \nDMI energy also explain the size of skyrmion in all three scenarios . For a give n DMI, skyrmions \nare smallest for DGd-Fe > 0 because cancellation in DMI leads to reduction in DMI effectiveness in \nthe sample . DGd-Fe < 0 scenario has the largest skyrmions because both ferromagnetic and \nantiferromagnetic are contributing to formation of a skyrmion, which means DMI is stronger \noverall. The trend of skyrmion’s radius in DGd-Fe > 0 scenario can also be explained by \ncancellation of DMI between ferromagnetic and antiferromagnetic pair. As DMI becomes larger, \nmore cancellation in DMI leads to s maller skyrmion. Thus, with large DMI, skyrmion decreases \nas DMI increases in the case of DGd-Fe > 0. \nTo determine the viability of using RE -TM alloys for skyrmion devices, simulations are also \ncarried out at 300 K. Samples of 50.7 nm x 50.7 nm x 4.8 nm are simulated using atomistic \nstochastic LLG equation in Eq. (2) . Since DMI is known decay away from the interface52-55, an \nexponential decay DMI is employed in the simulation. Fig. 4 shows the functional form of \nexpo nential decay DMI used in the simulation. In this model, DMI remains constant within 5 Å of \nthe top and bottom interface , and start to decay exponential at 5 Å away from the interface. \nFig. 5 summarizes the results of equilibrium spin configuration at 300 K. only ferrimagnetic \nstates are observed with DGd-Fe > 0. As discussed earlier, with DGd-Fe > 0, cancellation of DMI \nbetween ferromagnetic and antiferromagnetic pair leads to unfavorable conditions for skyrmion \nformation. For the case of DGd-Fe = 0 and DGd-Fe < 0, s mall skyrmions of near 10 nm are found in \nthe case of DGd-Fe = 0 and DGd-Fe < 0. Skyrmions this small are very promising for improving \ndensity and efficiency in skyrmion based devices. Fig. 6 shows a comparison between atomistic \nsimulation of GdFe and micromagnetic simulation of an equivalent ferromagnet. Using the same \nexponential decay DMI, m uch larger interfacial DMI is required to obtain skyrmion in the \nmicromagnetic simulation of an equivalent ferromagnet. This demon strates that internal spin \nstructure in a ferrimagnet is essential to prolong the effect of DMI away from the interface. This \nDMI robustness in a ferrimagnet can be explained by the antiferromagnetic coupling between \nthe two sublattices. Even without the p resence of DMI, the spins in the Gd sublattice are known \nto be canted at room temperature31. With the presence of DMI, the spins in the Gd sublattice \nare easily guided by DMI and leads to formation of skyrmions. Thus, antiferromagnetic couplings \ncan help to extend the influence of DMI, and increase stability of small skyrmions at room \ntemperature. \nConclusions \nEffect of interfacial DMI is investigated in amorphous ferrimagnetic GdFe using atomistic \nstochastic LLG algorithm. Three scenarios for the sign of DMI between Gd and Fe are considered. \nIt is revealed that for a ferrimagnet, if the DMI between ferromagnetic pair and \nantiferromagnetic pair has the same sign, it lead s to cancellation in DMI , and it is unfavorable \nfor skyrmion formations. If the DMI between ferromagnetic pair and antiferromagnetic pair has \nopposite sign, it is advantageous for skyrmion formation, and small skyrmions of near ~10 nm \nare found to be stable at room temp erature with exponential decay DMI. The antiferromagnetic \ncouplings in ferrimagnet are uncovered to help extend the influence of DMI in thicker sample s \nof ~ 5 nm . 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Nature 439, 419 -425 (2006). \n \nParameter Value \nGyromagnetic ratio (ϒ) 2.0023193 Gilbert Damping (α) 0.05 \nGd moment ( μGd) 7.63 μB \nFe moment ( μFe) 2.217 μB \nGd-Gd exchange constant (J Gd-Gd) 1.26 x 10-21 J \nFe-Fe exchange constant (J Fe-Fe) 3.82 x 10-21 J \nGd-Fe exchange constant (J Gd-Fe) -1.09 x 10-21 J \nTable. 1 Values of parameters used in the simulation. \n \nFigure 1 Amorphous structure of RE 25TM 75 from ab initio molecular dynamics calculations . Red \natoms are rare -earth, and blue atoms are transitional -metal. \nResults and Discussion \n \nFigure 2 Equilibrium spin configurations for various DMI (uniform DMI) at 0K for three \nscenarios of D Gd-Fe. Parameters used here are listed in Table 1 with Magnetic field is 0.01 T, \nanisotropy energy K is 0.3 x 105 J/m3(distributed within a 45 degree cone), and simulation space \nis 50.7 nm x 50.7 nm x 1.6 nm. For DFe-Fe = 0.25 x 10-22 J, DGd-Gd = 2.89 x 10-22 J, and | DGd-Fe| = 0 or \n0.85 x 10-22 J. For DGd-Fe = 0 and DGd-Fe < 0, the skyrmions size increases with DMI. On the other \nhand, for DGd-Fe > 0, the skyrmion size first increases with DMI at small DMI, then decreases at \nlarger DMI value. \n \nFigure 3 Skyrmions wall chirality for three scenarios of D Gd-Fe. For DGd-Fe = 0 and DGd-Fe < 0, the \nskyrmions wall is rotating counter -clockwise. On the oth er hand, for DGd-Fe > 0, the skyrmion wall \nis rotating clockwise. \n \nScenario EDMI(Gd-Gd) EDMI(Fe-Fe) EDMI(Gd-Fe) \nDGd-Gd, DFe-Fe > 0, DGd-Fe = 0 - - 0 \nDGd-Gd, DFe-Fe > 0, DGd-Fe > 0 + + - \nDGd-Gd, DFe-Fe > 0, DGd-Fe < 0 - - - \nTable. 2 Sign of total DMI energy E DMI computed from equilibrium spin configurations at 0 K. \n \nFigure 4 Plot of exponential decay DMI as function of distance from bottom interface (z) . In \nthis model, DMI remains constant within 5 Å of the top and bottom interface, as indicated by \nthe red line. At the center of the 4.8 nm sample, the strength of DMI decays exponentially as \nshown. \n \nFigure 5 Equilibrium spin configurations for various DMI (exponential decay DMI) at 300K for \nthree scenarios of D Gd-Fe. Parameters used here are listed in Table 1 with Magnetic field is \n0.01 T, anisotropy energy K is 0.3 x 105 J/m3(distributed within a 45 degree cone), and simulation \nspace is 50.7 nm x 50.7 nm x 4.8 nm. For DFe-Fe = 0.25 x 10-22 J, DGd-Gd = 2.89 x 10-22 J, and | DGd-Fe| = \n0 or 0.85 x 10-22 J. At 300K with exponential decay DMI, skyrmions are only found to exist with \nDGd-Fe = 0 and DGd-Fe < 0. \n \nFigure 6 Equilibrium spin configurations from atomistic simulation of GdFe at 300 K (left) and \nmicromagnetic simulation of an equivalent ferromagnet at 0 K (right) . \n \n" }, { "title": "1911.07467v1.Particle_size_controlled_magnetic_loss_in_magnetite_nanoparticles_in_RF_microwave_region.pdf", "content": "1 \n Particle size controlled magnetic loss in magnetite nanoparticles in RF-microwave region \nMudra Jadav, and S P Bhatnagar* \nDepartment of Physics, Maharaja Krishnakumarsinhji Bhavnagar University, Bhavnagar -364001 , India \n*E-mail: spb@mkbhavuni.edu.in \nReceipt date: 15th Nov 2019 \nPacs: 52.70Ds; 75.50.Mm; 75.50.Tt; 78.70.Gq \n \nAbstract: Frequency dependant complex magnetic \npermeability is used to understand RF -microwave \nbehaviour of magnetic nanoparticles in the frequenc y \nrange 250 MHz to 3 GHz. The stable dispersions of \nFe3O4 nanoparticles with mean size varying between 11 \nto 16 nm are prepared for this purpose. The effect of \nmean particle size and external static magnetic field \nover microwave absorption properties of ma gnetic \nfluid is studied. It is observed that frequen cy of \nferrimagnetic resonance ( 𝑓𝑟𝑒𝑠), frequency of maximum \nabsorpt ion ( 𝑓𝑚𝑎𝑥), loss tangent ( tan𝛿) and reflection \nloss ( 𝑅𝐿) can be controlled by modifying mean particle \nsize and strength of applied external static magnetic \nfield. This kind of study can be useful for radio-\nmicrowave devices like tunable attenuator, EM \nsheilder , and other applications like Hyperthermia. \nI. INTRODUCTION \nMagnetic fluids [1] are colloidal suspension of \nferro/ferrimagnetic nanoparticles coated with a \nsurfactant layer. Magnetic fluid is a smar t material \nwhich responds to external magnetic field along with \nits fluid like properties. Magnetic fluids have large \nnumber of technological applications in various fields \n[2] [3] [4]. Some studies on frequency dependence of \ncomplex magnetic permeability and occurance of \nferromagnetic resonance (FMR) are reported in [5 -8] \nfor magnetic fluids. Magnetic fluids are useful for \ndesigning radio -microwave devices due to their \nflexibility in shape and tunability with external \nmagnetic field. Many of the workers hi ghlighted their \npotential for radio -microwave applications such as in \nmodulator [9], Electromagnetic shielding [10], \ncontrolled impedance device [11], insulator device \n[12], nonreciprocal device [13], hyperthermia [14] [15] \n& thermal recovery technique [16 ], and in microwave \nabsorption & shielding using its composites [17][18]. \nMagnetic fluid parameters like particle size, shape, \ncomposition, surfactant, and non -magnetic carrier must be chosen to make them suitable for a particular \napplication. The modifica tios in these parameters can \naffect their properties. Modifications in particle size \nmodify their properties like magnetic [19], rheological \n[20], optical [21] and microwave absorption properties \n[22] [23] of magnetic fluid. In the reports [22 -23], \nresearc hers have measured ferromagnetic Resonance \n(FMR) in magnetic fluids at a fix frequency and applied \nmagnetic field. They have obtained broader linewidth \nand lower resonance field for larger particles in \ncomparision to the smaller ones. The effect of particl e \nsize over FMR and dispersion of resonance field was \nstudied theoratically in [24][25]. In a recent report [26], \nresearchers have studied temperature rise and specific \nabsorption rate (SAR) at 126 kHz for Fe3O4 nanoclusters \nof varying size between 250 nm to 640 nm. They have \nshown that larger extent of temperature rise and SAR \ncan be obtained for the nanocluster having highest \nsaturation magnetization and largest crystallite size. \nIn our previous report [27], the effect of particle \nconcentration, static m agnetic field and it’s orientation \non complex magnetic permeability of magnetic fluid \nwas studied. In this paper, the effect of particle size \nvariation over complex magnetic permeability, \nmicrowave absorption, and reflection loss in magnetic \nfluid is repor ted. Broadband measurements in the \nfrequency range 250 MHz to 3 GHz were carried out in \ncontrast to the fixed frequency measurements \nreported earlier [22,23,26]. The microwave properties \nwere studied as function of frequency as well as \nexternally applied static magnetic field of strength 0 to \n915 Oe. The fluid used was stable dispersion of single \ncore Fe3O4 nanoparticles in contrast to the multi core \nFe3O4 nanoclusters used in [26]. The field strengths \nused were comparable to anisotropy field (H A) while i n \n[22] [23], field strength used were much greater than \nHA. \nII. EXPERIMENTAL 2 \n A. Materials Preparation \nThe magnetite nanoparticles were synthesized by \ncoprecipitation of two salt solutions FeCl 3.6H 2O (SD \nfine chemicals) and FeSO 4.7H 2O (SD fine chemicals) i n \nthe presence of 25% ammonia solution (Merck). \nInitially the mixture of two salt solutions was digested \nfor 30 minutes at constant temperature and pH. The \nnanoparticles were coated with oleic acid (SD fine \nchemicals) surfactant and stabilized in low odor \nkerosene (SD fine chemicals) to prepare magnetic fluid. \nThe magnetic fluid was centrifuged at 8000 RPM for 20 \nmin in order to remove aggregates, if present. Different \npH values were selected at a constant temperature for \npreparing magnetic fluids. These fl uids were labelled as \nMF 1, MF 2, MF 3 and MF 4. Density of all fluids were \n0.91 gm/cc. The complete method of prepar ation is \ndiscussed elsewhere [ 28]. \nB. Methods \nX-ray diffraction (Philips X’pert MPD System) was \nused for structural characterization of pow der samples \nand diffraction data was analyzed by Reitveld \nrefinement using the programme Materials Analysis \nUsing Diffraction (MAUD) . The Transmission electron \nmicroscopy (TEM) (JEOL, JEM 2100) was used to \ndetermine particle size and size distribution. Op en \nsource software ImageJ was used for image analysis. \nThe magnetization measurements were taken using \nsearch coil method. A search coil and compensating \ncoil (2 cm long) were prepared by opposite winding of \n(36 SWG) wire with 500 turns on a nonmagnetic fo rmer \nwith inner diameter of 1 cm. Both of these coils former \nwere kept in an air core solenoid connected to power \nsupply. The differential output from coils was \nmeasured by digital storage oscilloscope (Aplab \nD36025M). A glass tube containing known amount of \nmagnetic fluid was quickly inserted into the search coil. \nThe flux change was observed by peak signal on the \noscilloscope screen. It’s calibration was done using a \nmagnetic fluid with known magnetization and \ncalibration constant was obtained. The peak i ntensities \nfor our sample fluids were converted into \nmagnetization using calibration constant. The \nmagnetization is detrmined as a function of magnetic \nfield . Magnetic field was measured using digital \ngaussmeter with axial hall probe (SES Instruments Pvt. \nLtd. DGM -204) Complex magnetic permeability of \nmagnetic fluid was determined using Vector Network \nAnalyzer (VNA) (Agilent 8714ES) in the frequency range \n250 MHz to 3 GHz. The transmission /reflection technique [ 29] was used to measure the scatteri ng \nparame ters. Nicolson -Ross [ 30] and Weir [ 31] \nalgorithm was used for calculation. A 50 Ω coaxial line \ncell was used as sample holder with 6.5 mm inner \ndiameter and 15 mm outer diameter and 14 mm \nlength. The coaxial line cell is made up of nonmagnetic \nmaterial. VNA was calibrated and checked using known \nstandards and known liquid. Measurements were \ntaken under the static magnetic field with field \nstrength between 0 -915 Oe. The field was produced \nusing an air core solenoid connected to a power supply \nand the sample holder was kept at centre of the \nsolenoid’s core. The schematic diagram of \nexperimental set -up is shown in figure 1. The direction \nof static field was parallel to the cell axis and EM wave \npropagation direction. The blank measurement was \ndone using air fil led sample holder with and without \nexternal static magnetic field and it is confirmed that \nthere is no effect of external magnetic field over the \nsample holder. \nIII. Results and Discussion \nFig.2 shows the X -ray diffraction spectra for the \nnanoparticles re fined using Reitveld refinement \nprogramme . The cubic crystal structure and spinel \nphase is confirmed by the X -ray spectra. For the inverse \nspinel arrangement of Fe3+, Fe2+ and O2- ions in \nmagnetite, O2- ions occupy lattice sites, Fe2+ ions \noccupy octahedra l voids, half of the Fe3+ ions occupy \ntetrahedral voids and the other half occupy octahedral \nvoids. Electron spins of Fe3+ ions at tetrahedral voids \nare aligned antiparallel to the electron spins of Fe3+ ions \nat octahedral voids. The total magnetic moment from \nFe3+ ions is zero. Electron spins of Fe2+ ions are aligned \nparallel to the spins of Fe3+ ions at neighbouring \noctahedral voids. These are responsible for the net \nmagnetization and ferri magnetic nature [ 32] of \nmagnetite. The crystallite size and lattic e parameter \nare found by reitveld fit and listed in table I. The \ndiscrepancy index for reitveld fit can be given by \nweighted profile R -factor (R wp). The R wp is found as \n2.16%, 2.64%, 2.59% and 2.44% for particle samples \nMF 1, MF 2, MF 3 and MF 4 respectiv ely. The lattice \nparameters found are slightly lower than the typical \nvalue 0.839 nm for bulk magnetite [ 33]. Due to the \nlarge surface area, the Fe2+ ions on surface can be \noxidized to form the maghemite layer on the surface. \nThis may be a possible reason for reduction of lattice \nparameter. But the presence of maghemite must be in \nvery low proportion and the corresponding XRD peaks 3 \n Table I Magnetic fluid parameters determined using XRD, TEM and Magnetization measurements. \n \nare not visible. Such reduction in lattice parameter is \nprevi ously reported in [ 34]. The crystallite size is \nsmalle st for MF 1 and largest for MF 4. The crystallite \nsize of magnetite is controlled in our experiment by \ncontrolling synthesis temperature and pH. The effect of \nthese parameters on crystallite size of nanoparticle in \ncoprecipitation method is di scussed in de tail in reports \n[28] [34]. \n \nFig.3 shows TEM images and particle size \ndistribution for all four samples. The size distribution is \nfitted by lognormal distribution and, mean size and \nstandard deviation are listed in table I. The mean size \nfound from TEM ana lysis ( 𝐷𝑇𝐸𝑀) is a physical or \nhydrodynamic size of particles. Fig.4 shows the \nmagnetization measurement data fitted to modified \nLangevin’s theory. The magnetization can be explained \nby Langevin’s theory of paramagnetism (relation 1) for \na monodispersed syst em. As magnetic fluid is a \npolydispersed system, Langevin’s theory is modified to \nconsider particle size distribution as described by \nrelation 2. In modif ied theory, Langevin function \n𝐿(𝑚𝐻\n𝑘𝐵𝑇) is weighted by lognormal size distribution \nfunction 𝐹(𝐷) given in relation 3. The modified \nLangevin theo ry is described in detail in [ 35]. \n \n𝑀\n𝑀𝑆=𝐿(𝛼)=coth 𝛼−1\n𝛼; 𝛼=𝑚𝐻\n𝑘𝐵𝑇;𝑚=𝑀𝑑𝑉 …(1) \n𝑀\n𝑀𝑆=𝐿(𝛼)𝐹(𝐷)𝑑𝐷 …(2) \n𝐹(𝐷)𝑑𝐷=1\n√2𝜋𝜎𝐷𝑒𝑥𝑝 [−(𝑙𝑛𝐷 −𝑙𝑛𝐷0)2\n2𝜎2 ]𝑑𝐷 …(3) \n \nwhere 𝑀𝑆 is saturation magnetization of magnetic \nfluid, 𝑚 is particle magnetic moment, 𝐻 is magnetic \nfield strength, 𝑘𝐵 is boltzman constant, 𝑇 is \ntemeparature, 𝑀𝑑 is saturation magnetization of the \nbulk material, 𝑉 is particle volume, 𝐷0 is mean particle \ndiameter and 𝜎 is standard deviation. The mean \nparticle size, standard deviation and saturation \nmagnetiz ation of magnetic fluid can be found by best \nfitting of the experimental data to the modified theory. \nThe values obtained for fitting parame ters \nsaturation magnetization ( 𝑀𝑆), mean diameter ( 𝐷0≈\n𝐷𝑀𝐴𝐺) and standard deviation ( 𝜎) for size di stribution \nof particles are listed in table I. The sizes D MAG are \nsmaller than 𝐷𝑇𝐸𝑀 as it is size of magnetic core often \ncalled magnetic size of particles. The hydrodynamic \nsize is always greater than the magnetic size of particles \nas it includes th e thickness of coating layer. The \nsaturation magnetization increases with particle size. \nSimilar results are repor ted in [ 28] [34] [36]. \n \nThe c omplex magnetic permeability ( 𝜇∗) has two \ncomponents, real ( 𝜇′) and imaginary ( 𝜇′′) and is given \nby 𝜇∗=𝜇′−𝑖𝜇′′. In the equilibrium state, magnetic \nmoments existing in magnetic fluid are all randomly \noriented. When magnetic fluid is influenced by EM \nwave (radio - microwave), magnetic moments get \npolarized by the magnetic field component of EM \nwave. The 𝜇′ component is a contribution from the \nmagnetization that is in phase with alernating magnetic \nfield and it depends on the extent of magnetic \npolarization . While the 𝜇′′ component is a contribution \nfrom the magnetization that is out of phase with \nalter nating magnetic filed and is related to loss. The \noccurrence of relaxation and resonance is expected. \nThere are two relaxation mechanisms Brownian and \nNeel’s mechanism. The particle to which moment is \nembedded physically rotates in the former case while \nmoment itself rotates inside the particle in the latter \ncase. The relaxation time for both of the mechanisms \nand the effective relaxation time ( 𝜏𝑒𝑓𝑓) can be \ncalculated as, \n Sample \nname XRD TEM Magnetization \n \nParticle \nsize \n(𝐷𝑋𝑅𝐷) \n(nm) Lattice \nparameter \n(nm) Particle \nsize \n(𝐷𝑇𝐸𝑀) \n(nm) Standard \ndeviation \n𝜎 Particle \nsize \n(𝐷𝑀𝐴𝐺) \n(nm) Standard \ndeviation \nσ Saturation \nMagnetization \n𝑀𝑆 (Oe) \nMF 1 10.60 0.8360567 11.86 0.21 10.5 0.26 141 \nMF 2 12.0 0.83753 12.80 0.35 11.72 0.22 145 \nMF 3 16.11 0.8373271 15.36 0.20 12.75 0.22 144 \nMF 4 17.09 0.8352231 16.11 0.22 13.34 0.24 163 4 \n 𝜏𝑒𝑓𝑓=𝜏𝐵𝜏𝑁\n𝜏𝐵+𝜏𝑁 …(4) \n 𝜏𝐵=3𝑉׳𝜂\n𝑘𝐵𝑇 ….(5) \n𝜏𝑁=𝜏0exp(𝜎).𝜎−12⁄ 𝑖𝑓 𝜎≥2 \n =𝜏0σ 𝑖𝑓 𝜎≪1 ….(6) \n \nwhere 𝜏𝑒𝑓𝑓 is effective relaxation time, 𝜏𝐵 is Brownian \nrelaxation time, 𝜏𝑁 is Neel relax ation time, 𝜂 is viscosity \nof carrier liquid, 𝑉′is hydrodynamic volume of particle , \n𝜏0 is precessional damping time (≈10-9sec), 𝜎=\n𝐾𝑉 𝑘𝐵𝑇 ⁄ , 𝐾 is anisotropy constant and 𝑉 is magnetic \nvolume. The effect of particle size on effective \nrelaxation time is discussed in [ 37]. The 𝜏𝑒𝑓𝑓 increases \nwith particle size. At absorption frequency ( 𝑓𝑚𝑎𝑥), 𝜇′′ \nattains a maximum where 𝑓𝑚𝑎𝑥 corresponds to 𝜏𝑒𝑓𝑓. \nThe 𝜇′′ peak signifies occurrence of energy loss \n(absorption), and of ten called as loss peak. In \nequilibrilium, a magnetic moment is oriented in the \ndirection of anisotropy field ( 𝐻𝐴). \n The incidence of EM wave causes a small \ndisturbance and magnetic mo ment starts to preccess \naround 𝐻𝐴. If external magnetic field ( 𝐻𝑒𝑥𝑡).) is applied, \nit will be added to 𝐻𝐴. When the frequency of \nprecessional motion matches with the frquency of EM \nwave, the precession would be continued by absorbing \nenergy from EM wave. It is called f erromagnetic \nresonance which le ads to strong energy absorption in \nthe syste m. At the resonance frequency ( 𝑓𝑟𝑒𝑠), 𝜇′=1. \nThe f res can be given by, \n \n𝑓𝑟𝑒𝑠=𝛾\n2𝜋(𝐻𝐴+𝐻𝑒𝑥𝑡) -(7) \n \nwhere 𝛾 is gyromagnetic rati o of electron, 𝐻𝐴 is \nanisotropy field given by 𝐻𝐴=4𝐾𝑀𝑆⁄ , 𝐻𝑒𝑥𝑡 is external \nstatic magnetic field. The frequency dependence of \ncomplex magnetic permeability for all four fluids in the \nabsence of any external field is shown in fig.5. As the \nfrquency increases, field alters it’s direction much \nfaster and the dipoles remain unresponded. So the \nextent of magnetic polarization and real component \n(𝜇′) decreases with frequency as observed in fig.5 a. As \nthe 𝜇′ component drops, the 𝜇′′ com ponent increases \nwith frequency and attain maximum (fig.5b). The initial \npermeability can be given by, \n \n𝜇𝑖𝑛𝑖=1+𝜒𝑖𝑛𝑖=1+n𝑚23𝑘𝐵𝑇𝜇0 ⁄ -(8) \n \nwhere 𝜒𝑖𝑛𝑖 is initial susceptibility, 𝑚 is magnetic \nmoment ; 𝑚=𝑀𝑆𝑉, 𝑀𝑆 is saturation magnetization, 𝑛 \nis particle number density, 𝜇0 is vacuum permeability . The initial susceptibility is proportional to particle \nvolume. The 𝜇𝑖𝑛𝑖 (@0.25 GHz) is expected t o increase \nwith particle size which c an be observed in our results \nfig.5 a. \n The 𝑓𝑟𝑒𝑠 is observed to be 1.28 GHz, 1.45 GHz, 1.62 \nGHz and 1.99 GHz for MF 1, MF 2, MF 3 and MF 4 \nrespectively ( fig.5). The 𝑓𝑚𝑎𝑥 is observed to be 1.26 \nGHz, 1.30 GHz, 1.42 GHz and 1.45 GHz for M F 1, MF 2, \nMF 3 and MF 4 respectively (fi g.5). Both t he 𝑓𝑟𝑒𝑠 and \n𝑓𝑚𝑎𝑥 increase as the particle size increases in the fluid. \nThe 𝑓𝑟𝑒𝑠 is directly proportional to anisotropy field (𝐻𝐴) \nwhen 𝐻𝑒𝑥𝑡=0 as in relation 7. The 𝑓𝑟𝑒𝑠 is increasing \nwith particle size leads to the possibility that anisotropy \nfield ( 𝐻𝐴)) and so the anisotropy constant ( 𝐾) are also \nincresing with size in the concerned size range. Some \nreports [ 38] [39] say that anisotropy for the \nnanoparticle is not purely of volumetric origin and \ndominated by surface contribution due to large surface \nto volume ratio and , thats why anisoptopy constant for \nnano materials are very often larger than that of bulk \nmaterial. According to tha t anisotropy constant \ndecre ases with particle size for t he nanoparticles . Our \nresults do not follow this approach . Here t he \nanisotropy constant, includes the effects from \nmagnetocrystalline nature, size, shape a nd \ninterparticle interaction [ 40] and is called effective \nanisotropy con stant 𝐾𝑒𝑓𝑓. The nano particle \nsynthesized here are not perfectly spherical, but they \nare slightly elongated which can be observed in TEM \nimages, so shape anisotropy contributes to constant \n𝐾𝑒𝑓𝑓 [41]. The fluid with large mean particle size mu st \nbe having large magnetic interactions between \nparticles. All these factor s contribute to the constant \n𝐾𝑒𝑓𝑓. In previous report [ 42], researchers have \nsuggested that oleic acid molecules covalently bonded \nto the particle surface effectively reduce s the surface \nspin disorder and the anisotropy is dominated by \nvolume contribution in oleic acid coated magnetite \nnanoparticles. The effective anisotropy constant ( 𝐾𝑒𝑓𝑓) \nincreases with particle size most probably due to shape \neffects and reduced sur face spin disorder . Our results \nsupport this idea proposed in [ 42]. The increase in \nconstant 𝐾𝑒𝑓𝑓 by increasing particle size is also \nreported in [20] for oleic acid coated magnetite \nnanoparticle. The enhencement in magnetic properties \nand in resona nce effect occurs by increasing particle \nsize. The resonance effect contributi on can be \nresponsible for the rise in loss peak and it’s shifting \ntoward higher frequencies. The theora tical study [24] \nsays that, 𝑓𝑟𝑒𝑠 and 𝑓𝑚𝑎𝑥 should increase as 𝜎 increases \nwhere 𝜎=𝐾𝑉 𝑘𝐵𝑇 ⁄ . Either increasing constant K or 5 \n particle volume 𝑉 can increase 𝜎 and it will lead to \nincreased 𝑓𝑟𝑒𝑠 and 𝑓𝑚𝑎𝑥. The interparticle interactions \nmay contribute to the 𝑓𝑟𝑒𝑠 only. According to a prev ious \nreport [ 43], the interparticle interaction s increase the \nfres considerably while the 𝑓𝑚𝑎𝑥 remains un affected . \nThis can be one possible reason for the large increment \nin 𝑓𝑟𝑒𝑠 and small increment in 𝑓𝑚𝑎𝑥 for the same \nchange in particle size . \n The complex magnetic permeability is determined \nunder the influence of static magnetic field with \nstrength 0 -368 Oe for all four fluids. Results are \npresented in fig.6 and 7. On the application of static \nmagnetic field, magnetic moments tr y to align in the \nfield direction and form chain structures. The magnetic \nfluid is a polydispersed system, having some smaller as \nwell as larger particles compared to the mean size. At \nlower field strength, larger particles will be the first \naffected and will align in chain structures. As the field \nstrength increases, more and more particles align in \nchains and few particles will be left to rotate freely. But \nthe magnetic moments can still overcome the \nanisotropy e nergy barrier ( 𝐾𝑉) and relax via Neel’s \nmechanism without the physical rotation of particles. \nAs the field strength increases, the energy barrier 𝐾𝑉 \nincreases and less magnetic moments are able to cross \nthe barrier and relax via Neel’s mechanism. This will \nlead the extent of magnetic polarizat ion and so the \nmagnetic permeability to decrease with field strength. \nIt can be observed that 𝜇′ and 𝜇′′ decreases with field \nstrength at the lower frequency end in fig.6 and 7 \nrespectively (for 𝐻𝑒𝑥𝑡>60 Oe). \nThe 𝑓𝑟𝑒𝑠 shifts to higher freq uency as the field \nstrength increases in each of the fluids (fig.6). It is \nexpected according to the relation 7. The field profile \nfor MF 4 appears to be much di fferent from MF 1 \n(fig.6). The 𝑓𝑟𝑒𝑠 shifts to 2.03 GHz at 368 Oe of field \nstrength from 1.28 Gz at absence of field in MF 1. While \nit shifts to 2.97 GHz at 368 Oe of field strength from \n1.99 GHz at absence of field in MF 4. For the same \nincrement of field stre ngth (368 Oe), shifting of the 𝑓𝑟𝑒𝑠 \nis larger in MF 4 compared to that in MF 1. The larger \n𝑓𝑟𝑒𝑠 spreadin g bandwidth in MF 4 signifies the larger \nvalue of 𝐻𝐴 and the corresponding constant 𝐾𝑒𝑓𝑓. For \nthe 60 Oe of applied field strength, there is a large rise \nin 𝜇′ curve for fluid MF 1 (fig.6) . While in case of fluid \nMF 4, the rise in 𝜇′ curve is comparatively small for 60 \nOe (fig.6) . When particle is influenced by magnetic field \nstrength comparable or greater than it’s anisotropy \nfield then only the magnetic moment rotates in the \ndirection of external field and aligned to form chain \nstructures. The large rise in the 𝜇′ curve in MF 1 is due to the alignment of moments. The report [44] suggests \nthat size and shape distribution of nanoparticles l eads \nto the wide distribution of anisotropy constant (𝐾𝑒𝑓𝑓) \nand of 𝐻𝐴 in the system. There must be more number \nof particles in MF 1 having 𝐻𝐴 comparable or less than \n60 Oe. In opposite to that in MF 4 there are very less \nnumber of particles having constant 𝐻𝐴 comparable or \nless than 60 Oe. \nThe lo ss peak and 𝑓𝑚𝑎𝑥 shifts to higher frequency as \nthe field strength increases in each of the fluids (fig.7). \nAs the field strength increases, contribution from \nresonance effect increases and the contribution from \nrelaxation decreases. The 𝑓𝑚𝑎𝑥 approches 𝑓𝑟𝑒𝑠 with \nincreasing field streng th. In case of MF 1 and MF 2, 𝜇′′ \npeak amplitude increases with field strength (up to 368 \nOe) (fig.7). While in MF 3 and MF 4, amplitude \nincreases up to 190 Oe and 118 Oe respectively, after \nthat it starts to decrease (fig.7). As the field strength \nincreases, initially the loss peak amplitude increases \nwith field strength because of the presence of aligned \nmagnetic moments. This increment will continue upto \na critical field strength. Beyond that as the fiel d \nstrength increases, barrier 𝐾𝑉 increases and less \nnumber of particles can participate in relaxation. So the \nloss peak amplitude decreases with field strength. The \ncritical field must be higher for MF 1 and MF 2 because \nof smaller mean particle size. Be cause the most fine \nparticles present in fluid will contribute to Neel’s \nrelaxation. \n The magnetic loss tangent (tan𝛿) can be given by, \ntan𝛿=𝜇′���𝜇′⁄ where 𝛿 is a loss angle between two \nmagnetic permeability components . The 𝜇′ and 𝜇′′ \ncompo nents corresponds to the loss less and lossy \nresponses of material respectively. The magnetic loss \ntangent is a ratio of lossy to the loss less response \ninvovled in complex magnetic permeability. It \nrepresents the loss -rate of energy for a dissipating \nsystem when applied energy in form of alternating \nelectromagnetic field [ 45]. The reflection loss ( 𝑅𝐿) is \nthe measure of the energy reflec ted back, can be \ncalculated as, \n \n𝑅𝐿 (𝑑𝐵)=20 𝑙𝑜𝑔|𝑍𝑖𝑛−𝑍0\n𝑍𝑖𝑛+𝑍0| …..(9 ) \n \nDetailed derivation for 𝑅𝐿 is given in [ 46]. For good \nmicrowave absorption properties of a material, it is \ndesirable to have high loss tangent and low reflection \nloss. T he tan𝛿 and 𝑅𝐿 are calculated for the absence o f \nstatic magnetic field for all four fluids and plotted in \nFig.8. The maximum tan𝛿 is attained at 1.44, 1.46, 1.46 6 \n and 1.47 GHz for MF 1, 2, 3 and 4 respectively. The \nmaximum tan𝛿 increases with mean particle size and is \nlargest for MF 4. The minim um 𝑅𝐿 is attained at 1.83, \n1.87, 1.87 and 1.94 GHz for MF 1, 2, 3 an d 4 \nrespectively. The minimum 𝑅𝐿 decreases with mean \nparticle size and the lowest for MF 4. The maximum \nloss tangent and minimum reflection loss is achieved in \nMF 4 due to the maximum complex magnetic \npermeability as a result of larger mean particle size. \nThe reflection loss is tabulated in table II for \nfrequencies 1, 2 and 3GHz for four fluids in the absence \nof external magnetic field. Reduction in RL due to the \nparticle size increment is largest at 3 GHz. \n \nTable II Reflection loss (RL) for four magnetic fluids \nin the absence of external magnetic field. \n \nSample \nname 𝑅𝐿(𝑑𝐵) \n f=1GHz f=2GHz f=3GHz \nMF 1 -1.17116 -1.99704 -0.91576 \nMF 2 -1.27275 -2.35231 -1.6774 \nMF 3 -1.19656 -2.379 67 -1.6818 \nMF 4 -1.33089 -2.80922 -2.26055 \n \n \n The tan𝛿 and 𝑅𝐿 are also calculated for the influence \nof static magnetic field of strength 0-915 Oe. The \nresults for tan𝛿 and RL are shown in figure 9 and 10 \nrespectively. The frequency and field dependence of \ntan𝛿 is similar to the 𝜇′′ component. The maximum \ntan𝛿 increases with field strength up to a critical \nstrength and then decreases with field strength. The \nminimum 𝑅𝐿 decreases as the field strength increases, \nafter a certain field strength, it seems that the minima \nis shifted to a higher frequency beyond our \ninstrumental range. It can be observed that for MF 4, \n𝑅𝐿 <-3dB in the approximate range 2.2 -3 GHz at field \nstrength 510 Oe. According to the relationship \nbetween relection lo ss and a bsorbed energy suggested \nin [ 47], when 𝑅𝐿 < -3dB, almost 50% of energy is \nabsorbed in the system. \n From the results it is clear that, this kind of fluid can \nbe used as wide bandwidth absorber. At a pa rticular \nfrequency, tan𝛿 and 𝑅𝐿 can be fine -tuned by \ncontrolling the field strength. \n The 𝑓𝑟𝑒𝑠, 𝑓𝑚𝑎𝑥, maxim um tan𝛿 is observed to \nincrease by 55.6%, 15% and 25.2% respectively and \nminimum 𝑅𝐿 is observed to decrease by 34.5 % by \nincreasing the mean size of Fe 3O4 nanopa rticles from \n11.8 nm to 16.1 nm in magnetic fluid. IV. Conclusion \nThe Magnetic fluids having Fe 3O4 nanoparticles of \nvarying mean size between 11 to 16 nm have been \nsynthesized using chemical co -precipitation method. \nThe frequency dependant complex magnetic \npermeability is reported for these four Magnetic fluids \nin the frequency range 250 MHz to 3 GHz. The initial \npermeability and frequency dependent complex \npermeability increases by increasing particle size in the \nfluid. The ferr imagnetic resonance frequenc y (𝑓𝑟𝑒𝑠), \nabsorption frequency ( 𝑓𝑚𝑎𝑥) and loss tangent (tan δ) \nincreases while reflection loss ( 𝑅𝐿) decreases with \nincreasing mean particle size in the fluid. Increasing \nparticle size leads to interparticle interactions and \nanisotropy energ y (KV) to increase which is responsible \nfor these results. The field dependence of these \nproperties have also been studied. By controlling the \nmean particle size and strength of static magnetic field, \nit is possible to fine tune the frequency of resonance \nand maximum absorption, reflection loss, absorption, \nand other dielectric properties of magnetic fluid which \nare usually desirable in radio -microwave devices and \nother applications like Hyperthermia. \nACKNOWLEDGEMENTS \nMJ acknowleges Inspire Fellowship Pr ogramme \n(IF140928), Department of Science and technology \n(DST), New Delhi for financial support. Authors thank \nCSIR -Central Salt and Marine Chemicals Research \nInstitute (CSMCRI), Bhavnagar for providing X -ray \ndiffraction and TEM analysis facility. \n \nREFERENCES \n[1] R. E. 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Batlle, Journal of Magnetism and Magnetic \nMaterials 316, e756 (2007). \n[43] P. C. Fannin , C. N. Marin, and I. Malaescu, J. \nPhys.: Condens. Matter 15, 4739 (2003). 8 \n [44] A. A. Mcghie, C. Marquina, K. O’Grady, and G . \nVallejo -Fernandez , J. Phys. D: Appl. Phys. 50, \n455003 (2017). \n[45] M. Gr een and X. Chen, Journal of Materiomics \n(In press) . doi:10.1016/j.jmat.2019.07.003 [46] Y. B. Feng , T. Qiu, and C. Y. Shen, Journal of \nMagnetism and Magnetic materials 318, 8 (2007). \n[47] S. M. Lee, International encyclopedial of \ncomposites (VHC publishers, New York , 2015) , p. \n404-430. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 9 \n Figure. 1 Schematic of experimental set -up. \n \nFigure.2 X -ray diffraction results for four samples of Fe 3O4 powder. \n \n \n \n \n \n10 \n \n \nFigure.3 Transmission Electron microscope (TEM) images for four samples of Fe 3O4 powde r. \n \n \n \nFigure.4 Magnetization measurement for four magnetic fluids. Hollow symbols represent experimental data \npoints and lines represent best fit to modified Langevin theory. Inset shows lognormal particle size distribution \nin four fluids found from theor y fit. \n \n \n11 \n Figure.5 (a) The real and (b) imaginary components of complex magnetic permeability is plotted with frequency \nfor four magnetic fluids in the absence of static field. \nFigure.6 The frequency dependence of real component of complex magnetic permea bility of MF 1, MF 2, MF 3 \nand MF 4 in the presence of static magnetic field with strength between 0 -368 Oe. \n12 \n \n \nFigure.7 The frequency dependence of imaginary component of complex magnetic permeability of MF 1, MF 2, \nMF 3 and MF 4 fluids in the presence o f static magnetic field with strength between 0 -368 Oe. \n \n \nFigure.8 (a) Loss tangent and (b) Reflection loss is plotted with frequency for four fluids in the absence of static \nfield. \n13 \n \n \nFigure.9 Frequency dependence of loss tangent (tan δ) for MF 1, MF 2, MF 3 and MF 4 fluids in the presence of \nstatic magnetic field with strength between 0 -915 Oe. \n \n \nFigure.10 Frequency dependence of reflection loss (RL) for MF 1, MF 2, MF 3 and MF 4 fluids in the presence \nof static magnetic field with strength between 0 -915 Oe. \n" }, { "title": "1704.04890v1.Strain_magneto_optics_of_a_magnetostrictive_ferrimagnet_CoFe2O4.pdf", "content": "Strain -magneto -optics of a magnetostrictive ferrimagnet \nCoFe 2O4 \n \nYu.P. Sukhorukov, A.V. Telegin , N.G. Bebenin, A.P. Nosov, V.D. Bessonov, A.A. Buchkevich \nM.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences , \nKovalevskaya str. 18, 620990 Yekaterinburg, Russia \n \nWe experimentally demonstrate that in magnetostrictive ferrimagnet ic single crystal of CoFe 2O4 \nthere is clear correlation between magnetostriction and magnetoreflecti on of unpo larized light in \nthe infrared range. The influence of magnetic field on specular reflection is likely to be indirect: \napplication of a magnetic field results in strong strain and deformation of the crystal lattice, \nwhich leads to the change in electron energy structure and hence reflecti on spectrum. \n \nI. INTRODUCTION \n \nOptical reflection and absorption in magnetics are sensitive to magnetic field and \ndeformation, which makes the optical measurements an effective tool for investigating \nelectronic, magnetic, and lattice subsystems as well as interactions between them. This \nsensitivity is exploited in many devices. \nA special part of solids is magnetostrictive materials in which application of a magnetic \nfield gives rise to large strain and deformation because of strong coupling of magnet ic mome nts \nwith crystal lattice. The electronic and optical properties of these materials can be more sensitive \nto magnetic field comparing to solids with weak magnetostri ction because magnetic field and \nstrain work together. \nMagnetooptical phenomena associated with magnetoelastic interactions are investigated \nby many authors (see e.g. [1, 2] and references therein) . This part of the solid state physics can \nbe called “strain -magneto -optics”. The effects mentioned were studied in polarized light. It is \nreasonable to find whether such effects can be observed in natural (unpolarized) light because \nthe use of natur al light is much easier than polarized light. In this work, we show that \nmagnetoelastic contribution clearly manifest s itself in magnetoreflection of natural light from \nferrimagnetic spinel of CoFe 2O4 in the infrared ( IR) spectral region . To our best knowledge, the \ninfluence of magnetostriction on magnetoreflection has not been observed yet . \nWe have chose n CoFe 2O4 (Curie temperature is TC = 812 K) as an object of investigation \nfor the following reasons. Firstly, this spinel is an insulator and therefore the interaction of light \nwith free charge carriers is absent. Secondly, CoFe 2O4 is highly transparent in the IR spectr al range . Finally, in CoFe 2O4 the magnetostriction constants λ100 is about 6×10-4 at room \ntemperature i.e. the magnetostriction of CoFe 2O4 is very strong [3]. \n \nII. SAMPLES AND EXPERIMENTAL TECHIQUE \n \nThe CoFe 2O4 single crystal was grown by floating zone melting with radiation heating. \nThe value of cubic crystal lattice parameter (a0 = 8.38 Å) defined from X -ray diffraction data \nwas found to be close to the data reported in [4]. Correspondence of chemical composition to \nCoFe 2O4 per formula unit was con firmed by X -ray microanalysis. The electrical resistivity was \nfound to be of above 105 Ω·cm at room temperature . Magnetization data were obtained with the \nhelp of Lake Shore 7400 vibrating sample magnetometer . The magnetos trictive properties were \ncharacterized by strain gauge technique using the (001) oriented plate -shaped samples with in -\nplane typical dimensions of 10×10 mm2 and thickness of d = 400 μm. The reflection coefficient \nR measurements were carried out using the plate -shaped samples with the same crystallographic \norientation but smaller typical dimensions of 4×4×0.22 mm3. The specular reflection coefficient \nwas calculated as R = Is/IAl, where Is and IAl are the intensit ies of the unpolarized light reflected \nfrom a sample and the Al mirror , respectively . Magnetoreflection is defined here as \nR/R = (R(H) - R(0))/R(0), where R(H) and R(0) are the values of reflection coefficient in an \nexternal magnetic field H and in zero magnetic field, respectively. The values of R and ∆R/R \nwere measure d at angles close to normal incidence of the light in the infrared spectral range from \n0.8 to 30 μm at a temperature of T = 295 K with the relative error of 0.2%. In all cases, a \nmagnetic field was applied in -plane to the sample surfa ce. \n \nIII. EXPERIMENTAL RESULTS \n \nA. Magnetization M(H) and magnetostriction (Δ l/l)100. \n \nThe magnetic field dependences of magnetization and magnetostriction are shown in Fig. \n1(a). At T = 295 K, the value of coercive field (Hc = 80 Oe) determined from hysteresis loops is \nclose to one reported in [4] for high-quality single crystal s of the same composition . When \nH||[100] and H||[010 ] a technical saturation is observed at H ≈ 0.9 kOe. If magnetic field is \nhigher the magnetization increase s linearly with H and reach es M = 82 emu/ g at H = 17 kOe, see \nFig. 1(a). This value of magnetization is close to the data reported in [4, 5]. \nThe magnetic field dependen ces of magnetostriction (Δl/l)100 (variation of length along \nthe [100] axis upon magnetic field) for H||[100] and H||[010] (see Fig. 1(b)) are similar to those reported in [3] for the un annealed CoFe 2O4 single crystals . The values of (Δ l/l)100 at saturation \nfor our samples exceed th e values reported earlier for non -stoichiometric and doped CoFe 2O4 \nsingle crystals [ 5-7]. For the H||[100] orientation , (Δl/l)100 is negative. The sharp increase in the \n(Δl/l)100 values starts at H = 1.5 kOe and reaches saturation (~ -624×10-6) in the applied field of \nH = 3 kOe. For the H||[010] , the sign of (Δ l/l)100 is positive and parabolic growth start s \nimmediately from H = 0. The saturation value of ~ +221×10-6 is reached in the field of \nH = 3 kOe. This value of (Δ l/l)100 is three times less than that for the H||[100] orientation . \nFor a ferromagnetic material with cubic crystal structure the relative elongation along an \naxis defined by the directional cosines βx,y,z can be expressed as [8] \n 2 2 2 2 2 2\n100 11131323x x y y z z x y x y y z y z x z x zl\nl \n, (1) \nwhere αx,y,z are the directional cosines of magnetization vector. Magnetic field is assumed to be \nsufficient for saturation. In our case αz = βy = βz = 0 and βx = 1. Therefore λ100 = -624×10-6 and \n(Δl/l)100 must be equal to -λ100/2 for H||[010] . The experimental data presented in Fig. 1 (b) \nindicate that sign of (Δl/l)100 for H||[010] is indeed positive but the (Δ l/l)100 value is less than \n|λ100| not twice , but three times. Therefore , our crystal may be considered as cubic although slight \ndistort ions of cubic lattice exist. \n-80080\n0100200\n-9 -6 -3 0 3 6 9-600-400-200M (emu/g)(a)\n \n H\n H\n (b)\nH(l/l)100 (10-6)\nH (kOe)\n \nFIG. 1. Magnetic field dependences of ( a) magnetization and ( b) magnetostriction ( Δl/l)100 for \nthe CoFe 2O4 single crystal at T = 295 K. \nB. The spectrum of specular reflection \n \nThe spectrum of the reflection in the infrared spectral region for the CoFe 2O4 single \ncrystal at T = 295 K shown in Fig. 2 (a) is similar to one reported for a polyc rystalline sample in \n[9]. The spectrum of the optical conductivity σopt calculated using the Kramers -Kronig relations \nfrom the experimental R(λ) spectrum is plotted in Fig. 2(b). Compari son of Figs. 2(a) and (b) \nshow s that at λ < 1.5 μm the reflection spectrum is formed by the absorption edge , by the phonon \nbands at λ1 = 16.4 μm ( E1 = 0.076 eV) and λ2 = 24.2 μm (E2 = 0.051 eV), and by the frequency \nindependent part of the reflection (R ~ 14.7%) in the wavelength range of 1.5 < λ < 7.5 μm. \nWeak peculiarities are seen at about λ = 2 - 3 μm. The E1 band is associated with the Co -O \nvibrations of ions in the octahedral sublattice , while the E2 band - with the vibrations of oxygen \nions in the tetrahedral sublattice [ 10]. The long -wave edge of the E1 band is distorted by \nadditional contribution from two weak phonon band s at λ3 ≈ 18.7 μm (0.066 eV) and \nλ4 ≈ 21.5 μm (0.058 eV) [11]. It should be noticed that these bands are clearly seen in the \ncalculated spectrum of optical conductivity. \n \nC. The spectrum of magnetoreflection \n \nExternal magnetic fi eld leads to substantial changes in reflectivity of the crystal - \nmagnetoreflection effect. The magnetoreflection of our single crystal in the saturated field of \n3.6 kOe varies from -1% to +4% depending on the spectral region. Figure 2(c) shows ∆ R/R for \nH||[100]. If H||[110] , the magnetoreflection is hardly detectable and is in fact within \nexperimental accuracy . When λ < 1.5 μm, the ∆R/R grows up with decreasing λ, which is likely \nto be due to the shift of the absorption edge in the magnetic field . The absorption edge is known \nto be formed by indirect interband transitions [12]; the energy gap is somewhat higher 1 eV, \nwhich corresponds to λ of order 1 μm. In zero magnetic field, t he absorption edge shifts to \nshorter wavelengths with lowering temperature (so -called \"blue\" shift ) [12, 13]. Positive sign of \n∆R/R indicates that a magnetic field give s rise to the red shift of the absorption edge. \nWhen 1.5 < λ < 7 μm , there are intense peak at λ = 2.96 μm and weak maximum at about \n6 μm. In the optical conductivity spectrum (Fig. 2b) , the peculiarity at λ = 2.96 μm (0.42 eV) is \nalso observed as well as in the absorption spectra of polycrystalline CoFe 2O4 [14]. Similar peaks \nare found in spectrum of many other complex oxides. They are usually referred to as MIR (mid \ninfrared) bands. Thus it is reasonable to suppose that the existence of the ∆ R/R bands in the 1.5 < λ < 7 μm spectral range is related to variation s of the intensity and position of the MIR \nband at λ = 2.96 μm and the fundamental edge under application of magnetic field. \nWhen λ > 7 μm and H||[100] the spectral dependence of ∆ R/R is characterized by the \nfeatures associated with the magnetic -field-induced shift of reflection minima near th e phonon \nbands. Near the first minimum of the reflectivity , the peculiarity is observed in narrow λ range \nfrom 11 to 12 μm. Similar peculiarity manifests itself m ore clearly between the first (16.4 μm) \nand second (24.2 μm) phonon band s. We think that the shifts and intensity variations of two \nweak phonon bands at λ3(Eu symmetry ) = 18.7 μm and λ4(T1u symmetry ) = 21.5 μm under \napplication of a magnetic field play the m ost important role. \n0 5 10 15 20 25 30-20240204060\n50100R/R (%)(c)\n \n (m)(a)\n \n R (%)\n (b)\nopt (-1cm-1)\n-4 -2 0 2 4024\n R/R\nH (kOe)\n \nFIG. 2. The reflection spectrum R at T = 295 K for the CoFe 2O4 single crystal (a), the spectral \ndependence of optical conductivity σopt calculated from the reflection spectrum using the \nKramers -Kronig relations (b), and the magnetoreflection spectrum ∆R/R for H||[100], \nH = 3.6 kOe (c). Inset: ΔR/R vs H for λ = 2.96 µm. \n \nIII. DISCUSSION \n \nOne can see that there is close connection between magnetoreflection and \nmagnetostriction: the magnetoreflection is large only if ( Δl/l)100 is large. The shape of magnetic \nfield dependence ΔR/R (Inset in Fig.2 (c)) is very similar to that of ( Δl/l)100. Taking into account that the red shift of the absorption edge, which is typical of magnetic semiconductor spinels like \nCdCr 2Se4 and HgCr 2Se4 [15], is not observed in CoFe 2O4 we may infer that interaction of charge \ncarriers with localized magnetic moments in CoFe 2O4 is not so strong as in the spinels \nmentioned. Therefore, the a nisotropy of optical properties of CoFe 2O4 is not relate d to the \nexchange interaction of charge carrier with localized magnetic moments which occurs for \nCdCr 2Se4 and HgCr 2Se4 [15, 16]. It is more likely that the influence of magnetic field on optical \nproperties of CoFe 2O4 is indirect: magnetic field gives rise to strong strain and hence \ndeformation of the crystal lattice which in turn results in the change of electronic spectrum. Our \nexperiments also shows that [100] and equi valent directions play special role. This conclusion is \nin accordance with the results of band calculations for the CoFe 2O4 [12], which indicates that the \nconduction band minimum lies at Γ point while the valence band tops are located at X point of \nBrillouin zone . \nAccording to [ 12], the conduction band at Γ point and valence band at X point are non-\ndegenerate. If the bottom of the band is situated at Γ point the shift Δεc of the bottom due to \ndeformation is known to be ∆εc = Ξu, where u = uxx + uyy + uzz is change in volume, stands \nfor deformation tensor and Ξ is deformation potential. If the valence band valleys are located on \n[100] and equivalent axes , then \nv d u xx uu . When deformation is due to magnetostriction \ndescribed by ( 1) the change in volume is absent i.e. u = 0, so that in sufficiently strong magnetic \nfield, the shift of t he fundamental edge is equal to \n100 vu if H||[100]. The \nmagnetoreflection ∆R(E)/R can be expressed as \n\n ln[ ]v\nvR E R E d R E R\nR R E dE \n , (2). \nIn the vicinity of E = 1 eV (λ ≈ 1 µm) the derivative d ln[R(E)]/dE is about 0.5 1/eV, Ξ u is \nusually 10 – 20 eV [17], λ100 ≈ 6.6×10-4, therefore we obtain that ΔR/R is about 0.3 – 0.7%. The \nexperimental value of ΔR/R at λ = 1 µm (Fig. 2) is 0.76%, so the calculated value of ΔR/R \nreasonably agrees with the experimental one if we take Ξ u = 20 eV. \nThe narrow peak of magnetoreflection is found at λ = 2.96 µm. This MIR band is \nobviously a manifestation of deep -level impurities. Unfortunately, o ur data are insufficient to \nmake a conclusion on nature of these impurities. However, t he sensitivity of the peak to the \nmagnetic field direction – and hence anisotropic strain - strongly suggest that the impurities are \nin a low -symmetry position. \nIt is stated above that w hen λ > 7 μm and H||[100] the magnetoreflection is due to the \nmagnetic -field-induced shift of reflection minima near th e phonon bands . At first glance, such \ninterpretation seems to be incorrect because the effect of magnetic field on crystal lattice must be extremely weak. However, in [18] i t was reported that the frequencies of optical phonons in \nCoFe 2O4 substantially depend on pressure. As the strain is controlled by the magnetic field \ndirection because of magnetostriction , we may infer that the shift of a reflection minimum does \nnot cause by magnetic field as such, but results from magnetic -field-induced strains. \n \nIV. CONCLUTION \n \nTo summarize, our study of optical properties of the CoFe 2O4 magnetostrictive spinel \nshowed that in infrared spectral r ange the significant magnetoreflection is observed in natural \n(unpolarized) light. The largest magnetoreflection has be en found near MIR band where it is as \nhigh as 4%. The clear correlation has been established between magnetoreflection ∆ R/R and \nmagnetostriction (Δ l/l)100: large magnetoreflection is observed only if magnetostriction is great. \nThe effect of magnetic field on optical properties of CoFe 2O4 is likely to be indirect: application \nof a magnetic field results in strong strain and deformation of lattice, which leads to the change \nof spectrum – strain -magneto -optics . The deformation potential for the top of the valence band is \nroughly estimated as Ξu = 20 eV. \n \nACKNOWLEDGMENTS \n \nThe work was cond ucted within the state assignment of the Federal Agency for Scientific \nOrganizations of the Russian Federation (theme “Spin” No. 0120146330) and the Ministry of \nEducatio n and Science RF (grant N o. 14.Z 50.31.0025). \n \n1. J. Ferre and G. A. Gehring , Rep. Prog . Phys . 47, 513 (1984). \n2. A. S. Moskvin, D. G. Latypov, and V. G. Gudkov, Fizika Tverdogo Tela 30, 413 (1988). [Sov. \nSolid State Physics ] \n3. R. M. Bozorth, E. F. Tilden, and A. J. Wiliams, Phys. Rev. 99, 1788 (1955). \n4. W. H. Wang and X. Ren, J. Cryst. Growth 289, 605 (2006). \n5. R. Sato -Turtelli, M. Kriegisch, M. Atif, and R. Grossinger, IOP Conf. Series: Materials \nScience and Engineering 60, 012020 (2013). \n6. R. C. Kambale, K. M. Song, C. J. Won, K. D. Lee, and N. Hur, J. Cryst. Growth 340, 171 \n(2012). \n7. D. Bonnenberg , E. L. Boyd , B. A. Calhoun, V. J. Folen, W. Gräper, ; A. P. Greifer, C. J. \nKriessman, R. A. Lefever, T. R. McGuire, M. Paulus, G. H. Stauss, R. Vautier, and H. P. J. \nWijn, Magnetic and Other Properties o f Oxides and Related Compounds in “Landolt -Bornstein” edited by K. H. Hellwege and A. M. Hellwege, (Springer -Verlag , Berlin, 1970), \nvol. III/4b, p. 367. \n8. S. V. Vonovsky, Magnetism (Wiley, New York, 1974) . \n9. M. I. Danil’kevich , G. V. Litvinivich, and V. I. Naumenko . J. Appl. Spectrosc . 24, 38 (1976). \n10. R. D. Waldro n, Phys. Rev. 99, 1727 (1955). \n11. R. Bujakiewicz -Koronska, L. Hetmanczyk, B. Garbarz -Gios, A. Budziak , A. Kalvane , K. \nBormanis , and K. Druzbicki , Cent . Eur. J. Phys . 10, 1137 (2012). \n12. B. S. Holinsworth, D. Mazumdar, H. Sims, Q. -C. Sun, M. K. Yurtisigi, S. K. Sarker, A. \nGupta, W. H. Butler, and J.L. Musfeldt, Appl. Phys. Lett. 103, 082406 (2013) . \n13. C. Himcinschi , I. Vrejoiu , G. Salvan , M. Fronk , A. Talkenberger , D. R. R. Zahn , D. Rafaja , \nJ. Kortus , J. Appl . Phys. 113, 084101 (2013). \n14. A. Rahman, A. Gafur, A. R. Sarker , Int. J. Innovative Research in Advanced Engineering 2, \n99 (2015). \n15. M. I. Auslender and N. G. Bebenin, Solid. State Commun . 69, 961 (1989). \n16. M. I. Auslender, E. V. Barsukova, N. G. Bebenin, B. A. Gizhevskii, N. N. Loshkareva, Yu. \nP. Sukhorukov, and N. M. Chebotaev, Zh. Eksp. Teor. Fiz . 68, 139 (1989) . [Sov. Phys. \nJETP ]. \n17. K. Seeger, Semiconductor Physics (Springer -Verlag , Berlin , Heidelberg GmbH , 2004) , p. \n538. \n18. Zhongwu Wang, R. T. Downs, V. Pischedda, R. Shetty, S. K. Saxena, C. S. Zha, Y. S. Zhao, \nD. Schiferl, and A. Waskowska, Phys . Rev. B 68, 094101 (2003). " }, { "title": "2312.11990v1.Evidence_for_coexistence_of_spin_glass_and_ferrimagnetic_phases_in_BaFe12O19_due_to_basal_plane_freezing.pdf", "content": " \n \nCOMMUNICATION \n Please do not adjust margins \nPlease do not adjust margins Received 00th January 20 xx, \nAccepted 00th January 20 xx \nDOI: 10.1039/x0xx00000x \n Evidence for coexistence of spin -glass and ferrimagnetic phases in BaFe 12O19 \ndue to basal plane freezing \nKeshav Kumar,a Shrawan Kumar Mishra,a Ivan Baev,b Michael Martins ,b and Dhana njai Pandey*a\nWe present here result s of low-temperature magnetization and x -\nray magnetic circular dichroism studies on single crystals of \nBaFe 12O19 which reveal for the first time emergence of a spin glass \nphase , in coexist ence with the long -range ordered ferrimagnetic \nphase , due to the freezing of the basal plane spin comp onent . \nHexaferrites constitute an important family of compounds used \nin several technological applications such as permanent magnets \nin motors, credit cards, sonars, computer memories, spintronic \ndevices , and microwave communications1. Recent years have \nwitnessed revival of interest in these compounds following the \ndiscovery of type-II multiferroicity in the Y - and Z -type \nhexaferrites with strong magnetoelectric coupling around roo m \ntemperature2–4. Further, t he M -type hexaferrites have also \nevinced a lot of attention due to the discovery of several exotic \nquantum critical phenomena such as quantum paraelectricity \n(QPE)5,6, quantum electric dipole liquid state (QEDL)7,8, \nquantum tunneling of magnetization9, quantum electric dipole \nglass8, and magnetic quantum critical point10. Based on low -\ntemperature dc magnetization (M(T)), ac susceptibility ( (ω, T)) \nand x -ray absorption spectra (XAS)/x -ray magnetic circular \ndichroism (XMCD) studies on single crystal s of BaFe 12O19 \n(BFO) , we report here another novel phenomenon resulting from \nfreezing of the transverse (basal plane ) component of the spins \ninto a spin glass state at low temperatures. Our results show that \nBaFe 12O19 falls in the category of the geometrically frustrated \nordered compounds11–16 showing exotic spin liquid, spin ice, and \nspin glass transitions even in the absence of any apparent \nsubstitutional disorde r, with one very significant difference. \nUnlike the spin -glass phases in other geometrically frustrated \ncompounds where it emerges from the high -temperature \nparamagnetic phase, the spin -glass phase of BFO emerge s from \nthe long -range ordered (LRO) ferrimagn etic (FIM) phase which \ncontinue s to coexist with the spin -glass phase. We have used flux -grown crystals of B FO in the present \ninvestigation. The details of crystal growth, characterization and \nphysical property measurements are given in electronic \nsupplementary information ( ESI). The as -grown crystals are \nhexagonal platelet -shaped with well -developed facets as sho wn \nin Fig. 1(a). The crystallinity and symmetry of the crystals were \nchecked using Laue diffraction pattern collected in the reflection \ngeometry using a polychromatic beam incident along the c -axis \nof the hexagonal unit cell (see Fig. 1(b)). The presence o f closely \nspaced diffraction spots along six symmetry -related directions \nnot only confirms the crystallinity but also confirms the \nhexagonal symmetry of the as -grown crystals. The \na. School of Materials Science and Technology, Indian Institute of Technology \n(Banaras Hindu University), Varanasi, India -221005. \nb. Universität Hamburg, Institut fü r Experimentalphysik Luruper Chaussee 149, D -\n22761 Hamburg, Germany . \nElectronic s upplementary information (ESI) available : Powder synthesis, crystal \ngrowth and characterization detail, see DOI: 10.1039/x0xx00000x \nFig. 1: (a) Photograph of as -grown crystal, (b) Laue pattern of BaFe 12O19 \nsingle crystal with x -ray beam along [00l] direction, and (c) kagome bilayer \nconfiguration linked via pyrochlore slabs with different nearest neighbour \nbond lengths r 1, r2, and r 3.COMMUNICATION Journal Name \n2 | J. Name ., 2020 , 00, 1-3 This journal is © The Royal Society of Chemistry 20 xx Please do not adjust margins \nPlease do not adjust margins magnetoplumbite structure of the BFO in the P63/mmc space \ngroup was confirmed by Rietveld technique using x -ray powder \ndiffraction pattern collected on calcined powder samples (see th e \nESI). The refined structural parameters and selected bond \nlengths are given in table S2 & S3 of ESI. \n As per the classical Gorter model17, the magnetic structure of \nbarium hexaferrite comprises 3d5Fe3+ spins at the 2a, 2b, and 12k \nWyckoff sites of the P6 3/mmc space group with spin up \nconfiguration and the spins at the 4f iv and 4f vi Wyckoff sites wit h \nspin down configuration, giving rise to an overall ferrimagnetic \nstructure with a net magnetic moment of 20μ B per formula unit17. \nThis Ising like picture for the 3d5Fe3+ spins is, however, \nquest ionable, since a magnetic transition has been reported in the \nab-plane with strong spin -phonon coupling18. The variation of \nM⊥C(T) and M//C(T), measured during warming cycle on a zero-\nfield cooled (ZFC) crystal, with a magnetic field of 100 Oe \napplied perpendicular ( ⊥) and parallel (//) to the c -axis of the unit \ncell, respectively , shown in Fig. 2 reveals that M⊥C(T) increases \nsteadily with decreasing temperature upto ~4 0K and then shows \na peak ~40K, whereas M//C(T) decreases continuously with \ndecreasing temperature. This confirms a magnetic transition at \n~40K . Our results suggest that the spins are not fully aligned \nalong the c -axis of BFO at low temperature s but have a \nsignificant component transverse to the c -axis in the basal plane \n(00l) due to the canting of the spins away from the c -axis. \n In order to confirm the canting of the 3d5Fe3+ spins away \nfrom the c -axis, we investigated the angle -dependent XMCD \nsignals using XAS spectra recorded on a single crystal of B FO at \n30K in the normal and grazing incidence (GI) geometries where \nthe angle (θ) between the direction of propagation vector of the \ncircularly polarized soft x -ray beam and c-axis is 00 and 150, \nrespectively. The incident -flux-normalized x -ray absorption \nspectra (XAS) obtained in the normal and GI geometries using \nleft circularly and right circularly polarized x -ray photons \ncorresponding to the Fe L 2, 3 edges, labeled as σ + and σ -, are \nshown in Figs. 3(a) and (b), respectively. The XMCD spectra ( σ \n= σ + - σ-) at iron L 2, 3 edges for the normal and grazing angle \nincidence of polarized x -ray photon are shown in the same figure \nbelow the XAS spectra. From the angle -dependent XAS and \nXMCD spectra, spin magnetic moment c an be calculated using \nthe following spin sum rule equation19,20. \n mspin + 7mTθ = - (6P−4Q) nh\nR ………...(1) \nin which P = ∫(σ+−σ−)dω \nL3, Q = ∫ (σ++σ−)dω] \nL3+L2, R = \n∫ (σ++σ−)dω \nL3+L2, mspin is the total spin magnetic moment in \nunits of μB/formula unit, nh is the number of Fe 3d holes, mTθ = \nμB/ħ with < Tθ> being the expectation value of magnetic \ndipole operator, and L 3 and L 2 represent the integration range \nover energies of the two absorption edges. Using equation (1), \nwe obtained ( mspin + 7mT00)≈0.134 μB/ion for the magnetic \nmoment parallel to the c -axis and ( mspin + 7mT150)≈0.06 μB/ion for \nthe transverse component of the moment. The observation of \nsignificant XMCD signal for the GI geometry clearly suggests \nthat the 3d5Fe3+ spins are canted away from the c -axis. In order \nto further confirm the spin canting, we also analysed the XAS \nspectra and XMCD signals in the GI geometry recorded with dc \nfield ( H=100Oe ) applied parallel to the beam direction (see Fig.3(c)). The significant enhancement of the XMCD signal s in \nthe presence of dc field further confirms that the 3d5Fe3+ spins \nare indeed canted away from the c -axis of BFO . Thus, both the \ndc magnetization M(T) and XMCD studies reveal that a finite \ncomponent of the magnetic moments, which are primarily \naligned parallel to the c -axis, lies in the ab -plane (i.e., basal plane \n(00l)) perpendicular to the c -axis. \n In an isostructural compound SrCr 9xGa12-9xO19 (SCGO) with \nmagnetoplumbite structure, it has been shown that the spins in \nthe ab -plane undergo exotic spin liquid21 and spin glass22,23 \ntransitions for Ga content 00 \nwe have checked that the damping values obtained using \nboth methods were consiste nt. The value of the intrinsic \ndamping was found to be slightly higher than the values \ngiven in part of the literature (4 –5 10-3) [38–41]. This \ncould be explained by the growth conditions . Indeed, the \nstudied carried out by Xu et al. [42] shows that the \nmagnetic damping of sputtered CoFeB is very sensitive \nand decrease s when the argon pressure increase s. They \nreport ed a damping value of 1310-3\n when Ar pressure \nwas 3 mTorr . The annealing also affect s the \ndamping [38,41] . Another possibility might be to \nattribute this slight difference to the aluminium oxide \nlayer that caps the CoFeB magnetic layer probed by FMR \nin order to obtain the intrinsic damping. \nAs we can see in Fig. 3, the magnetic damping increases \nstrongly for both capping layers, Pt as well as Ir . This \nphenomenon is a well-known feature of damping \nenhancement due to spin pumping effect [19,26,29] , and \ncan be characterised by the following relation: \n \n∆𝛼=𝛼−𝛼0=𝑔𝜇𝐵𝑔𝑒𝑓𝑓↑↓\n4𝜋𝑀𝑆𝑡𝐹𝑀 (4) \nwhere g is the Landé factor (2.11 for CoFeB) , 𝑔𝑒𝑓𝑓↑↓ is the \nreal part of the effective spin mixing conductance, 𝜇𝐵 is \nthe Bohr magneton and 𝑡𝐹𝑀 is the ferromagnetic (CoFe B \nin our case ) layer thickness . Replacing our experimental \nvalues in eq. (4) , we estimated the value of 𝑔𝑒𝑓𝑓↑↓ for the \nIr/CoFeB interface to be ar ound 3 0 nm-2, and 3 2nm-2 for \nthe Pt/CoFeB interface. These values are in the typical \norder of magnit ude of effective spin mixing conductances \nobtained for ferromagnetic/Pt systems (for Py/Pt : 21 to 30 nm-2 [23,29,31,33] ; for Co/Pt: 80nm-2 [26]) and epitaxial \nFe/Pt : 26 nm-2 [43]. Especially, for the CoFeB/Pt \ninterface, some reported values are 40 nm-2 [44], 54 nm-2 \nand 47 nm-2 for the opposite stacking order, Pt/CoFeB , \nin [45] and 50.7 nm-2 in [46]). In the case of Ir it has bee n \nreported for NiFe/Ir interfaces so far, 13 nm-2 in [17] and \n25.2 nm-2 in [18]. Those values are effective value since \nthey include interface contributions such a spin memory \nloss [18,26] . Let us point out here that we do not consider \nthe imaginary part of the spin mixing conductance in our \nwork, since the Kittel fittings performed do not sho w a \nvalue of the gyromagnetic ratio differing from the one of \nelectrons, as predicted for metallic systems [19,47] . \n \nSPIN DIFFUSION LENGT H AND SPIN HALL \nANGLE DETERMINATION . \nFinally, the spin pumping voltage measured normalized \nby the resistance of the FM/NM slab gives the charge \ncurren t produced by ISHE. In this geometry t he charge \ncurrent measured can be expressed as \nfollows [26,27,29] : \n𝐼𝐶=𝜃𝑆𝐻𝐸𝑙𝑠𝑓𝑤𝐽𝑆𝑒𝑓𝑓tanh (𝑡𝑁𝑀\n2𝑙𝑠𝑓) (5) \nwhere 𝑤=10 μm is the width of the device . Here, 𝐽𝑆𝑒𝑓𝑓 \nis the effective spin current density injected in the NM \nlayer and it follows the relationship [26,27] : \n𝐽𝑆𝑒𝑓𝑓=𝑒𝑔𝑒𝑓𝑓↑↓𝛾2ℎ𝑅𝐹2\n4𝜋𝛼2𝐴(𝜔) (6) \nwhere 𝐴(𝜔)= 𝛾𝜇0𝑀𝑒𝑓𝑓+√(𝛾µ0𝑀𝑒𝑓𝑓)2+4𝜔2\n(𝛾µ0𝑀𝑒𝑓𝑓)2+4𝜔2 represents the \ninfluence of the magnetic dynamics on the injected spin \ncurrents, as it was shown by Ando et al [27]. Figure 4 \nshows the raw data of spin pumping voltage (Vsp) as a \nfunction of the applied magnetic field (µ0H) for an \nexcitation frequency of 15 GHz . We can observe that the \nvoltage is a purely symmetric lorenztian around resonance \nfield and it changes its sign upon chan ging the sign of the \napplied field. All these are features of a n ISHE spin \npumping voltage. Furthemore, that is also verified in the \ncase of CoFeB/Pt bilayer. \n \n \n \n \n4 \n \nFigure 4: Spin pumping voltage Vsp for Si/SiO 2/ CoFeB (5nm)/Ir \n(6nm) and Si/SiO 2/CoFeB(5 nm)/Pt(6 nm) as a function of the \napplied field absolute value ( |H|) for an excitation frequenc y of \n15 GHz . Results are shown for positive and negative static \napplied fields. Symbols represent experimental measurements \nwhereas the full lines correspond to a fit of the data by a \nLorentzian function. A constant offset was subtracted . The raw \nvoltage is purely symmetrical , getting rid of the spurious signals . \nFigure 5 shows the evolution of the charge current \nproduced as a function of the iridium thickness in \nCoFeB/Ir bilayer s for various frequenc ies. Using these \nresults, the spin diffusion length for Iridium , 𝑙𝑠𝑓𝐼𝑟, can be \ndeduced from equation (5) . Thus, 𝑙𝑠𝑓𝐼𝑟 obtained for each \nfrequency is displayed in Figure 6. Our results show \nconsistent value s of 𝑙𝑠𝑓𝐼𝑟=1.3±0.1 nm, and 𝑙𝑠𝑓𝑃𝑡=2.4±\n0.3 nm (red dashed line s). \n \n \nFigure 5: Produced c harge current (Ic) as a function of the \niridium thickness (tIr) for frequencies ranging from 4 to 26 GHz. \nThe symbols represent experimental values whereas the solid \nlines show the fitting obtained thanks to eq. (5) \nThe value of spin diffusion length obtained for platinum is \nin agreement with the values found in the literature : the \nexperimental values reported using SP -FMR set -up range \nfrom 0.5 nm [32], to 10 nm [23], with numerous values \nin between [10,48] . The obtained value 𝑙𝑠𝑓𝐼𝑟=1.3 nm is \ntwice larger than the one presented in earlier \nstudies [17,18] with similar FMR -based methods, and \nclose to the one reported by spin -orbit torque technique, \n~1 nm [7]. That difference might be due to different Ir \nresistivity . However, we would like to note that lsf values reported only by spin pumping FMR measurements (not \nspin pumping voltage measurements) consider an \nexponential decay of damping with and argument \n(2lsf/tNM). However, the tNM damping evolution is not \nreliable to estimate lsf as it was pointed out in ref. [26]. \nWe can observe that discrepanc y with results in Fig. 3 \nwhere ld is close to lsf estimated by charge current \ndependence in Fig. 5 and 6 but we have use d a different \nexponential argument ( ld/tNM). This is likely to explain the \ndifference with the two previous studies [17,18] . \nFurther, 𝑙𝑠𝑓𝐼𝑟 = 1.3 nm can be compared to the usual range \nof thicknesses where iridium is used, especially in the \ncase of synthetic ferrimagnets where the iridium spacer is \nused to maximize the RKKY coupling, around 0.5 nm or \n1.5 nm (1st and 2nd peaks) [2,8,9] . From the experimental \nvalues of spin diffusion length and resistivity of the HM \nlayer , we can compute its spin resistance 𝑟𝑠=𝜌𝑙𝑠𝑓. The \nresistivities measured for Pt and Ir are t he following: \n𝜌𝑃𝑡=245 nΩ.m and 𝜌𝐼𝑟=250 nΩ.m. We thus have the \nspin resistance 𝑟𝑠,𝑃𝑡=0.59 fΩ.m2 and 𝑟𝑠,𝐼𝑟=0.32 fΩ.m2. \nThe value of 𝑟𝑠,𝑃𝑡 is very close to the experimental result \npublished in ref . [26] as well as close to the theoretical \nvalue reported by Liu et al. [49]. We can also use the \nremark from reference [26], stating that in the case of Pt, \ngiven the results reported in the literature, the product of \nthe effective spin Hall angle and the spin diffusion length , \n𝜃𝑆𝐻×𝑙𝑠𝑓, is a quantity that is nearly independent on the \ntechnique or the setup used, and its effective value is \nestimated to be close to 0.1 9 nm . It is therefore possible to \nobtain the effective spin Hall angle of platinum, leading to \na value o f 𝜃𝑆𝐻𝑃𝑡≈7.6%. \n \n \n \n \nFigure 6: Spin diffusion length 𝑙𝑠𝑓 as a function of the frequency \nin iridium and platinum deduced from the fit shown in figure 5 . \n \nTo determine 𝜃𝑆𝐻𝐼𝑟 accurately and independently, the value \nof the effective spin current is needed . In order to do so, it \nis mandatory to estimate the strength of the radio \nfrequency excitation field, ℎrf, and its frequency \ndependence , as well as the 𝑔𝑒𝑓𝑓↑↓ factor for the CoFeB/Ir \ninterface. The latter was previously estimat ed to 30 nm². \nAfter accurate measurements of the transmission l ine and \nof the scattering matrices of the devices corresponding to \nour samples, we have concluded that the frequency \n5 \n dependence of hRF with respect to the frequency of the \nsignal is the same fo r the iridium and the platinum based \nsamples. This was expected, since the values of the \nconductivities are found to be very close for both \nmaterials. \nNow , we defined the quantity: \n \nℑ𝑆𝑃=𝐼𝐶∙𝛼2\n𝑙𝑠𝑓∙𝑔𝑒𝑓𝑓↑↓∙𝐴(𝜔) (7) \n \nWe can then plot the ratio ℑ𝑆𝑃𝐼𝑟\nℑ𝑆𝑃𝑃𝑡 of this parameter given in \neq. (7) for different frequencies experimentally measured \nas displayed in Fig. 7 . This ℑ𝑆𝑃𝐼𝑟\nℑ𝑆𝑃𝑃𝑡 value is most likely to give \nthe right estimate of the actual spin Hall angles ratios, \nsince it can be interpreted as: \n \nℑ𝑆𝑃𝐼𝑟\nℑ𝑆𝑃𝑃𝑡=𝜃𝑆𝐻𝐼𝑟\n𝜃𝑆𝐻𝑃𝑡tanh (𝑡𝐼𝑟(2𝑙𝑠𝑓𝐼𝑟) ⁄ )\ntanh (𝑡𝑃𝑡(2𝑙𝑠𝑓𝑃𝑡⁄ )) (8) \n \nThe limit obtained for 𝑡𝑁𝑀≫𝑙𝑠𝑓𝑁𝑀 is the ratio of spin Hall \nangles. Indeed, we can represent this ratio as a function of \nthe nonmagnetic materials thickness as shown in Fig. 8 . \nWe can observe a very large discrepan cy between the \nvalue of the ratio given in eq. (8) and the one obtained \nexperimentally for tNM = 2nm. \n \n \n \nFigure 7: ratio of the spin pumping currents based on eq.( 7) \nas a function of the frequency . The dashed lines are guidelines \ntowards the value obt ained at high frequency. \nNumerous elements can explain the difference between \nthe model given in eq. (8) and the experimental results at \nlow thickness. First, we can question the validity of the \nassumptions used in our study. We have considered that \nthe re sistivity, the spin diffusion length, and the spin Hall \nangle of the materials were independent of the \nnonmagnetic material thickness . However, this \napproximation does not hold for very low thicknesses, \nwhich is where the model and the experimental results do \nnot match. Furthermore, at very low thicknesses, the \nroughness and the quality of the interface plays a larger \nrole than for thick layers. The errors on the thicknesses \nand on the ratios are expected to be larger than for thicker \nsamples. Nevertheless , a good agreement for nonmagnetic \nmaterials thickness superior to 2nm is obtained, and we \ncan estimate the ratios of effective SHE efficiencies to be \n𝜃𝑆𝐻𝐼𝑟\n𝜃𝑆𝐻𝑃𝑡=0.26. This approach lets us evaluate the values of lsf \nand θSH with prec ision. \nBesides , using our determination of the spin Hall angle of \nplatinum at 7.6%, we can estimate the spin Hall angle of \niridium to be around 2% . Literature provides a large range \nof values for Pt that span to more than an order of \nmagnitude, ranging fr om 0.33 to 0.0067 [22,23,32 –\n35,47,48,24 –31]. Our result for Ir is in good a greement \nwith what was found in ref. [17], with a 2% value , and \ntwice the one reported by spin -orbit torque in ref. [7]. \nThe method that we present here enables to make a \ncomparison by getting rid of many artefacts that seem to \nbe the cause of a broadening of the results obtained in the \nliterature. \nWe can use the works in ref. [50] to evaluate the \nefficiency of these two materials for spin -to-charge and \ncharge -to-spin conversion applications . For the generation \nof a charge current, the figure of merit proposed is the \nproduct 𝜆𝐼𝑆𝐻𝐸∗=𝜃𝑆𝐻×𝑙𝑠𝑓. We find a value of 0.186 nm \nfor Pt, and 0.026 nm for Ir, suggesting that iridium is a \npoor candidate for further spin pumping applications. \nHowever, if we consider the figure of merit to assess the \nspin current generatio n, which is mandatory for spin orbit \ntorque (SOT), given by the formula 𝑞∗=0.38×𝜃𝑆𝐻\n𝑙𝑠𝑓, we \nfind a value of 12 10-3 nm-1 for Pt and ~610-3 nm-1 for Ir. \nTherefore, it appears that even though Pt is the best \nmaterial amongst those studied in both cases, Ir is good \ncandidate, with half the ability of Pt to generate efficiently \na spin current. \n \n \n \nFigure 8: Evolution of the corrected spin pumping currents ratio \nas a function of the non-magnetic materials thickness (Black \nsquares) and the exp ected dependence (red line) according to \neq. (8) . \n \nCONCLUSION \nIn this paper, we have described a n approach that enables \nthe measurement of the spin Hall angle of a material with \nrespect to another one. We report reliable values of spin \ndiffusion lengths of 1.3 0.1 nm for iridium, 2.4 0.3 nm \n6 \n for platinum from the NM thickness dependence of the \ncharge current (and not from damping evolution) . The \nspin mixing conductances for both interfaces CoFeB/Ir \nand CoFeB/Pt have been estimated around 30 and 32 nm-\n2, respectively . The spin Hall angle of Ir has the same sign \nas the one of Pt and represents 26% of its value. We could \nobtain a θSH value of 7. 6% for Pt, from which we could \ndeduce a θSH value of 2% for Ir. Even though this \nprocedure does not give by itself the value of the spin Hall \ncharacteristics of a material, it gives information about \nmaterials in same conditions, and enable s a comparison \nbetween various materials. This can be an opportunity to \nunify the results concerning spin diffusion length s and \nspin Hall angles, given the large dispersion in the results \nreported in the past decade. The spintronic parameters we \nare report ing for Ir will appeal for more applications \nexploiting this material in new spin -orbitronic devices \nsuch as combined spin -orbit torque and spin transfer \ntorque effects in magnetic tunnel junctions [51] . This is \nby combining two major effects in spintronics, RKKY \nand SHE. \n \nAKNOWLEDGEMENTS \nT. F. thanks the ANRT and the company Vinci \nTechnolog ies for funding his PhD , under the CIFRE \nconvention No. 2016/1458. All authors acknowledge \nsupport from Agence Natio nale de la Recherche (France) \nunder contract N° ANR -18-CE24 -0008 (MISSION) and \nANR -19-CE24 -0016 -01 (TOPTRONIC), from the French \nPIA project “Lorraine Université d’Excellence”, reference \nANR -15IDEX -04-LUE , from Region Grand Est, \nMetropole du Grand Nancy, I nstitut Carnot ICEEL , from \nthe “FEDER -FSE Lorraine et Massif Vosges 2014 -2020”, \na European Union Program , and from . 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Grimaldi1, C.-H. Lambert1, T. Devolder2, and P. Gambardella1* \n1Department o f Materials, ETH Zurich, 8093 Zu rich, Switzerland \n2Centre de Nanosciences et de Nanotechnologies, CNRS, Universit é Paris -Sud, Universit é Paris -Saclay, \n91405 Orsay Cedex, France \n*email : giacomo.sala@mat.ethz.ch ; pietro.gambardella@mat.ethz.ch \n \nABSTRACT \nMeasurements of the transverse Hall resistance are widely used to investigate electron transport, \nmagnetization phenomena, and topological quantum states. Owing to the difficulty of probing transient \nchanges of the transverse resistance, the vast majority of Hall effect experiments are carried out in \nstationary conditions using either dc or ac currents. Here we present an approach to perform time -\nresolved measurements of the transient Hall resistance during current -pulse injection with sub-\nnanosecond temporal resolution. We apply this technique to investigate in real -time the magnetization \nreversal caused by spin -orbit torques in ferrimagnetic GdFeCo dots. Single -shot Hall effect \nmeasurements show that the current -induced switching of GdFeCo is widely distributed in time and \ncharacterized by significant activation delays , which limit the total switching speed d espite the high \ndomain -wall velocity typical of ferrimagnets. Our method applies to a broad range of current -induced \nphenomena and can be combined with non -electrical excitations to perform pump -probe Hall effect \nmeasurements. \n 2 \n INTRODUCTION \nThe broad family of Hall effects includes phenomena of ordinary, anomalous1, planar2,3, \ntopological4,5, and quantum6–8 origin. These effects have become standard tools for benchmarking the \nphysics of metallic, semiconducting, and topological materials as well as the func tionality of electronic \nand spintronic devices. The anomalous Hall effect (AHE), for example, allows for probing the \nemergence of magnetic ally-ordered phases1,9–11, field-12 and current -induced magnetization reversal13–\n15, domain wall motion16, and spin -orbit torques (SOTs) 17–19. Measurements of the transverse resistance \nalso provide insight into magnetoresistive phenomena, such as the planar Hall effect and spin Hall \nmagnetoresistance, which can be used to track the response of antiferromagnets and magnetic insulators \nto applied magnetic fields, currents, and heat20–22. Extending these measurements to the time domain \nwould enable access to the dynamics of a vast range of electronic and magnetic systems . As is well -\nknown, the ordinary and planar Hall effects are widely employed in sensors for the detection of magnetic \nfields and microbeads23–25, and have a frequency bandwidth extending up to several GHz (Refs. 26,27). \nHowever, there are only few examples of time-resolved (\ns-ns) measurements of the magnetization \ndynamics using the Hall effect , which are limited to observations of laser -induced heating28 and the \ntransit of domain walls29,30. \nHere , we present an all-electrical technique suitable for systematic real-time measurement s of \nany kind of transverse magneto resist ance in devices with current flowing in -plane. The key idea consists \nin disentangling the tiny magnetic Hall signal from the large non -magnetic background by minimiz ing \nthe current leakage in the sensing arms of the Hall cross. This approach, which relies on the counter -\npropagation of electric pulses , is well adapted for radio -frequenc ies and proves particularly useful for \nfast excitations, e.g., ns - and sub -ns-long pulses. We demonstrate the capability of this technique by \nstudying the magnetization dynamics triggered by SOTs17 in ferr imagnetic GdFeCo dots patterned over \na Pt Hall bar. In our detection scheme, the ns -long pulses do not only generate the pe rturbation on the \nmagnetization but also serve as the tool for tracking the magnetic response, including single -shot \nswitching events. This capability opens up the possibility of performing systematic time -resolved Hall 3 \n measurements of current -induced excitations in a broad variety of planar devices and provides access to \nstochastic events. \n Ferrimagnets have recently attracted considerable attention due to the enhanced SOT \nefficiency31–33 and the extraordinary high current -induced domain -wall velocity34–36 attained, \nrespectively, at the magnetization and angular -momentum compensation points . These properties make \nthem promising candidates for the re alization of fast and energy -efficient spintronic devices36. However, \nthe current -driven magnetization dynamics in these systems has been investigated only using magneto -\noptical pump -probe methods36,37, which do not provide information on stochastic events . Our time -\nresolved AHE measurements show that the reversal of the magnetization in GdF eCo evolves in different \nphases , which compris e an initial quiescent state, the fast reversal of the magnetization, and the \nsubsequent settling in the new equilibrium state without ringing effects . Despite the high domain -wall \nvelocity attained by ferrimagnets, we find that the total switching time is severely affected by an initial \nactivation phase, during which the magnetization remains quiescent . We associate this phase, which has \nnot been reported so far in ferrimagnets , with the time required to nucleat e a reversed domain assisted \nby Joule heating . The single -shot AHE traces reveal the existence of broad distribution s of the nucleation \nand reversal time s and disclos e the stochastic character of the SOT -induced dynamics , which is not \naccessible to pump -probe techniques. Our measurements further show that the domain nucleation time \ncan be substantially reduced by increasing the current amplitude, leading to a minimum of the critical \nswitching energy for pulses of reduced l ength. \nRESULTS \nTime -resolved anomalous -Hall-effect measurements . \nElectrical t ime-resolved measurements using the Hall effect , or any form of transverse magneto -\nresistance, suffer from the difficulty of generating a detectable Hall signal without spoiling the signal -\nto-noise ratio . The main obstacle is the current shunting into the sensing line of the Hall cross , caused \nby the finite electric potential at its center . When a pulse reaches the cross, a portion of the current flows \nthrough the transverse arms (along ±y in the top panel of Fig. 1 a), thus producing a spurious electric \npotential associated with the resistance of the leads. This potential is much larger than the signal of 4 \n magnetic origin and hinders its detection. A limitation remains e ven in differential measurements \nbecause the unavoidable asymmetry of the leads introduces a finite differential offset23 that can satura te \nthe dynamic range of the Hall voltage amplification stage. These problems do not exist i n standard dc \nmeasurements as the current leakage is countered by the high input impeda nce of the measuring \ninstrument . At high frequency, however, impedance matching requires a low resistance (50 Ohm) at the \ninput port of the instrument , usually an os cilloscope. \nThe approach that we introduce here consists in injecting two counte r-propagating rf pulses with \namplitude |𝑉P\n2| and opposite polarity , as depicted in the bottom panel of Fig. 1 a. Provided that these \npulses reach the center of the cross at the same time and have the same amplitude , a virtual gr ound is \nforced there. The virtual ground limits the spread of the current because the voltage drop on the entire \nsensing line (Hall arm, cable , and input impedance of the oscilloscope) is ideally zero. The synchrony \nof the two balanced pulses, generated by a balun power divider, is ensured by the symmetry of the paths \nconnecting the balun to the device, as schematized in Fig. 1b. Thanks to the opposite polarity of the \npulses, the current flows along the x direction, with double magnitude relative to the current produced \nby a single pulse of amplitude 𝑉P\n2, and sign determined by the polarity of the pulses. The current generates \ntime-dependent transverse Hall voltages , 𝑉+ and 𝑉−, which are pre -amplified and acquired by a sampling \noscilloscope triggered by an attenuated portion of the original pulse. If no change of the magnetization \noccurs during the pulses, the magnetic signal mimics the shape of the pulse. A deviation from this \nreference signal is the signature of ongoing magnetization dynamics. In the specific case discussed \nbelow, the tra nsverse voltage stems from the A HE and its change over time gives access to the out-of-\nplane component of the magnetization. We note that, in the more general situation of asymmetric Hall \ncrosses, our technique allows for compensating detrimental resistance offsets by tuning the relative \namplitude of the counter propagating pulse s. This capability is unique to our approach and cannot be \nimplemen ted in time-resolved differential Hall measurements30. We also remark that the main additional \ncomponent to the setup required by our approach is the balun divider, which is a simple and affordable \ncircuit element. More details about the electric circuit, including the rf and dc sub -networks, sensitivity, 5 \n resistance offsets compensation, and time-resolution are discussed in the Methods and in Supplementary \nNote s 1, 2, and 5 . \nSwitching dynamics of ferrimagnetic dots . \nWe adapted this concept to investigate the SOT -induced magnetization switching of 15-nm-thick, 1 -\nm-wide Gd 30Fe63Co7 dots with perpendicular magnetization , patterned on top of a 5 -nm-thick Pt Hall \nbar (see Fig. 1c,d, Methods , and Supplementary Note 3). The compensation temperature of the \nferrimagnetic dots is below room temperature, such that the net magnetization and AHE are dominated \nby the magnetic moments of Fe and Co. Therefore, in our room -temperature measurements the current -\ninduced switching in the presence of an in -plane static magnetic field has the same polarity as in \nperpendicularly -magnetized ferromagnets with a Pt underlayer14,17. Specifically , the parallel alignment \nof current and field favours the down state of the magnetization, whereas the antiparallel orientation \npromotes the up state, which correspond to negative and positive an omalous Hall resistance, \nrespectively. \nThe differential signal 𝑉+−𝑉− is determined by the magnetization orientation, which changes \nwith time during a switching event . Figure 2a shows the switching trace s obtained by measuring 𝑉+−\n𝑉− during the reversal of a GdFeCo dot for different pulse amplitudes. In order to minimize spurious \ncontribution s to the magnetic signal, a background signal was recorded by fixing the magnetization in \nthe initial state, either “up” or “down” , and subtracted from the data . The down -up and up -down \nswitching traces obtained by averaging over 1000 pulses are shown as red and blue lines, respectively. \nThe black lines represent a reference trace obtained by subtracting two background measurements \ncorresponding to the magnetizatio n pointing up and down. This reference trace describes the maximum \nexcursion of the Hall voltage during a current pulse (see Supplementary Note 4 for more details ). The \ndeviation of the switching traces from the top and bottom reference levels corresponds to the change of \nthe out-of-plane magnetization driven by the SOTs during the 20 -ns-long current pulse. Dividing the \nswitching traces by the corresponding reference trace provides the normalized magnetic time trace s \nshown in Fig. 2b -e. In these average measurements, the transition between the top and bottom reference \nlevels of the switching trace is sufficiently clear such that the normalization by the reference trace is not 6 \n strictly required. The latter, however, is important to highligh t the switching in single -shot \nmeasurements, which will be presented later on . \nThe measurements in Fig. 2 b-e allow us to electrically probe the time-resolved SOT -induced \ndynamics in planar devices , which so far has been achieved only by X-ray and magneto -optical \ntechniques36–39. We find that the switching dynamics of the ferr imagnetic dots comprises three phases: \nan initial quiescent state , the reversal phase , and the final equilibrium state, with the magnetization \nremaining constant both before and after the reversal. Both the quiescent and reversal phase present \nstochastic components. The observation of a long quiescent phase challenges the common assumption \nthat the magnetization reacts instantaneously to the SOT owing to the orthogonality between the initial \nmagnetization direction and the torque40–42, unlike the spin -transfer torq ue between two collinear \nmagnetic layers43. Instead, our measurements show that the duration of this phase can be comparable to \nthe pulse length. The quiescent phase is a characteristic of the thermally -activated regime, in which \nthermal fluctuations assist the switching a nd lead to a stochastic delay time . Because of the relatively \nhigh perpendicular anisotropy of the ferrimagnetic dots (see Supplementary Note 3 ), the thermal \nactivation plays a role up to current density of the order of 1.5 × 1012 A m-2, similar to the switching of \nhigh-coercivity ferromagnetic nanopillars by spin transfer torque44. By increasing the pulse amplitude \nor the in -plane field, the duration of quiescent phase is significantly reduced as the switching dynamics \napproaches the intrinsic regime (see Fig. 2b -e and the following section s). \nSingle -shot measurements \nAlthough the averaging process improves the quality of the traces, it conceals the stochastic nature of \nthe dynamics . Here , we show that our technique provides sufficient signal -to-noise contrast to detect \nindividual reversal events in Hall devices. By using the procedure outlined above , we measured single -\nshot switching traces for different in-plane magnetic fields and pulse amplitudes, as shown in Fig. 3 for \nthree representative voltages . The single -shot traces are qualitativ ely similar to the average traces. \nHowever, the duration of the quiescent and transition phase s varies significantly from trace to trace. By \nfitting each trace to a piecewise linear function, we define 𝑡0 as the duration of the initial quiescent phase \nduring which the normalized Hall voltage remains close to 1 (0 ) before the up -down (down -up) reversal 7 \n (see Methods ). In the following, w e refer to 𝑡0 as the nucleation time, arguing that the quiescent phase \nis associated with the reversal of a seed domain38,45,46, in analogy to measurements performed on \nferromagnetic tunnel junctions47. Additionally, we designate the duration of the transition between the \nup-down or down -up magnetization levels as the transition time ∆𝑡 (Ref. 48). The total switching time is \nthus given by 𝑡0+∆𝑡. \nTo gain insight into the stochastic variations of 𝑡0 and ∆𝑡, we recorded a set of 1000 individual \ntraces for several values of the applied in -plane field B and voltage V. Figure 4 shows the statistical \ndistributions of 𝑡0 and ∆𝑡 obtained at representative fields and pulse amplitudes. The comparison \nbetween the single -shot statistics in Fig . 4 and the averag ed traces in Fig. 2 reveals that the duration of \nthe quiescent phase is systematically underestimated in t he average measurements relative to the mean \n𝑡̅0, whereas the duration of the transition phase is systematically overestimated relat ive to the mean Δ𝑡̅̅̅. \nThe deviation of the times deduced from the av erage measurements relative to 𝑡̅0 and Δ𝑡̅̅̅ can reach up \nto -25% and 60%, respectively. The quantitative disagreement is determined by the superposition of \nwidely -distributed nucleation events. As shown by the average curves at the bottom of Fig. 3, t he large \nspread of the nucleation events anticipate s the starting point of the average dynamics and, at the same \ntime, broaden s the apparent switching duration . Therefore, only single -shot measurements can \naccurately quantify the full switching dynamics, including the variability of events as well as the \nduration of the nucleation and transition phases , and their distributions . \nThe data reported in Fig. 4 show that 𝑡0 approximately follows a nor mal distribution, as expected \nfrom random events . In contrast , ∆𝑡 has a significant positive skew with the mean Δ𝑡̅̅̅ shifted towards \nthe shorter times. Moreover, 𝑡̅0 and its standard deviation decrease strongly upon increasing either the \npulse amplitude or the field, whereas Δ𝑡̅̅̅ shows only a moderate dependence on the voltage . These \ndistinct statistical distributions and dependenc ies are the signature of different physical processes \nunderlying the initial phase and the transition phase of the reversal . Doubling the pulse amplitude or \nfield leads to a ~10-fold reduction of 𝑡̅0, consistently with an activated domain nucleation process that \nis promoted by SOT s and assisted by the in -plane field47 and thermal fluctuations . \nIn contrast with 𝑡̅0, the effe ct of the in -plane field on Δ𝑡̅̅̅ is negligible. This observation supports \nthe interpretation of Δ𝑡 in terms of domain -wall depinning and propagation time, since, for the fields 8 \n used in this study, the domain wall mobility is saturated at the maximum value expected for Néel \nwalls49,50. On the other hand, stronger pulses are expected to ease the depinning of domain walls and \nincrease their speed, in accordance with the reduction of Δ𝑡̅̅̅ at larger volta ges. Consistent with our \nanalysis, ∆𝑡 can be interpreted as the time required for the seed domain to expand across the entire area \nof the dot. Therefore, the inverse of ∆𝑡 provides an upper limit to the domain wall velocity in our devices. \nThe average domain wall velocity estimated from the mean of the distributions reaches several hundreds \nof m/s, whereas the peak velocity can be as large as 4 km/s . Such a high speed is in line with the velocities \nestimated by measuring the domain wall displacements in GdFeCo following the injection of current \npulses34–36. Further improvements of the domain wall velocities have been demonstrated by tuning the \nstoichiometry and transient temperature of GdFeCo so as to approach the angular momentum \ncompensation point35. Our measurements demonstrate that the nucleation phase , characterized by a long \ndelay time 𝑡0, is the real bottleneck of the SOT -induced switching dynamics of ferrimagnets. Therefore, \nthe efficient operation of ferr imagnetic devices based on SOTs requires strategies to reduce the initial \nquiescent phase and mitigate the associated stochastic effects. \n \nIntrinsic and thermally activated switching regimes . \nMeasurements of the threshold switching voltage 𝑉c as a function of the pulse duration 𝑡𝑃 evidence the \nexistence of two switching regimes40, as shown in Fig. 5 (see also Supple mentary Note 6). Above \napproximately 5 ns, 𝑉c changes weakly with 𝑡𝑃, which is a signature of the thermally -assisted reversal40,44 \nand reveals the importance of thermal effects for the typical pulse lengths and amplitudes used in this \nstudy (𝑡𝑃= 20 ns ). On the other hand, the critical voltage increases abruptly for 𝑡𝑃 ≲ 3 ns, as expected \nin the intrinsic regime where the switching speed depends on the rate of angular momentum transfer \nfrom the current to the magnetic layer. Indeed, in this regime, 𝑉c scales proportionall y to 1/𝑡𝑃 (see \nSupplementary Fig. S7). Switching with 𝑡= 300 ps (equivalent average domain wall speed > 3.3 km/s, \nunder the assumption 𝑡0≈0) demonstrates that the quiescent phase can be suppressed by strongly \ndriving the magnetization. In this case, the SOTs alone are sufficient ly strong to drag the magnetization \naway from the equilibrium position and induce the nucleation of a domain against the energy barrier 9 \n without substantia l thermal aid. Finite element simulations support this point by showing t hat the \ntemperature rise times in our devices are larger than 2 ns. \nImportantly , the suppression of the quiescent phase requires more intense pulses but does not \nimply a larger energy consumption because the threshold energy density decreases by more than 4 times \nupon reducing 𝑡𝑃 from 20 ns to < 1 ns (see Fig. 5). This favorable trend highlights the advantage of \nusing materials for which the fast dynamics does not require excessively large current densities. We \nnote that the current densities used in this study are compatible with previous results obtained on GdCo \n(Ref. 36). In that work the current density at 3 00 ps is approximat ely 1.05 × 1012 A m-2, whereas in our \ndevice s with three times larger GdFeCo thickness the threshold current density reaches 3.6 × 1012 \nA m-2. For 20 -ns-long pulses, this value reduces to 0.82 × 1012 A m-2. On the other hand, a more stringent \ncomparison of our findings with the measurements reported in Ref. 36 is not straightforward because the \ndevice geometries, the materials and their magnetic properties are dis similar. \n \nSensitivity and temporal resolution . \nFinally, w e present considerations on the sensitivity and time resolution of our technique that apply to \nall conductors with a finite transverse resistivity 𝜌xy. In all generality, we assume that 𝜌xy≠0 only in \na finite region of the Hall cross (the “magnetic dot”). The Hall voltage generated by two counter -\npropagating voltage pulses of opposite amplitude 𝑉P/2 and −𝑉P/2 is given by 𝑉+−𝑉−= 𝑓𝜌xy\n𝑡𝑉P\n𝑅I, \nwhere t is the thickness of the dot, 𝑅I the resistance of the injection line, and f a sensitivity factor (<1) \nthat depends on the ratio between the area of the dot and the Hall cross as well as on the inhomogeneous \ncurrent distribution within the device . An equivalent circuit model of the Hall cross and sensing \napparatus shows that the differential Hall signal S measured at the input ports of the oscilloscope is the \nresult of the amplified voltage partition between the two branches of the sensing line, e ach having a \nresistance 𝑅S, and the input resistance of the amplifier 𝑅A: \n𝑆=2𝐺𝑉H\n2𝑅A\n𝑅A+ 𝑅S\n2 , (1) \nWhere G is the gain of the amplifier stage. The total noise superimposed to the signal reads 10 \n 𝑁≈2(𝐺𝑁in+ 10𝑁𝐹\n10𝐺𝑁in+10𝑉R\n28), (2) \nwhere the first term represents the amplified sum of the Johnson and pulse generator noises (𝑁in), the \nsecond term the noise introduced by the amplifier with noise figure 𝑁𝐹, and the third term the vertical \nresolution of the oscilloscope with 8 bits and acqui sition range 𝑉R (see Supplementary Note 1 for a \ndetailed derivation of Eq s. 1 and 2). On the basis of Eqs. 1 and 2, we estimate a signal -to-noise ratio \n𝑆\n𝑁≈ 2.2 and ≈ 66 for the single -shot and average traces measured with 𝑉P= 2.2 V , respectively. These \nvalues are in fair agreement with the actual 𝑆\n𝑁 that characterizes the traces in Figs. 2 and 3. The main \ncontributions to the noise are the 𝑁𝐹 of the amplifiers (54%) and the resolution of the oscilloscope \n(30%). The 𝑆\n𝑁 can thu s be improved by means of amplifiers with lower 𝑁𝐹 (1-2 dB, against the 6 dB of \nour current setup ) and oscilloscopes with higher vertical resolution (up to 10 -12 bits) or better vertical \nrange. \nThe temporal resolution is determined by the sampling rate a nd bandwidth of the oscilloscope \nas well as by the acquisition mode. In this work, all the traces were acquired in the interpolated real -\ntime mode, which allows for a nominal temporal resolution of ≈ 100 ps, sufficient to track the dynamics \nof ns -long pulses . Using an oscilloscope with a higher sampling rate could improve the time resolution \ndown to about 10 ps. The minimal duration of the pulses that can be used to excite the magnetization , \non the other hand, is determined by the impedance matching and symmetry of the circuit. In our case, \nthe minimal pulse length is limited to a few ns by the inductive coupling between the wire bonds that \nconnect the sample, which gives rise to over - and under -shoots in the transverse voltage a t the rising and \nfalling edges of a pulse (see Supplementary Note 4). This problem can be solved by using optimally -\nmatched rf probes to connect the sample. Ultimately, i t is of primary importance that the two branches \nof the injection (sensing ) lines have equal lengths in order to guarantee the synchronization of the \ninjected (sensed ) signals. For symmetric branches , the relative delay of the balanced pulses at the center \nof the Hall cross is determined by the balun divider and is of the order of 1 ps (Ref. 51). Such a time lag \nlimits the duration of the shortest measurable pulses. \n 11 \n DISCUSSION \nWe have demonstrated a technique to perform time-resolved measurements of the Hall effect and \ntransverse magneto resistive signals in devices with current flowing in-plane and applied it to investigate \nwith sub -ns resolution the switching dynamics of ferr imagnetic dots induced by SOTs . Our results show \nthat the current -induced magnetization reversal in GdFeCo is characterized by strong stochastic \nfluctuations of the ti me required to nucleate a domain . The quiescent phase that precedes the nucleation \nis a dynamical characteristic that ferrimagnets share with ferromagnets and that has not be en reported \npreviously for these materials . The observation of this phase , whose duration and variability are \ndetermined by the applied current and in -plane field , implies that the switching process is thermally \nactivated. The corresponding switching delay depends on the combination of two effects. For a given \nstrength of the S OTs and in -plane field, the average duration of the quiescent phase 𝑡̅0 is mainly \ndetermined by the temperature dependence of the magnetic anisotropy and the rate of increase of the \ntemperature47. In this scenario, 𝑡0 does not change between switching events and its standard deviation \nshould be of the order of the pulse rise time . In addition to this deterministic process, 𝑡0 is influenced \nby stochastic thermal fluctuat ions, which cause the spread reported in Fig. 4 . \nUpon reducing the length of the pulses and increasing their amplitude, the nucleation time can \nbe suppressed to below 1 ns, which results in a minimum of the critical switching energy . Following the \ninitial nucleation phase , the transition between two opposite magnetization states is both fast and \nmonotonic, compatible with the extremely large domain -wall velocity reported for ferrimagnets. \nHowever, the reversal is also highly non -deterministic and characterized by a spread of transition times , \nwhich deserves further investigation . Overall, our data show that the switching delay time can be rather \nlong in ferrimagnets, unlike the subsequent domain wall motion, which is very fast. The coexistence of \nthese slow and fast phases should be considered in future studies of ferrimagnets to correctly quantify \nthe switching speed . \nThe sensi tivity of the time-resolved Hall measurements is sufficient to perform both average \nand single -shot measurements , thus providing access to reproducible and stochastic processes. This dual \ncapability combined with the straightforward implementation of our s cheme and the widespread 12 \n availability of Hall experimental probes makes our technique useful for a broad range of studies. The \ntemporal evolution of the transverse voltage can be induced directly by the current, as in this work, or \nby a different stimulus, like magnetic fields, light or heat, using a pump -probe scheme with a variable \ndelay time between excitation and counter -propagating voltage pulses . In the latter case, the electric \ncurrent serves uniquely as the probing tool and its duration, amplitude, and waveform can be arbitrarily \nchosen. As any form of Hall effect or transverse magnetoresistance equally fit s our detection scheme , \npotential applications include time -resolved investigations of electrically - and thermally -generated spin \ncurrents and spi n torques in magnetic materials, switching of collinear and noncollinear \nantiferromagnets, as well as time -of-flight detect ion of skyrmion and domain walls in racetrack devices . \nTime -resolved Hall effect measurements can also probe the emergence or quenchi ng of symmetry -\nbreaking phase transitions in driven systems. F urther , as the Hall response is a quint essential signature \nof chiral topological states , real -time detection can provide insight into edge transport modes as well as \ncurrent -induced transitions between quantum Hall and dissipative states. \nMETHODS \nDevice fabrication. \nThe Hall cross es and the dot s were fabricated by lithographic and etching technique s. First, the full stack \nsubstrate/Ta(3)/Pt(5)/ Gd 30Fe63Co7(15)/Ta(3)/Pt(1) (thicknesses in nm) was grown by dc magnetron \nsputtering on Si/SiN(200) substrate, pre -patterned by e-beam lithography, and subsequently lift ed off. \nA Ti hard mask was defined by a s econ d step of e -beam lithography, electron evaporation , and lift -off. \nThe hard mask protected the circular area s corresponding to the dot s during the Ar -ion milling that was \nused to etch the layers above Pt(5) and define the Hall cross es. Finally, Ti(5) /Au(50) contact pads were \nfabricated by optical litho graphy and electron evaporation, followed again by lift -off. \nElectrical setup. \nWith reference to Fig. 1, the pulses are produced by a reverse -terminated pulse generator (Kentech \nRTV40 ) with variable pulse length (0.3 -20 ns, rise time < 0.3 ns) and adjustable polarity, and fed to a \ndirectional coupler, which delivers a small portion ( -20 dB) of the signal directly to the oscilloscope \n(trigger). The balanced -unbalanced (balun) power divide r (200 kHz – 6 GHz , Marki Microwave BAL -\n0006 ) splits the signal into two balanced pulses, with very similar amplitude. Next, the pulses travel to \nthe Hall cross through identical paths. The four bias-Tees next to it combine the rf and dc sub-networks \nof the circuit, allowing both time -resolved (oscilloscope) and static (lock -in amplifier) measurements. \nPrior to detection, the transverse Hall potentials are amplified by amplifiers (Tektronik PSPL 5865 ) with \n26.5 dB voltage gain, 30 ps rise time and 30 kHz – 12 GHz bandwidth. The oscilloscope is also a \nTektronik instrument, with 2.5 GHz bandwidth, 20 GS a/s sampling rate , and 50 Ohm ac -coupled input \nimpedance. A lock-in amplifier (Zurich Instruments MFLI) generates a small low -frequency sinusoidal \ncurrent (𝐼out, 100 -200 µA, 10 Hz) and demodulates the corresponding static anomalous Hall voltage \n(𝑉in). The Hall cross lies on a custom -built printed -circuit board with SMA connections and is contacted 13 \n electrically by Al wire bonds. The device is located betwee n the pole pieces of an electromagnet, whose \nmagnetic field 𝐵 can be varied in amplitude and direction within the 𝑥𝑧 plane. \nFits of the time -resolved Hall voltage traces . \nWe fit the individual normalized switching traces with a piecewise linear function of the form: \nUP−DOWN: 𝑦(𝑡)={1, 𝑡<𝑡0\n1−𝑡−����0\n∆𝑡,𝑡0<𝑡<𝑡0+∆𝑡\n0,𝑡>𝑡0+∆𝑡 \nDOWN−UP: 𝑦(𝑡)={0, 𝑡<𝑡0\n𝑡−𝑡0\n∆𝑡,𝑡0<𝑡<𝑡0+∆𝑡\n1,𝑡>𝑡0+∆𝑡 \nfor up-down and down -up switching, respectively . 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Appl. 10, 044060 \n(2018). \n \nACKNOWLDEGEMENTS \nThis work was funded by the Swiss National Science Foundati on (Grant S No. 200020 -172775 and No. \nPZ00P2 -179944 ), the Swiss Government Excellence Scholarship (ESKAS -Nr. 2018.0056) and the ETH \nZurich (Career Seed Grant SEED -14 16 -2). \nAUTHOR CONTRIBUTIONS \nP.G., E.G., G.S., and T.D. conceived the experiments. G.S. and V.K. developed the setup and the \nmeasurement protocol. C.-H.L. deposited the samples. G.S. fabricated the device, performed the \nmeasurements , and analyzed the results. G.S. and P.G. wrote the manuscript. All authors discussed the \ndata and commented on the manuscript. \nCOMPETING INTEREST \nThe authors declare no competing financial interests. \nADDITI ONAL INFORMATION \nSupplementary information is available in the online version of the paper. \nCorrespondence and requests for materials should be addressed to G.S. ( giacomo.sala@mat.ethz.ch ) and \nP.G. ( pietro.gambardella@mat.ethz.ch) . \n 17 \n FIGURES \n \nFigure 1. Experimental setup for t ime-resolved Hall effect measurement s. a, The injection in a Hall \ncross of a single pulse with amplitude 𝑉P causes current (𝐽) shunting in the transverse sensing line (along \n𝑦 in the upper panel ). In contrast, t wo pulses with opposite polarity (𝑉P\n2) that meet at the center of the \nHall cross impose a virtual ground, thereby forcing the current to propagate along the main cha nnel \n(along 𝑥 in the bottom panel ). b, Schematics of the rf setup. The initial pulse is fed to a balun divider, \nwhich splits the signal into two half pulses with opposite polarity that reach the device at the same \ninstant. The current -induced transverse Hall potentials are amplified and detected by the oscilloscope, \ntriggered by an attenuated portion of th e initial pulse. Note that the electric paths traversed by 𝑉+ and 𝑉− \nare symmetric and have equal length in the real setup . The dc sub -network (lock -in amplifier and bias -\nTs, dashed lines ) allows for the static characterization of the device. c, The devi ce is a 1 -µm-wide \nferrimagnetic GdFeCo dot at the center of a Pt Hall cross, as shown by the false -color scanning electron \nmicrograph. The in-plane magnetic field 𝐵x is collinear to the current. The scale bar corresponds to 1 \nµm. d, Out-of-plane hysteresis loop of a GdFeCo dot measure d by the anomalous Hall effect. \n \n \n18 \n \nFigure 2 . Switching dynamics of ferrimagnetic dots. a, Reference (in black) and switching traces of \nPt/GdFeCo dots for 20 -ns-long voltage pulses of increasing amplitude, showing up -down (blue lines ) \nand down -up (red lines ) reversals. The curves are averages of 1000 events. The in-plane magnetic field \nis 125 mT. b,c, Normalized down -up and up -down switching traces at different pulse amplitude s \ncorresponding to the traces in a. The current density in the Pt layer corresponding to a pulse amplitude \nof 1.4 V is ≈ 5. 2 × 1011 A m-2. d,e, Normalized down -up and up -down switching traces at different in-\nplane fields, for pulses with 1.6 V amplitude. In all the measurements the current was positive, whereas \nthe field was positive (negative) in c,e (b,d). \n \n \n \n \n \n \n \n 19 \n Figure 3. Single -shot Hall effect measurements. Normalized single -shot traces of Pt/GdFeCo dots for \n20-ns-long pulses. The pulse amplitude is 1.4, 1.8, and 2.2 V in a, b and c, respectively. The in-plane \nmagnetic field is 125 mT. The pulse amplitude in a is close to the threshold switching voltage (see \nSupplementary Note 3). The black lines are fits to the traces with a piecewise linear function . The \nbottom -most curve in each graph is the average of the 10 traces above , fitted with the cumulative \nGaussian function (red) . \n \n \n \n \n \n \n \n \n \n \n20 \n \n \nFigure 4. Distribution of nucleation and transition times. a,b Percentage distributions of the \nnucleation time 𝑡0 and transition time ∆𝑡 for different amplitudes of 20 -ns long voltage pulses, extracted \nfrom the fits of the single -shot traces. At 1.4 V, the magnetization does not switch in 22.5% of the events ; \nthese events are not included in the plot. The in-plane field is 125 mT . c,d Same as a,b, for different in-\nplane field s at a constant pulse amplitude of 1.8 V. At 100 mT, the magnetization does not switch in \n9.8% of the events. At 200 mT, the left -most bin includes 24% of the events. This is likely an artifact of \nthe fits due to the limited signal -to-noise ratio of the traces, which causes difficulties in fitting the \ndynamic s close to the rising edge of the pulse . To ease the comparison, in all of the g raphs the binning \nsize is 250 ps, larger than the temporal resolution of 100 ps. \n \n 21 \n Figure 5. Switching with short pulses. Threshold switching voltage ( black dots, left scale ) and energy \ndensity ( red dots, right scale ) as a function of the pulse length. The critical switching voltage is \ndetermined by after-pulse probability measurements as the voltage at which the device switches in 50% \nof the trials (see S uppleme ntary Note s 3 and 6). The applied in -plane field is 100 mT. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n22 \n SUPPLEMENTARY INFORMATION \n \nTable of content s \nNote 1. Sensitivity of the technique \nNote 2. Temporal resolution of the technique \nNote 3. Sample characterization \nNote 4. Measurement protocol and analysis of raw signals \nNote 5. Compensation of resistance offsets \nNote 6. Switching with short pulses \nSupplementary References \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 23 \n Supplementary Note 1. Sensitivity of the technique \nThe smallest detectable signal is determined by the relative amplitude of the time -resolved anomalous \nHall signal and the noise of the electric circuit. In what follows, we estimate the sensitivity of our \ntechnique by calculating the anomalous Hall voltage generated by the magnetic dots (see Fig. S1a), the \ntime-resolved amplified voltage , and the superimposed noise. \nThe anomalous Hall voltage 𝑉𝐻 depends on the transverse anomalous Hall resistance 𝑅𝑥𝑦 and \non the current 𝐼𝑥, thus it can expressed as \n𝑉𝐻=𝑅𝑥𝑦𝐼𝑥=𝑅𝑥𝑦𝑉𝑃\n𝑅𝐼, \nwhere 𝑉𝑃 is twice the amplitude 𝑉𝑃/2 of the positive (or negative) pulse in Fig. S1b , and 𝑅𝐼 the \nresistance of the injection line. 𝑅𝑥𝑦 is directly proportional to the anomalous Hall resistivity 𝜌𝑥𝑦, but the \ncomparison with values reported in the literature is not immediate because of geometrical reasons. First, \nthe current distribution is highly inhomogeneous in the Hall cross. Second, most of the current flows \nthrough the Pt layer but a small portion enters also the GdFeCo dot and propagates vertically. Third, the \nanomalous Hall effect do es not extend over the entire cross but is limited to the dot area. This geometry \nis very different from the typical experimental configuration used to measure 𝜌𝑥𝑦, namely a multi layer \nHall bar, where the current spreads out in the magnetic layer , which is continuous and extend s to the \ntransvers e probes, i.e., the sensing line. To account for these differences, we introduce three geometrical \nparameters. We define the filling factor 𝐹= 𝜋(𝐷\n2𝑤)2 as the ratio between the areas of the dot and the \ncentral portion of the cross (see Fig. S1a), arguin g that the Hall signal scales with the magnetic area. In \naddition, we introduce the sensitivity factor 𝜀, which represents the finite sensitivity of the probes to \nvariations of the electric potential in the cross1,2. 𝜀 is determined by the dimension of the Hall cross and \nSupplementary Figure S1 . Electrical model of the Hall cross. a, Scanning electron micrograph of \nthe 1 -µm-wide dot and Hall cross , and associated resistors. 𝑤 and 𝐷 are the width of the Hall cross \nand the diameter of the dot, respectively. b, Equivalent electric circuit of the Hall cross, with the \nassociated resistors, voltage sources , and amplifier. The dashed rectangle corresponds to the \nschematic in c, which repre sents the equivalent model of the amplifier. d, Model of the Johnson \nnoise: every resistor is replaced by an ideal noise -free resistor and a noise voltage source. \n24 \n by the position of the dot with respect to its center. Finally, we add a parameter 𝛿 that takes into account \nthe non-uniform contribution to the anomalous Hall voltage across the thickness of GdFeCo. The value \nof 𝛿 is determined by the specific current distribution within the volume of the dot and by the relative \nweight of volume and interface as sources of the anomalous Hall voltage. Therefore, given the thickness \n𝑡 of the Pt layer, the anomalous Hall voltage reads \n𝑉𝐻= 𝜀𝛿𝐹𝜌𝑥𝑦\n𝑡𝑉𝑃\n𝑅𝐼. (1) \nThis voltage is then amplified and measured in real time. Figure S1b presents the equivalent electric \ncircuit of the Hall cross. We model the device with four resistors ( 2× 𝑅𝐼\n2,2 ×𝑅𝑆\n2) connected through a \ncentral node. The resistors represent the two branches of the injec tion line with total resistance 𝑅𝐼, along \nwhich the pulses are injected, and the two branches of the sensing line with resistance 𝑅𝑆 used to probe \nthe anomalous Hall voltage . Since the resistance difference between the two branches of the sensing \n(injection) line is a few Ohm at most, we assume for simplicity that the branches are equal in pairs. For \nthe sensing line, this hypothesis is equivalent to considering an ideal offset -free transverse voltage. The \nanomalous Hall effect can be modelled by two vo ltage supplies of opposite sign ( ±𝑉𝐻\n2) placed along the \ntwo sensing branches. At the centre of the cross, the counter -propagating pulses enforce a virtual ground. \nThen, the differential Hall signal 𝑆 measured at the input ports of the oscilloscope is th e result of the \namplified voltage partition between 𝑅𝑆\n2 and the input resistance of the amplifier 𝑅𝐴: \n𝑆=2𝐺𝑉𝐻\n2𝑅𝐴\n𝑅𝐴+ 𝑅𝑆\n2 (2) \nHere, the amplifier is treated as a simple ideal amplifying stage with gain 𝐺 and 𝑅𝐴= 50 Ohm input and \noutput impedances (see Fig. S 1c). The input resistance of the oscilloscope is also 𝑅𝑆= 50 Ohm. \nThe measured amplified signal is accompanied by noise, which originates mainly from the \nJohnson noise of the resistors ( 𝑁𝐽), the noise of the pulse generator ( 𝑁𝑃) caused by its output impedance, \nthe resolution of the oscilloscope 𝑁𝑆𝐶, and, above all, the noise figure ( 𝑁𝐹) of the amplifiers. Additional \nnoise sources are the passive electric devices present in the circuit (bias -Tees, couplers, balun divider). \nMoreover, the wire bonds and our printed circuit board pick up electromagnetic disturbances from the \nenvironment. However, these extra noise contributions are negligible compared to 𝑁𝐽, 𝑁𝑃, 𝑁𝑆𝐶, and 𝑁𝐹. \nWe model the Johnson noise by replacing each r esistor in Fig. S1b with the equivalent Thevenin circuit, \ncomprising of an ideal resistor of the same resistance 𝑅 and voltage source 𝑁𝐽= √2 𝑁𝑟𝑚𝑠=\n √8𝑘𝐵𝑇∆𝑓𝑅, with 𝑘𝐵𝑇≈4.1×10−21J the thermal energy and ∆𝑓≈50 MHz the bandwidth (20 ns \npulses). Th e resulting equivalent noisy circuit is sketched in Fig. S1d. It can be simplified by condensing \nthe contributions of all the resistors into a single effective resistance 𝑅𝑒𝑓𝑓: \n𝑅𝑒𝑓𝑓=\n( 2𝑅𝐴𝑅𝐼\n2𝑅𝐴+𝑅𝑆+2𝑅𝐼\n 𝑅𝐼\n2+(𝑅𝐴+ 𝑅𝑆\n2)𝑅𝐼\n2𝑅𝐴+𝑅𝑆+2𝑅𝐼 ) 2\n𝑅𝐼\n2+ \n( 𝑅𝐴+2𝑅𝐴𝑅𝐼\n8𝑅𝐴+4𝑅𝑆+2𝑅𝐼\n𝑅𝐴+ 𝑅𝑆\n2+(𝑅𝐴+ 𝑅𝑆\n2)𝑅𝐼\n4𝑅𝐴+2𝑅𝑆+𝑅𝐼 ) 2\n𝑅𝑆\n2. \nThen, the input noise to each amplifier is \n𝑁𝑖𝑛=√8𝑘𝐵𝑇∆𝑓𝑅𝑒𝑓𝑓+𝑅𝐴\n𝑅𝐴+ 𝑅𝑆\n2 𝑁𝑃. \nConsidering also the Johnson noise of the oscilloscope’s input impedance 𝑅𝑆𝐶 and the digital -to-\nanalogue quantization, the total noise superimposed to the signal reads 25 \n 𝑁=2(𝐺𝑁𝑖𝑛+ 10𝑁𝐹\n10𝐺𝑁𝑖𝑛+ √8𝑘𝐵𝑇∆𝑓𝑅𝑆𝐶+10𝑉𝑅\n28), (3) \nwhere the first term represents the amplified sum of the Johnson and pulse generator nois es, the second \nterm the noise introduced by the amplifier, and the third term the Johnson noise of the input impedance \n𝑅𝑆𝐶 of the oscilloscope. Strictly speaking, the last contribution in Eq. (3) is not noise, but the intrinsic \nfinite sensitivity of the oscilloscope. This resolution is determined by the number of bits (8) and divisions \n(10), and the voltage range ( 𝑉𝑅). Finally, the factor 2 is due to the mathematical subtraction of the \namplified 𝑉+ and 𝑉−. \nEquations (1) -(3) can be used to estimate th e signal -to-noise ratio ( 𝑆/𝑁) and the sensitivity of \nthe setup. In our case: 𝐷 = 1000 nm, 𝑤 = 1500 nm, 𝑅𝐼 = 360 Ohm, 𝑅𝑆 = 806 Ohm, 𝑅𝑆𝐶 = 50 Ohm, 𝑅𝐴 \n= 50 Ohm, 𝐺 = 20 (26 dB), 𝑁𝐹 = 6 dB, 𝑁𝑃 = 9 µV, 𝑉𝑅 = 7 mV, 𝜀 = 0.4 (Ref. 2). These values lead to: \n𝑅𝑒𝑓𝑓 = 16 Ohm, 𝑁𝑖𝑛 = 6.1 µV, 𝐹 = 0.35. We a ssume a pulse with amplitude 𝑉𝑃 = 2.2 V, an anomalous \nHall resistivity 𝜌𝑥𝑦 = 10 µOhm cm (Refs. 3,4), and 𝛿 = 0.21, the latter being chose n to match the \nexperimental 𝑅𝑥𝑦 = 0.6 Ohm. Thus, we obtain 𝑉𝐻 = 3.6 mV and 𝑆 = 7.9 mV. This last result is in good \nagreement with the experimentally measured value (cf. Fig. 2a). The total noise amounts to 𝑁 = 1.8 mV. \nThis value is to a large extent (up to 54%) determined by the noise figure of the amplifier, which \nintensifies the Johnson noise 𝑁𝑖𝑛 of the circuits. This noise is expected to become more severe as the \npulse length is reduced, i.e., the bandw idth is enlarged. For a 1 -ns long pulse, it may increase by 4 -5 \ntimes. The second largest contribution (30%) originates from the signal quantization, whereas the \ncontribution of 𝑅𝑆𝐶 is negligible. On the basis of these figures, we estimate that the sign al-to-noise ratio \nof a single measurement is of the order of 𝑆/𝑁 ≈ 4.4. Since the time traces are obtained by subtraction \nof two measurements (see Supplementary Note 4) , the 𝑆/𝑁 of the individual time trace (single -shot \nmeasurements ) reduces to ≈ 2.2. By averaging over 1000 switching traces, the ratio can be improved \nby a factor of 30, which gives 𝑆/𝑁 ≈ 66. This estimate matches reasonably well the actual signal -to-\nnoise ratio of the average traces in Fig. 2a (bottom panel, 2.2 V pulse amplitude) , which have about 6.5 \nmV and 0. 15 mV signal and root-mean -square noise amplitude, respectively. \nThese considerations explain why our technique is advantageous. Without the compensation of \nthe pulses at the centre of the Hall cross, the transverse signals 𝑉+ and 𝑉− are of the same order of \nmagnitude as the injected pulse, e.g., 1 V. The magnetic signal is thus a tiny variation on the order of a \nfew mV on top of the large background. In such conditions, a much higher range 𝑉𝑅 is required to \naccommodate the entire signal into the available divisions of the oscilloscope. As a consequence, the \nfinite vertical resolution becomes dominant over the rest of the noise and masks the magnetic signal. \nSourcing the oscilloscope with the differential signal 𝑉+−𝑉− would definitely improve the resolution \nby removing part of the background. Still, this approach would not solve completely the problem, \nbecause of the asymmetries between the sensing branches. In contrast, our technique minimizes the \ncurrent spread and hen ce allows for exploiting the full acquisition range of the oscilloscope to probe \nonly the magnetic signal. \nThis analysis suggests also a few directions for further improvements. In the first place, the \ndevice geometry and the materials (thickness, resistiv ity) could be designed to maximize 𝑉𝐻. For \nexample, the anomalous Hall resistance could be enhanced by increasing the ratio between the width of \nthe sensing arms and the dot diameter5,6, so as to increase the fact or 𝜀. Likewise, the central area of the \ncross should be made the smallest possible, compatibly with the dot size. This optimization becomes \nfundamental when downscaling the devices to sub -µm dimensions. However, the device optimization \nis not free from constraints because the anomalous Hall voltage, the current density required to induce \nthe magnetization switching, the geometry of the Hall cross, and its resistance are not independent. For \ninstance, the device miniaturization, which would enlarge 𝑉𝐻, would also increase the resistance of both \nthe injection and sensing lines, hence the Johnson noise. Therefore, an alternative option is the 26 \n optimization of the setup. At the present stage, the critical source of noise in our circuitry is the voltage \nampli fier. With all other parameters fixed, amplifiers with a 1 dB noise figure are expected to provide \n𝑆/𝑁 = 3.5 for the single -shot traces. Additionally, the subtraction of 𝑉+ and 𝑉− prior to detection by the \noscilloscope should improve the 𝑆/𝑁 by permi tting the reduction of 𝑉𝑅. If 𝑉𝑅 is reduced to the minimum \nof our oscilloscope (2 mV), then the 𝑆/𝑁 would further increase to 5.4. The subtraction could be done \nwith an additional balun used in the opposite configuration, namely, with the input signa ls 𝑉+ and 𝑉− \nconnected to the inverting and non -inverting ports of the device. \n \nSupplementary Note 2. Temporal resolution of the technique \nAs described in the main text, the temporal resolution is determined by the sampling and by the \nacquisition mode (real time, interpolated real time, etc.). In this work, the traces were acquired in the \ninterpolated real -time mode, which allows for a nomina l temporal resolution of ≈ 100 ps, sufficient to \ntrack the dynamics of ns -long pulses. For shorter pulses, the nominal resolution could be improved to a \nfew ps by using a faster oscilloscope . We note that the other elements of the circuit and the cabling m ay \ndistort the shape of the electrical excitation if their transfer function does not match the required \nfrequency range, but they do not influence the temporal resolution. Instead, it is of primary importance \nto ensure the equal length and symmetry of the injection (sensing) lines of the circuits to guarantee the \nsynchronization of the injected (sensed ) signals. \nThe shortest traces that we could reliably measure correspond to 2 -3 ns-long pulses. This \nlimitation has a different “extrinsic” origin than the c ircuit components , namely the geometry of the Hall \ncross, which was not specifically desig ned for transmitting rf pulses , and, above all, the use of wire \nbonds to contact the device, which are inductively coupled . As a consequence, the raw traces have edge \nspikes w ith about 1 ns FWHM (see Fig. S4 c) that complicate the analysis of the magnetic traces for \npulses shorter than 1 ns. The replacement of the wire bonds with rf probes would improve the \ntransmission of sub -ns pulses. We stress that these limitations affect the length of the puls es, but not the \ntemporal resolution, which remains 100 ps and ca n be independently improved . \n \nSupplementary Note 3. Sample characterization \nFigure S2a reports the hysteresis loops of a Gd 30Fe63Co7 device as probed by static measurements of the \nanomalous Hall resistance, with field applied perpendicular to the plane (polar angle = 0°) and almost \nin plane (89°). The sense of rotation of the hysteresis loop indicates that the magnetizatio n is dominated \nby the transition metals Fe and Co . The GdFeCo layer has perpendicular magnetic anisotropy, with an \neffective anisotropy field of the order of 300 mT. The saturation magnetization was estimated to be 25 \nkA/m using SQUID magnetometry performe d on a full film sample. The device can be reliably switched \nbetween the up and down states by bipolar electric pulses in presence of an in -plane magnetic field \ncollinear with the current direction, as typical of spin -orbit torques (see Fig. S2b). \nWe note that the GdFeCo devices studied here belong to a batch of samples with variable Gd \nconcentration that cross the magnetization compensation temperature. However, we found that the \nfabrication steps alter the properties of the devices with respect to those of the full films. This undesired \nchange is one of the limitations of amorphous ferrimagnets, which are particularly sensitive to standard \noperations such as the ion milling and the resist baking. These issues have already been observed by \nother gro ups (see e.g., Ref. 7–9) and are possibly caused by the selective oxidation or migration of t he \nrare-earth atoms10. Our estimate, based on the variation of the magnetization compensation te mperature \nwith the Gd concentration (about 30 K every 1%) , is that the magnetization compensation temperature \nis around 250 K. Because of Joule heating during pulsing, we are confident that the magnetization of \nour devices is always “FeCo -like” for the tim e-resolved Hall effect experiments reported in this work, \nwhich were all performed in ambient conditions . 27 \n In order to determine the working point required to induce the switching, we measured the \nprobability of switching as a funct ion of in-plane magnetic field and pulse amplitude . To this aim, we \nused the dc sub -network of the circuit shown in Fig. 1 in the main text . The procedure was the following. \nWe applied a sequence of set -reset rf pulses with identical length and amplitude but opposite polarity. \nThe variation of the transverse dc resistance before and after each pulse was compared with the \nanomalous Hall resistance to assess the outcome of the pulse: if the variation was larger than 75 % of \nthis reference, we considered that the pulse succeeded in switching the magnetization. Every pulse \nsequence comprised 50 set -reset pairs of pulses , and the switching probability was defined by the ratio \nof successful pulses to 50. We repeated this procedure for different pulse lengths, amplitudes , and fields , \nas reported in Fig. S3a-d. As expect ed, the minimum voltage for 100 % switching decreases as the field \nor the pulse length are increased \nSupplementary Note 4. Measurement protocol and a nalysis of raw signals \n \nIn an ideal scenario, the signal measured by the oscilloscope should approximately resemble a \n“rectangle”, that is, it should replicate the temporal profil e of the applied electric pulse . In such a case, \nthe amplitude of the signal (height of the rectangle) would already represent the measurement of the \nmagnetization state. If the magnetization was in equilibrium, the amplitude would remain constant, to a \nhigh or low level in dependence of the up or down orientation of the magnetizat ion. During the \nswitching, instead, the trace would transition from one level to the other. However, spurious \nnonmagnetic contributions alter the ideally -rectangular profile of the measured signal. These \ncontributions have multiple origins. First, the edge of the pulses have large spikes caused by the \ninductive coupling between the wire bonds and the electric contacts of the PCB. Second, the device \nitself, which is not adapted to radio frequencies, distorts the pulses and hence the measured signal. In Supplementary Figure S2 . Sample characterization. a, Hysteresis loops measured by the \nanomalous Hall resistance with the field applied out of plane (0°) and in plane (89°) . b, Switching \nof the magnetization by a sequence of positive set (V > 0) and negative reset (V < 0) pulses with \nlength of 20 ns and amplitude of 1.6 V. The in -plane field was 150 mT. Note that the switching \namplitude is smalle r than the anomalous Hall amplitude in a because of the tilt induced by the applied \nfield (cf. with the red trace in a at 150 mT). \n28 \n addition, the voltage amplifier introduce s high-frequency oscillations. Finally, as shown in Fig. S4a,b, \nthe voltage difference between the inverted (I) and non -inverted (NI) pulses at the output of the balun \ndivider is several mV, which is about 1% of the pulse amplitude. This component is quite small with \nrespect to the input pulse. Yet, the unbalance causes a residual current leakage through the transverse \narms which adds a small voltage offset (comparable to or smaller than the magnetic signal). Therefore, \nthe magnetic signal is better extracte d from the raw traces by comparing measurements of a reference \nand the switching and removing the non -magnetic part. In fact, every trace of the same type as in Fig. \n2b-e of the main text results from the combination of two measurements. The procedure that we adopt \nto isolate the magnetic signal is the following11. \nFirst, in the presence of a positive in -plane magnetic field, we acquire a background signal by \nrepeatedly injecting identical pulses with the same current direction and amplitude . In these conditions, \nthe magnetization remains in the equilibrium state, which is determined by the field direction and the \nsign of the spin -orbit torques defined by the current polarity. The latter equals the polarity of the pulse \ntravelling along +𝑥, that is , from left to right in Fig. 1a in the main text. T he differential voltage 𝑆 \nmeasured during each pulse is nominally always the same, but we average over multiple pulses, typically \n5000, to reduce the noise. Then, we repeat the same step for the op posite field direction and the same \ncurrent polarity, to acquire the background signal corresponding to the opposite equilibrium state (see \nFig. S4c). By definition, all the undesired contributions do not change with the magnetic configuration \nof the devic e, hence they can be removed by subtracting the two signals. Their difference yields the net \nmagnetic contrast: reference trace = Background ( 𝐵 < 0) – Background ( 𝐵 > 0). This is the black trace \nshown in Fig. S4d as well as in Fig. 2a. Since for 𝑉 > 0 a nd 𝐵 < 0 (𝐵 > 0), the magnetization remains in \nSupplementary Figure S3 . Switching probability . a-d, After -pulse switching probability as a \nfunction of pulse amplitude, for different pulse lengths and in -plane fields. 29 \n the up (down) state, corresponding to positive (negative) anomalous Hall voltage, the reference trace so \ndefined has positive sign. \nNext, we acquire the signal corresponding to the switching of the magnetization by slightly \nvarying the procedure, that is, by delivering a train of set-reset pulses with alternating polarity. Now, at \neach pulse the current direction changes and so does th e magnetization. For example, for positive field, \nthe positive current causes the up -down switching, whereas the successive negative current induces the \ndown -up reversal. By averaging over 1000 pulses of the same polarity, we acquire the green signal \nshown in Fig. S4 c (a positive in -plane field is applied). It coincides initially with the signal for \nBackground ( 𝐵 < 0) (magnetization up) and during the pulse it transitions to the signal for Background \n(𝐵 > 0) (magnetization down). Therefore, similarly to t he reference trace, the signal 𝑆 associated to a \nswitching event is combined with one of the two backgrounds: switching trace = S ( 𝐵 > 0) – Background \n(𝐵 > 0). The application of this procedure leads to the blue trace in Fig. S4d as well as to the trac es in \nFig. 2. The ≈ 0 mV (≈ 5 mV) trace level identifies the uniformly -magnetized down (up) state, whereas \nany deviation of the traces from the top and bottom levels correspond to a tilt of the magnetic moments \nor to a multi -domain configuration. Finally, the normalization of the switching trace to the reference \ntrace provides the purely -magnetic time trace s (cf. Fig. 2b -e in the main text ). The same identical \napproach is used for detecting single -shot events, with the only difference that, instead of aver aging, \nevery single switching signal is recorded . The procedure that we adopt to measure and remove the \nbackground signal is very similar to that reported in Ref. 12. Therefore, our measurement protocol is \ncomparable to that of standard time-resolved Hall measurements . \nFinally, w e note that the reference trace can be acquired by using protocols different from ours, \nwhich is adapted to the specific case of spin -orbit torque switching. For example, the background signals \nSupplementary Figure S4 . Analysis of raw signals. a, Inverted (I) and non -inverted (NI) pulses at \nthe outputs of the balun divider, for a 20 ns, 1.6 V input pulse; the sign of the I pulse has been \ninverted for comparison. b, Close -up of the difference between the I and NI pulses (I - NI). Inset: \nfull voltage difference between the two pulses. c, Average raw electric signals corresponding to the \nbackground , for the two in-plane field directions, and to the switching (for positive field) . d, \nReference and switching traces obtained by subtraction of the signals in c. \n30 \n of perpendicularly -magnetized samples could also be acquired by fixing the magnetiz ation with out -of-\nplane fields . If the polarity of the current has an effect, a reference could also be obtained by comparing \nbackground signals measured with opposite curre nt polarity. Alternatively, the signal measured with a \nlow-amplitude pulse could be used as background: under the assumption that the low amplitude does \nnot produce magnetic changes, the corresponding trace could be subtracted from a higher -amplitude \ntrace after proper rescaling. In antiferromagnets, repeated pulses produce a memristive -like switching. \nThen, the background trace could be obtained after applying a sequence of repeated pulses that saturate \nthe read -out signal to the maximum (or minimum) level . Therefore, in general, t he measurement \nprotocol can be adapted to the specific application . \nSupplementary Note 5. Compensation of resistance offsets. \nOur technique does not imply a more complex circuit or measurement protocol than traditional \ndifferential Hall measurements. For comparison, we consider the work by Yoshimura et al. (Ref. 12). In \nour setup, we included DC components to simultaneously access the static electric and magnetic \nproperties of the devices. Once the DC subnetwork, which is not necessary for time -resolved \nmeasurements, is removed, the sole difference between the differential Hall measurement presented in \nRef. 12 and our technique is the balun divider. The balun is a simple, small, and affordable component \nthat does not require any power supply and easily fits into any electrical setup. \nAs an additional advantage, our technique allows for compensating possible resistive offset s \nthat are caused by the imperfect fabrication or are intrinsic to asymmetric devices. To prove this point, \nwe have measured the raw electrical signals corresponding to the “up” and “down” magnetiz ation states \nin a Hall bar device with two off -centered Hall crosses (see Fig. S5a). In contrast to the symmetric Hall \ncross considered in the manuscript, in this device the electric potentials determined by the two pulses at \nthe center of the right Hall c ross are different because of the asymmetric resistance load. As a result, the \ncurrent does flow in the transverse arms and the signals measured on the oscilloscope pre sent a finite \noffset (see Fig. S5 b). Since this offset is not negligible, to acquire the signals we could not use the the \nmaximum vertical resolution of the oscilloscope . Such problems can be circumvented by correcting the \npulses amplitudes to enforce the virtual ground at the position of the Hall cross. In the specific case \ndiscussed here, w e added a 4 dB attenuator along the direction of the negative pulse. Thanks to this \nadjustment, the vertical offset was removed f rom the raw signal, which allowed us to exploit the highest \nvertical resolution of the oscilloscope. Therefore, our techni que d oes not require the device under test \nto be longitudinally symmetric. Although we do not have at our disposal devices with asymmetric \ntransverse Hall arms, we believe that transverse resistance offsets could be compensated in the same \nway as for the longitudinal offset. Since comme rcial attenuators provide attenuation steps as small as \n0.5 dB (= 0.944), the amplitude of the pulses can be tuned with rather large precision. This capability is \na specificity of our technique, for no such countermeasures can be taken in standard differen tial Hall \nmeasurements. \n \n \n \n \n \n 31 \n \n \n \n \n \n \nSupplementary Note 6. Switching with short pulses. \nThe measurements presented in the main text were performed with 20-ns-long pulses . These relatively \nlong pulses allow us to clearly identify the different phases of the dynamics. In Fig. S6 we present \nadditional average time -resolved measurements performed with 5 -ns-long pulses. At the largest pulse \namplitude the nucleation time is reduced down to about 800 ps. This decrease is consistent with the \nafter-pulse probability measureme nts shown in Fig. S7a, which shows the switching probability \nmeasured as a function of the pulse amp litude and length for a constant in -plane field of 100 mT . The \nplot demonstrates that d eterministic switching can be o btained with pulses as short as 300 ps , which \nimplies quenching of the nucleation time at sufficiently high pulse amplitudes. From Fig. S7a we \nextracted the threshold switching voltage, defined as the voltage at which the device switches in 50% of \nthe trials, and plotted it against 1/𝑡𝑃 in Fig. S7b (see also Fig. 5 in the main text). Below approximately \n5 ns, the voltage increases linearly with the inverse of 𝑡𝑃, which is a signature of the intrinsic regime \nwhere the switching speed depends on the rate of angular momentum transfer from the current to the \nmagnetic layer. On the other hand, the different dependence for 𝑡𝑃> 5 ns reveals the importance of \nthermal e ffects for the typical pulse lengths used in this study ( 𝑡𝑃= 20 ns). \n Supplementary Figure S5. Compensation of resistance offsets. a) Hall bar with off -centered Hall \ncrosses. The anomalous Hall effect is measured in the right Hall cross. The negative pulse moving \nfrom right to left is attenuated by 4 dB compared to the positive pulse. The scale bar corresponds to \n4 µm. b) Raw differential Hall voltage 𝑉𝐻= 𝑉+−𝑉−, with uncompensated (UN) and compens ated \n(C) resistance offset, corresponding to the up and down magnetization states for current pulses that \ndo not induce switching. \n32 \n \nSupplementary Figure S6. Switching with 5 -ns pulses. Normalized average traces showing the up -\ndown magnetization switching with 5 ns -long pulses of different amplitude. Both the current and the \nin-plane 125 mT field were positive. \nSupplementary Figure S7. Switching as a function of pulse length. a) Dependence of the switching \nprobability on the pulse amplitude for different pulse lengths. Each point is the result of 50 trials. \nThe applied in -plane field was 100 mT. Note that these measurements were performed on a different \ndevice than that used for t he time -resolved measurements but they were fabricated at the same time \nfrom the same layer. b, Threshold switching voltage (black dots, left scale) and energy density (red \ndots, right scale) as a function of the inverse pulse length. The critical switching voltage is \ndetermined from a as the voltage at which the device switches in 50% of the trial s. \n33 \n Supplementary R eferences \n1. Cornelissens, Y. G. & Peeters, F. M. Response function of a Hall magnetosensor in the diffusive \nregime. J. Appl. Phys. 92, 2006 –2012 (2002). \n2. Webb, B. C. & Schültz, S. Detection of the magnetizati on reversal of individual interacting \nsingle -domain particleswithin Co -Cr columnar thin -films. IEEE Trans. Magn. 24, 3006 –3008 \n(1988). \n3. Hartmann, M. & McGuire, T. R. Relationship between Faraday Rotation and Hall Effect in \nAmorphous Rare -Earth —Transition -Metal Alloys. Phys. Rev. Lett. 51, 1194 –1197 (1983). \n4. Honda, S., Nawate, M., Ohkoshi, M. & Kusuda, T. Hall effect and magnetic properties in GdFe \nand CoCr sputtered films. J. Appl. Phys. 57, 3204 –3206 (1985). \n5. Kikuchi, N., Okamoto, S., Kitakami, O., S himada, Y. & Fukamichi, K. Sensitive detection of \nirreversible switching in a single FePt nanosized dot. Appl. Phys. Lett. 82, 4313 –4315 (2003). \n6. Alexandrou, M., Nutter, P. W., Delalande, M., De Vries, J., Hill, E. W., Schedin, F., Abelmann, \nL. & Thomson , T. Spatial sensitivity mapping of Hall crosses using patterned magnetic \nnanostructures. J. Appl. Phys. 108, (2010). \n7. Le Guyader, L., El Moussaoui, S., Buzzi, M., Chopdekar, R. V., Heyderman, L. J., Tsukamoto, \nA., Itoh, A., Kirilyuk, A., Rasing, T., Kim el, A. V. & Nolting, F. Demonstration of laser induced \nmagnetization reversal in GdFeCo nanostructures. Appl. Phys. Lett. 101, (2012). \n8. El-Ghazaly, A., Tran, B., Ceballos, A., Lambert, C. H., Pattabi, A., Salahuddin, S., Hellman, F. \n& Bokor, J. Ultrafast magnetization switching in nanoscale magnetic dots. Appl. Phys. Lett. 114, \n(2019). \n9. Kirk, E., Bull, C., Finizio, S., Sepehri -Amin, H., Wintz, S., Suszka, A. K., Bingham, N. S., \nWarnicke, P., Hono, K., Nutter, P. W., Raabe, J., Hrkac, G., Thomson, T. & Heyderman, L. J. \nAnisotropy -induced spin reorientation in chemically modulated amorphous ferrimagnetic films. \nPhys. Rev. Mater. 4, 074403 (2020). \n10. Hansen, P. Magnetic amorphous alloys. Handb. Magn. Mater. 6, 289 (1991). \n11. Grimaldi, E., Krizakova, V., Sala, G., Yasin, F., Couet, S., Sankar Kar, G., Garello, K. & \nGambardella, P. Single -shot dynamics of spin –orbit torque and spin transfer torque switching in \nthree -terminal magnetic tunnel junctions. Nat. Nanotechnol. 15, 111 –117 (2020). \n12. Yoshimura, Y., Kim, K., Taniguchi, T., Tono, T., Ueda, K., Hiramatsu, R., Moriyama, T., \nYamada, K., Na katani, Y. & Ono, T. Soliton -like magnetic domain wall motion induced by the \ninterfacial Dzyaloshinskii –Moriya interaction. Nat. Phys. 12, 157 –161 (2016). \n " }, { "title": "1006.1983v1.NMR_study_on_the_stability_of_the_magnetic_ground_state_in_MnCr____2_O____4_.pdf", "content": "NMR study on the stability of the magnetic ground state in\nMnCr 2O4\nDong Young Yoon, Soonchil Lee\nDepartment of Physics, Korea Advanced Institute of Science and Technology,\nDaejeon 305-701, Republic of Korea\nYoon Seok Oh, Kee Hoon Kim\nCeNSCMR, Department of Physics and Astronomy,\nSeoul National University, Seoul 151-747, South Korea\nAbstract\nThe canting angles and \ructuation of the magnetic ion spins of spinel oxide MnCr 2O4were stud-\nied by nuclear magnetic resonance (NMR) at low temperatures, which has a collinear ferrimagnetic\norder below TCand a ferrimagnetic spiral order below Ts1:3.\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. The temperature\nhysteresis of the spiral volume fraction indicates that the spiral and collinear ferrimagnetic\n7phases are generally mixed below Ts.\nThis work was supported by National Research Foundation of Korea (NRF) grants: (Nos.\nKRF-2008-313-c00290 and 2009-0078342).\n1M. Schmidt, W. Ratcli\u000b II, P.G. Radaelli, K. Refson, N.M. Harrison, and S.W. Cheong, Phys.\nRev. Lett. 92, 056402 (2004).\n2N. Tristan, J. Hemberger, A. Krimmel, H-A. Krug von Nidda, V. Tsurkan, and A. Loidl, Phys.\nRev. B 72, 174404 (2005).\n3V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. Huang, M. Miller, A.J. Schultz, and S.E.\nNagler, Phys. Rev. Lett. 100, 066404 (2008).\n4Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, and Y. Tokura, Phys. Rev. Lett.\n96, 207204 (2006).\n5P. W. Anderson, Phys. Rev. 102, 1008 (1956).\n6D. H. Lyons, T. A. Kaplan, K. Dwight, and N. Menyuk, Phys. Rev. 126, 540 (1962).\n7J. M. Hastings and L. M. Corliss, Phys. Rev. 126, 556 (1962).\n8T. W. Houston and A. J. 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": "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. 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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": "1806.01167v1.Current_induced_domain_wall_motion_in_compensated_ferrimagnet.pdf", "content": " 1Current-induced domain wall motion in compensated ferrimagnet \nSaima A Siddiqui1, Jiahao Han1, Joseph T Finley1, Caroline A Ross2 and Luqiao Liu1 \n1Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, \nCambridge, MA 02139 \n2Department of Materials Science and Engineeri ng, Massachusetts Institute of Technology, \nCambridge, MA 02139 \n \nDue to the difficulty in detecting and manipulati ng magnetic states of antiferromagnetic materials, \nstudying their switching dynamics using electrical methods remains a challenging task. In this work, by \nemploying heavy metal/rare earth -transition metal alloy bilayers , we experimentally studied \ncurrent-induced domain wall dynamics in an antiferro magnetically coupled system. We show that the \ncurrent-induced domain wall mobility reaches a maximum close to the angular momentum compensation. \nWith experiment and modelling, we further reveal the internal structures of domain walls and the \nunderlying mechanisms for their fast motion. We show that the chirality of the ferrimagnetic domain \nwalls remains the same across the compensation points, suggesting that spin orientations of specific sub-\nlattices rather than net magnetization determine Dzyaloshinskii-Moriya interaction in heavy \nmetal/ferrimagnet bilayers. The high current-induced domain wall mob ility and the robust domain wall \nchirality in compensated ferrimagnetic material opens new opportunities for high-speed spintronic \ndevices. \n 2Antiferromagnetic materials have fast intrinsi c magnetization dynamics and are insensitive to \nmagnetic fields, making them potential candidates for th e next generation of dense, high-speed spintronic \ndevices [1-5]. However, their magnetic state is difficu lt to manipulate and detect electrically. Therefore, \nstudying their high frequency switching dynamics using electrical methods remains challenging. In contrast, ferrimagnetic materials, many of which ha ve antiparallel aligned su blattices, provide another \npossible platform for realizing fast device operation [6,7], with the advantage that their magnetization \nstate can be detected or altered even near the compensation point because th eir sensitivity to current-\ninduced spin torque and their ma gneto-electric [8-12] or magneto-optical [13,14] response does not \ndisappear. Rare earth-transition metal (RE-TM) alloys are well known ferrimagnets in which the RE and \nthe TM sublattices align antiparallel with each othe r reducing the net angular and magnetic moments. \nHigh frequency magnetic resonance and picosecond ma gnetic switching by optical pulses have been \ndemonstrated in thin films of these materials [6,7 ,15,16], motivating their application in ultrafast \nspintronic devices. Recently it was demonstrated that electrical current could be used as an efficient \nswitching mechanism for RE-TM ferrimagnets even at the compensation point via spin-orbit torque [8-\n12]. However, limited by the quasi-static measurem ent techniques, the electrically driven switching \ndynamics in these materials is yet to be explor ed. Experiments on current-induced domain wall (DW) \nmotion not only provide a convenient way to study the time-dependent switching dynamics of multi-domain magnets [17-19], but also can lead to useful high density memory and logic devices [20,21]. Very \nrecently, it was shown using magnetic field driven experiments that angular momentum compensated \nferrimagnetic materials possess great velocity advant ages [22], which provides th e possibility of reaching \nhigh operational speed in devices based on those material s. However, it remains unclear if the same speed \nmaximum is retained in current-induced DW motion. Particularly, different from field driven case, \nadditional factors such as Dzyaloshinskii-Moriya in teraction (DMI) and the domain wall chirality play \nimportant roles in current-induced experiment [23-25 ]. It is an open question how DMI evolves in the \npresence of two oppositely aligned sublattices, and whet her it supports the needed chirality for efficient \nspin orbit torque induced DW motion at the co mpensation points. To answer these questions, we 3experimentally study the fast current-induced DW dynamics in compensated ferrimagnets by \ncharacterizing the DW motion in Pt/Co 1-xTbx wires with various chemical compositions and reveal the \nphysical mechanisms behind the phenomena. \nA series of Pt (3 nm)/Co 1-xTbx (2-3 nm)/SiN x (3 nm) samples was deposited using magnetron \nsputtering. The Pt underlayer provides the spin-orbit torque while SiN x is used as an insulating capping \nlayer. Figure 1(a) shows the coercive fields ( Hc) and the saturation magnetizations ( Ms) of unpatterned \nfilms at different compositions. Because of the la rge bulk perpendicular magnetic anisotropy, the easy \naxes of the samples are oriented out-of-plane. Co 1-xTbx reaches its magnetization compensation point at x \n= 0.34, where Ms is minimum and Hc diverges. We note that this compensation composition for Pt/Co 1-\nxTbx is slightly different from what has been observed previously for Ta/Co 1-xTbx samples [8], probably \ndue to the extra contribution from the proximity-indu ced magnetization in Pt [26]. Due to the unequal \ngyromagnetic ratios of RE and TM elements, th e concentration where the magnetization reaches \ncompensation is different from that with zero tota l angular momentum. Using the literature values of g \nfactors of Co (~2.2) [27] and Tb (~1.5) [28,29] atoms, we can al so estimate that the angular momentum \ncompensation point is around x ≈ 0.25. \nThe deposited films were patterned into micron size wires and Hall bar structures for DW motion \nmeasurement and anomalous Hall resistance ( RAH) characterization, respectively. The schematic structure \nof the Hall bar is shown in Fig. 1( b) along with the measurement set-up. RAH changes sign between \nCo0.67Tb0.33 and Co 0.64Tb0.36 [Fig. 1(c)], which is consistent with a transition from being Co-dominated to \nbeing Tb-dominated in magnetic moment [8,30]. Fi gure 2(a) shows the schematic of the set-up for \nstudying DW motion in 2-5 µm wide Co 1-xTbx magnetic wires. All the velocity measurements are done at \nroom temperature. First, a large extern al magnetic field along the out-of-plane z direction ( Hz) was \napplied to saturate the magnetization. Next, a 1~ 100 ms duration magnetic field pulse in the opposite \ndirection was applied to nucleate and initiate DW propagation from the large contact pad region. A \nmagneto-optical Kerr effect (MOKE) microscope w as utilized to track the position of the DWs. By 4sweeping Hz on a series of samples, we verified that th e yellow and green regions in our MOKE image \nrepresent domains where the Co sublattice points along the - z and + z direction, respectively, for all \nchemical compositions studied. This is consistent w ith the observations that the TM sublattice dominates \nthe Kerr signal for TM-RE alloys in the visible light regime [14,15]. For convenience, we will label the \ntwo different domains as ↓ and ↑ domains in the follo wing discussion, where ↓ and ↑ denote the \norientations of the Co sublattice. After the initia l nucleation of DWs, electrical current pulses of 20-ns \nduration were then applied to the wires to move the DW. The initial and the final positions of the DWs \nwere measured with MOKE microscopy and the velo city was calculated from the DW displacement and \nthe total pulse length. Figure 2(b) gives an exampl e of the positions of DWs after multiple current pulses \nof the same width. Samples with different channel widths (2 μm - 5 μm) were tested, but no dependence \nof velocity on sample width was observed (see Supplementary Fig. S1 [31]). \n The DW velocity as a function of the applied current density in a series of Pt/Co 1-xTbx wires \n(x = 0.17 - 0.41) is summarized in Fig. 2(c). In our experiment, both ↑↓ and ↓↑ DWs move along the \ndirection of the charge current (see Supplementary Fig S2 [31]), similar to the direction of DW motion \nobserved previously in Pt/ferromagnetic systems [23, 24], which supports the essential role of spin-orbit \ntorque. To exclude the contributions from differences in threshold current densities between samples, we \nfocus on the regions where the velocity and the curre nt density roughly satisfy a linear relationship. The \nDW mobility of all the samples, defined as the ratio between the velocity and the current density, is \ndetermined by fitting the slopes of the linear regions in Fig. 2(c). The results are summarized in Fig. 2(d) \nand show that the DW mobility varies by more than an order of magnitude depending on the chemical \ncomposition. Starting from the Tb-dominant samples, the mobility increases with Co concentration and \nreaches a maximum around x ≈ 0.21 ~ 0.26, which agrees with the estimated angular momentum \ncompensation point range, considering the error bars in our experimental data and the angular momentum \ncompensation calculation. We note that the fact that DW attains its maximum sp eed when the net angular 5momentum rather than the magnetization reaches zer o is also consistent with recent study on the \nfield-induced DW motion [22], despite different driving forces. \nThe DW mobility [~ 5 × 10-10 m3/(A•s)] obtained in our compensated CoTb sample is much \nhigher than what was observed previously with ferr omagnetic layers (e.g., Co FeB [23,32] and Co/Ni/Co \n[24]) where the mobility is 0.2 × 10-10 m3/(A•s) – 1 × 10-10 m3/(A•s), and is comparable to the values \nfound in synthetic antiferromagnet multilayers [33]. Compared with these previous experiments, the \ncurrent densities used in our experiment are relatively low (< 5 × 1011 A/ m2 vs 1~5 × 1012 A/ m2). To \nachieve even higher absolute values of DW velocity, larger current densities are required. We found that \nabove a certain current density (defined as the ma ximum current density), nucleation of new domains \nstarts to occur, which puts an upper limit on the a pplicable current. This is similar to a previous \nobservation in the Ta/CoFeB/MgO system, where the breakdown of DW motion was attributed to current-\ninduced weakening of the perpendicular anisotropy [32]. The decrease in anisotropy was observed in the \ntemperature-dependent vibrating sample magnetometry (see Supplementary Fig. S3 [31]). It is also noted \nthat while magnetic anisotropy decreases rapidly du e to heating effect, the changes of magnetization \nremains small (less than 10% of room temperature valu e) before the films lose coercivity (Supplementary \nFig. S3(b) [31]), suggesting that the drift of magnetization is insignificant during the current pulse \napplication. \nTo understand the evolution of DW velocity as a function of net moment, we modelled the DW \nmotion for ferrimagnetic materials with two unequal subl attices. Previously it was demonstrated that the \nmagnetic dynamics of ferrimagnets could be describ ed with the Landau–Lifshitz–Gilbert equation by \nreplacing the regular gyromagnetic ratio and da mping coefficient with the effective values, ߛ and \nߙ: ௗෝ\nௗ௧ൌെ ߛ ෝൈሬሬሬԦߙ ෝൈௗෝ\nௗ௧െߛ ఏಹ\nଶఓ బெ௧ሺෝൈෝൈෝሻ. Here, ܯൌܯ ଵെܯ ଶ,, \n ߛൌሺ ܯ ଵെܯ ଶሻ/ሺܵ ଵെܵ ଶሻ, and ߙ ൌሺ ߙ ଵܵଵߙ ଶܵଶሻ/ሺܵ ଵെܵ ଶሻ with M1,2, S1,2 and α1,2 representing \nthe magnetization, angular moment per unit volume and damping coefficient of the two sublattices [6,7]. 6 ෝ denotes the unit vector along the direction of ሬሬሬԦଵെሬሬሬԦଶ (Néel vector). t, ෝ, ߠு and j are the thickness \nof the magnetic film, the orientation of spins inject ed into the ferrimagnet, spin Hall angle and applied \ncurrent density, respectively. As is shown in Supp lementary Note 3 [31], under this replacement of ߛ \nand ߙ, the DW velocity of a ferrimagnetic wire can be derived as: ݒൌݒௌ\nඥ1ሺ ݆ ௌ⁄݆ሻଶ ൘ , where \nݒௌൌߛ ∆ܪ and ݆ௌൌସఓ బఈெ௧\nగఏ ಹܪெூ represent the saturation velocity and saturation current \ndensity, respectively. Here ܪெூ is an in-plane effective field, originating from DMI (see discussions \nbelow) and ∆ is the domain wall width. This expression of DW velocity is similar to that of the \nferromagnetic systems except that ߛ and ߙ are utilized [25]. It can be seen that unlike a typical \nferromagnet whose highest DW velocity is limited by the chirality stabilizing force -- the DMI effective \nfield, a ferrimagnet does not have a speed limit because ߛ and ߙ diverge at the compensation point. \nThis ensures the linear relationship between j and v exists throughout the whole current range. DW \nvelocities in ferrimagnets with differe nt net angular moment are calculated and compared in Fig. 2(e). It \nshows that the compensated ferrimagnetic material has velocity advantages at large or intermediate \ncurrent densities, which is consistent with our experimental observations. \n DMI plays an important role in stabilizing the DW chirality and alleviating the velocity reduction \ncaused by Walker breakdown. In a ferromagnet, the effective field from DMI directly determines the \nhighest DW velocity that can be reached [25]. Fo r a compensated ferrimagnet, as discussed above, the \nDW velocity is no longer restrained by ܪெூ. However, a non-zero DMI is still critical to ensure that a \nNéel type of DW is favorable, for which the spin or bit torque has the highest efficiency (Supplementary \nNote 3 [31]). So far little is known about th e DMI at the interface between heavy metals and \nferrimagnetic alloys. In particular, it is not clear how the DW chirality varies when the net magnetization \nor net angular momentum goes through zero as the composition varies. To characterize DMI in \nferrimagnetic Co 1-xTbx, we measured current-induced DW velocities as a function of in-plane field ( Hx) \nalong the wire direction. The results from a Co 0.79Tb0.21 sample are illustrated in Fig. 3(a) and 3(b). It can 7be seen that under positive (negative) Hx, the ↓↑ DW in Co 0.79Tb0.21 moves faster (slower) compared with \nthe zero field case. The trend is opposite for ↑↓ DWs, and the DW velocities even change direction \nat ܪ௫ൌേ1000 Oe. The fact that the motion for one type of DW is enhanced while the other is suppressed \nis consistent with the Néel wall characteristics, where the applied ܪ௫ strengthens (or weakens) the \neffective DMI field [Fig 3(c)]. This is in contrast with other DW configurations (e.g., a Bloch wall), \nwhere a symmetric variation of the DW velocity under ܪ௫ is expected. \nTo answer the question of whether the DW chang es its chirality at the compensation points, we \nplot the dependence of DW velocity as a function of Hx in Fig. 3(e)-3(g) for three different Co 1-xTbx \nsamples. Since the angular momentum and magnetization compensation points are at x = 0.25 and 0.34 \nrespectively, the samples with x = 0.21, 0.33, and 0.41 in Fig. 3(e)-3(g) represent three different cases: \nCo-dominant in both angular momentum and magnetization, Co-dominant in magnetization and Tb-\ndominant in angular momentum, and Tb-dominant in both angular momentum and magnetization, \nrespectively. First, we find that there is no qualitativ e change in the DW motion characteristics across the \nangular momentum compensation point [Fig. 3(e) and 3(f)]. Under an Hx field, the Co 0.67Tb0.33 sample \nexhibits similar behavior to the previously discussed Co 0.79Tb0.21 sample, where the velocity of ↓↑ (↑↓) \nDWs increases (decrease s) under small positive Hx. However, across the magnetization compensation \npoint, the opposite trend was seen [Fig. 3(e)], where the velocity of the ↓↑ (↑↓) DWs decreases (increases) \nunder the same positive Hx field. The sign reversal in the v vs Hx slopes across the magnetization \ncompensation point can be explained by the schema tic DW structures shown in Fig 3(c) and 3(d). A \npositive Hx field will stabilize the ↓↑ DW in magnetically Co-dominant samples, while it will destabilize \nthe ↓↑ DW in Tb-dominant samples. Therefore, the si gn reversal reflects that the left-handedness is \nmaintained throughout all our Pt/Co 1-xTbx samples, suggesting that the DMI is correlated with the spin \norientations of specific sub-lattices rather than the net magnetization . The in-plane magnetic fields which \novercome DMI and result in zero domain wall velocity for the above three compositions are summarized \nin Fig. S5 [31]. It is noted that HDMI does not simply scale following the expected relationship of ܪெூ ൌ 8ܯ/ܦ ߤݐ∆, where D, ݐ ,ߤ and ∆ representing the interfacial DMI energy density, film thickness, \nvacuum permeability and DW width [25]. Instead, samp les with higher Tb concentration tend to have \nlarger HDMI , which could be attributed to the strong spin orbit coupling associated with rare earth element. \nWe note that besides allowing for fast DW movement , the strong DMI and robust chirality exhibited in \nour compensated ferrimagnet provide the possibility of engineering skyrmion structures with zero total \nangular momentum. Because of the cancelation of the side deflections from the skyrmion Hall effect of \ntwo sublattices, these compensated skyrmions are expected to have greatly enhanced mobility [4,34]. \nTo summarize, we experimentally investigated the fast domain wall dynamics in Co 1-xTbx \nferrimagnetic samples. We found that the domain wa ll mobility reaches a maximum in samples close to \ncompensated angular momentum and it is higher than those in the ferromagnetic electrodes. The high \ndomain wall velocity in a compensated ferrimagnetic mate rial is consistent with our theoretical modelling, \nwhere it is shown that the absence of velocity saturation ensures a high mobility. By measuring the \ninfluence of in-plane field on the domain wall velocity , we further demonstrated that the domain walls \nhave chiral internal structures which are stabilized by the Dzyaloshinskii-Moriya interaction and the same \nchirality is maintained across the compensation points. Thus we identifies that the Dzyaloshinskii-Moriya \ninteraction in ferrimagnetic materials is related to the spins of the sublattices in contrast to the net \nmagnetization. Our study on current-induced domain wall motion in ferrimagnets opens the opportunity \nto electrically probe the fast domain wall dynamics in angular-momentum compensated systems. The \nlow magnetic moment, large electrical and optical r esponse, as well as the possi bility of reaching high \nspeed dynamics makes it highly attractive to employ ferrimagnets for spintronic applications. \n 9This research was partially supported by the Na tional Science Foundation un der grant 1639921, and the \nNanoelectronics Research Corporation (NERC), a wholly-owned subsidiary of the Semiconductor \nResearch Corporation (SRC), through Memory, Logi c, and Logic in Memory Using Three Terminal \nMagnetic Tunnel Junctions, an SRC-NRI Nanoelectr onics Research Initiative Center under Research \nTask ID 2700.001. \n 10FIG. 1. (a) Saturation magnetization and coercive fields of Pt/Co 1-xTbx thin films from vibrating sample \nmagnetometry. (b) Schematics of the device geometry and electrical setup for Hall resistance \nmeasurement. (c) Anomalous Hall resistance of 4- μm wide Pt/Co 1-xTbx Hall bars. The coercive fields of \nthe patterned structures differ from that of the c ontinuous films due to domain nucleation and pinning \nprocesses at wire edges. \n \n 11\n \nFIG. 2. (a) Electrical set-up for the domain wall motion measurement. The yellow and the green regions \nrepresent ↓ and ↑ domains in the MOKE microscope image of 4- μm wide Co 0.69Tb0.31 wire. (b) Domain \nwall motion in Co 0.69Tb0.31 wire with consecutive current pulses. Bl ue and red dotted lines show the initial \npositions of ↓↑ and ↑↓ domain walls in the top MOKE image, resp ectively. (c), Domain wall velocity as a \nfunction of current density for Pt/Co 1-xTbx at x = 0.17, 0.21, 0.26, 0.31, 0.33, 0.38 and 0.41 (from bottom \nto top panel). The error bars reflect standard devi ations from multiple measurements. (d) Domain wall \nmobility extracted from the dotted lines in (c) for Pt/Co 1-xTbx. (e) Calculated current-induced domain wall \nvelocity for a series of ferrimagnetic samp les with different net angular momentum, Seff. 12 \nFIG. 3. Domain wall velocity as a function of current density at different longitudinal fields along the \nlength of the Pt/Co 0.79Tb0.21 sample for (a) ↓↑ and (b) ↑↓ domain walls. Schematic illustration of the \ndomain wall texture for ↓↑ and ↑↓ Néel domain walls in (c) magnetically Co-dominant and (d) \nmagnetically Tb-dominant samples. Blue and green a rrows represent the magnetizations from Co and Tb \nsublattices, respectively. The chirality of the ↓↑ and ↑↓ domain walls remains the same as the composition \nchanges from the Co- dominant to Tb -dominant side. The effective fi elds from Slonczewski-like torque \n(HSL) on Co and Tb sublattices are show n by yellow and brown arrows, respectively. It can be seen that \nHSL on the two sublattices work constructively to m ove domain walls. The domain wall motion direction \nremains the same in both samples. The black ↓ and ↑ arrows show the domain orientation as detected in \nthe MOKE measurement. The length of the blue and green arrows below the domain wall region reflects \nthe influence of external field Hx on domain wall chirality. Domain wall velocity as a function of in-plane \nfield for samples of (e) Pt/Co 0.79Tb0.21, (f) Pt/Co 0.67Tb0.33 and (g) Pt/Co 0.59Tb0.41 wires, respectively. Red \nsquares (negative current) & red ci rcles (positive current) represent ↓↑ domain walls and blue triangles \n(negative current) & blue stars (positive current) represent ↑↓ domain walls, respectively. Red solid lines \nand blue dashed lines are the linear fit of the experimental data for ↓↑ and ↑↓ domain walls. There is a \nsign reversal in the slopes of the red an d blue lines between (e), (f) and (g). \n 13References \n[1] A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and T. Rasing, Nature 429, 850 (2004). \n[2] T. 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Woo et al. , Nat Commun 9, 959 (2018). \n " }, { "title": "1806.04881v1.Low_magnetic_damping_of_ferrimagnetic_GdFeCo_alloys.pdf", "content": "1 \n Low magnetic damping of ferrimagnetic GdFeCo alloys \nDuck-Ho Kim1†*, Takaya Okuno1†, Se Kwon Kim2, Se-Hyeok Oh3, Tomoe Nishimura1, \nYuushou Hirata1, Yasuhiro Futakawa4, Hiroki Yoshikawa4, Arata Tsukamoto4, Yaroslav \nTserkovnyak2, Yoichi Shiota1, Takahiro Moriyama1, Kab-Jin Kim5, Kyung-Jin Lee3,6,7, and \nTeruo Ono1,8* \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan \n2Department of Physics and Astronomy, University of California, Los Angeles, California \n90095, USA \n3Department of Nano-Semiconductor and Engineering, Korea Univers ity, Seoul 02841, \nRepublic of Korea \n4College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan \n5Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea \n6Department of Materials Science & Engineering, Korea University , Seoul 02841, Republic \nof Korea \n7KU-KIST Graduate School of Converging Science and Technology, K orea University, Seoul \n02841, Republic of Korea \n8Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, \nOsaka University, Osaka 560-8531, Japan \n \n† These authors contributed equally to this work. \n* E-mail: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp 2 \n We investigate the Gilbert damping parameter for rare earth (RE)–\ntransition metal (TM) ferrimagnets over a wide temperature rang e. Extracted from the \nfield-driven magnetic domain-wall mobility, was as low as 7.2 × 10-3 and was almost \nconstant across the angular momentum compensation temperature 𝑻𝐀, starkly \ncontrasting previous predictions that should diverge at 𝑻𝐀 due to vanishing total \nangular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to \nthe total angular momentum but is dominated by electron scatter ing at the Fermi level \nwhere the TM has a dominant damping role. \n 3 \n Magnetic damping, commonly described by the Gilbert damping par ameter, \nrepresents the magnetization relaxation phenomenon, describing how quickly magnetization \nspins reach equilibrium [1–3]. Understanding the fundamental or igin of the damping as well \nas searching for low damping materials has been a central theme of magnetism research. \nSeveral theoretical models for magnetic damping have been propo sed [4–11] and compared \nwith experiments [12–20]. Ultra-low damping was predicted in fe rromagnetic alloys using a \nlinear response damping model [11] and was demonstrated experim entally for CoFe alloys \n[20]. However, the majority of these studies have focused only on ferromagnetic systems. \nAntiferromagnets, which have alt ernating orientations of their neighboring magnetic \nmoments, have recently received considerable attention because of their potential importance \nfor spintronic applications [21– 30]. Antiferromagnetic spin sys tems can have much faster \nspin dynamics than their ferromagnetic counterparts, which is a dvantageous in spintronic \napplications [21, 25, 31–39]. However, the manipulation and con trol of antiferromagnets is \nchallenging because the net magnetic moment is effectively zero . Recently, antiferromagnetic \nspin dynamics have been successfully demonstrated using the mag netic domain-wall (DW) \ndynamics in ferrimagnets with finite magnetization in the vicin ity of the angular momentum \ncompensation temperature, at which the net angular momentum van ishes [38]. This field-\ndriven antiferromagnetic spin dyn amics is possible because the time evolution of the \nmagnetization is governed by the commutation relation of the an gular momentum rather than \nthe commutation relation of the magnetic moment. \nMotivated by the aforementioned result, in this letter, we inve stigate the magnetic \ndamping of ferrimagnets across th e angular momentum compensatio n temperature, which \nwill allow us to understand magnetic damping in antiferromagnet ically coupled system. We 4 \n selected rare earth (RE)–transition metal (TM) ferrimagnets for the material platforms \nbecause they have an angular momentum compensation temperature 𝑇 w h e r e \nantiferromagnetic spin dynamics are achieved [38, 40, 41]. The magnetic-field-driven DW \nmotion was explored over a wide range of temperatures including 𝑇, and the Gilbert \ndamping parameter was extracted from the measured DW mobility a t each temperature by \nemploying the collective coordina te model initially developed f or ferrimagnetic spin \ndynamics [38]. Contrary to the previous prediction that the Gil bert damping parameter would \ndiverge at 𝑇 due to the vanishing of the total angular momentum [42, 43], w e found that the \nGilbert damping parameter remained nearly constant over a wide range of temperatures \nacross 𝑇 with the estimated value as low as 7.2 × 10-3, which was similar to the reported \nvalues of TM-only ferromagnets [20]. These results suggested th at Gilbert damping was \nmainly governed by electron scattering at the Fermi level, and hence, the 4f electron of the \nR E e l e m e n t , w h i c h l i e s f a r b e l o w t h e F e r m i l e v e l , d i d n o t p l a y an important role in the \nmagnetic damping of RE–TM ferrimagnets. \nFor this study, we prepared perpendicularly magnetized ferrimag netic GdFeCo films \nin which the Gd and FeCo moments were coupled antiferromagnetic ally. Specifically, the \nfilms were 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN on an intrinsic Si substrate. The \nGdFeCo films were then patterned into 5-µm-wide and 500-µm-long microwires with a Hall \ncross structure using electron beam lithography and Ar ion mill ing. For current injection, \n100-nm Au/5-nm Ti electrodes were stacked on the wire. A Hall b ar was designed to detect \nthe DW velocity via the anomalous Hall effect (AHE). \nWe measured the magnetic DW motion using a real-time DW detecti on technique [38, \n40, 41, 44, 45] [see Fig. 1(a) for a schematic]. We first appli ed a magnetic field of –200 mT 5 \n to saturate the magnetization al ong the –z direction. Subsequen tly, a constant perpendicular \nmagnetic field 𝜇𝐻, which was lower than the coercive field, was applied along +z direction. \nNext, a d.c. current was applied along the wire to measure the anomalous Hall voltage. Then, \na current pulse (12 V , 100 ns) was injected through the writing line to nucleate the DW in the \nwire. The created DW was moved along the wire and passed throug h the Hall bar because of \nthe presence of 𝜇𝐻. The DW arrival time was detected by monitoring the change in the Hall \nvoltage using a real-time oscillo scope. The DW velocity could t hen be calculated from the \narrival time and the travel dis tance between the writing line a nd Hall bar (500 µm). \nFigure 1(b) shows the averaged DW velocity 〈𝑣〉 as a function of the perpendicular \nmagnetic field 𝜇𝐻 for several temperatures 𝑇∗. Here, we used the d.c. current density of \n|𝐽|ൌ1.3×1010 A / m2 to measure the AHE change due to DW motion. Note that 𝑇∗ i s a n \nelevated temperature that considers Joule heating by d.c. curre nt [46]. To eliminate the \nundesired current-induced spin-transfer-torque effect, we avera ged the DW velocity for 𝐽 \nand –𝐽, i.e., 〈𝑣〉ൌሾ𝑣ሺ𝐽ሻ𝑣ሺെ𝐽ሻሿ/2. Figure 1(b) shows that 〈𝑣〉 increases linearly with \n𝜇𝐻 for all 𝑇∗. Such linear behavior can be described by 〈𝑣〉ൌ𝜇ሾ𝜇𝐻െ𝜇 𝐻ሿ, where 𝜇 \nis the DW mobility and 𝜇𝐻 is the correction field, which generally arises from \nimperfections in the sample or complexities of the internal DW structure [47, 48]. We note \nthat 𝜇𝐻 can also depend on the temperature dependence of the magnetic properties of \nferrimagnets [45]. Figure 1(c) shows 𝜇 as a function of 𝑇∗ at several current densities \n(|𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A / m2). A sharp peak clearly occurs for 𝜇 a t 𝑇∗ൌ241.5 K \nirrespective of |𝐽|. The drastic increase of 𝜇 is evidence of antiferromagnetic spin dynamics \nat 𝑇, as demonstrated in our pre vious report [38, 40, 41]. \nThe obtained DW mobility was theoretically analyzed as follows. The DW velocity 6 \n of ferrimagnets in the precessional regime is given by [38, 39] \n 𝑉 ൌ 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ\nሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶ𝜇𝐻, ሺ1ሻ \nwhere 𝑉 is the DW velocity, 𝜆 is the DW width, 𝜇𝐻 is the perpendicular magnetic field, \n𝛼 is the Gilbert damping parameter, 𝑀 and 𝑠 are the magnetization and the spin angular \nmomentum of one sublattice, respectively. The spin angular mome ntum densities are given \nby 𝑠ൌ𝑀 /𝛾 [49], where 𝛾ൌ𝑔 𝜇/ℏ is the gyromagnetic ratio of lattice 𝑖, 𝑔 i s t h e \nLandé g factor of lattice 𝑖, 𝜇 is the Bohr magneton, and ℏ is the reduced Plank’s constant. \nThe Gilbert damping is in principle different for two sublattic e s , b u t f o r s i m p l i c i t y , w e \nassume that it is the same, which can be considered as the aver age value of the damping \nparameters for the two sublattices weighted by the spin angular momentum density. We note \nthat this assumption does not alter our main conclusion: low da mping and its insensitivity to \nthe temperature. Equation (1) gives the DW mobility 𝜇 a s 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ/\nሼሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶሽ, which can be rearranged as \n 𝜇 ሺ𝑠ଵ𝑠 ଶሻଶ𝛼ଶെ𝜆ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ𝛼𝜇 ሺ𝑠ଵെ𝑠 ଶሻଶൌ 0 ሺ2ሻ \nUsing Eq. (2) to find the solution of 𝛼, we find \n 𝛼 േൌ𝜆ሺ𝑀ଵെ𝑀 ଶሻേඥሾ𝜆ଶሺ𝑀ଵെ𝑀 ଶሻଶെ4 𝜇ଶሺ𝑠ଵെ𝑠 ଶሻଶሿ\n2𝜇ሺ𝑠ଵ𝑠 ଶሻ. ሺ3ሻ \nEquation (3) allows us to estimate 𝛼 for the given 𝜇. We note that for each value of 𝜇, 𝛼 \nca n h av e t w o v a lu e s, 𝛼ା and 𝛼ି because of the quadratic nature of Eq. (2). Only one of \nthese two solutions is physically sound, which can be obtained using the following energy \ndissipation analysis. 7 \n The energy dissipation (per unit cross section) through the DW dynamics is given by \n𝑃ൌ2 𝛼 ሺ 𝑠 ଵ𝑠 ଶሻ𝑉ଶ/𝜆 2𝛼ሺ𝑠 ଵ𝑠 ଶሻ 𝜆Ωଶ [38, 39], where Ω is the angular velocity of the \nDW. The first and the second terms represent the energy dissipa tion through the translational \nand angular motion of the DW, respectively. In the precessional regime, the angular velocity \nis proportional to the translational velocity: Ωൌ ሺ𝑠ଵെ𝑠 ଶሻ𝑉/𝛼ሺ𝑠 ଵ𝑠 ଶሻ𝜆. Replacing Ω b y \nthe previous expression yields 𝑃ൌ𝜂 𝑉ଶ w h e r e 𝜂ൌ2 ሺ 𝑀 ଵെ𝑀 ଶሻ/𝜇 is the viscous \ncoefficient for the DW motion: \n 𝜂 ൌ2\n𝜆ቊ𝛼ሺ𝑠ଵ𝑠 ଶሻ ሺ𝑠ଵെ𝑠 ଶሻଶ\n𝛼ሺ𝑠ଵ𝑠 ଶሻቋ . ሺ4ሻ \nThe first and the second terms in parenthesis capture the contr ibutions to the energy \ndissipation from the translational and angular dynamics of the DW, respectively. The two \nsolutions for the Gilbert damping parameter, 𝛼ା and 𝛼ି, can yield the same viscous \ncoefficient 𝜂. The case of the equal solutions, 𝛼ାൌ𝛼 ି, corresponds to the situation when \nthe two contributions are identical: 𝛼േൌሺ 𝑠 ଵെ𝑠 ଶሻ/ሺ𝑠ଵ𝑠 ଶሻ. For the larger solution 𝛼ൌ\n𝛼ା, the energy dissipation is dominated by the first term, i.e., through the translational DW \nmotion, which should be the case in the vicinity of 𝑇 where the net spin density ሺ𝑠ଵെ𝑠 ଶሻ \nis small and thus the angular velocity is negligible. For examp le, at exact 𝑇, the larger \nsolution 𝛼ା is the only possible solution because the smaller solution is zero, 𝛼ିൌ0, and \nthus unphysical. For the smaller solution 𝛼ൌ𝛼 ି, the dissipation is dominated by the second \nterm, i.e., through the precessional motion, which should descr ibe cases away from 𝑇. \nTherefore, in the subsequent analysis, we chose the larger solu tion 𝛼ା in the vicinity of 𝑇 \nand the smaller solution 𝛼ି far away from 𝑇 and connected the solution continuously in \nbetween. 8 \n The other material parameters such as 𝑀ଵ, 𝑀ଶ, 𝑠ଵ, and 𝑠ଶ a r e e s t i m a t e d b y \nmeasuring the net magnetic moment of GdFeCo film, |𝑀୬ୣ୲|, for various temperatures. \nBecause 𝑀୬ୣ୲ includes contributions from both the Gd and FeCo sub-moments, the sub-\nmagnetic moments, 𝑀ଵ a n d 𝑀ଶ, could be decoupled based on the power law criticality [see \ndetails in refs. 38, 40]. The spin angular momentums, 𝑠ଵ and 𝑠ଶ, were calculated using the \nknown Landé g factor of FeCo and Gd (the Landé g factor of FeCo is 2.2 and that of Gd is \n2.0) [50–52]. \nFigures 2(a)–(c) show the temperature-dependent DW mobility 𝜇, sub-magnetic \nmoment 𝑀, and sub-angular momentum 𝑠, respectively. Here, we used the relative \ntemperature defined as ∆𝑇 ൌ 𝑇∗െ𝑇 to investigate the Gilbert damping near 𝑇. The \nGilbert damping parameter 𝛼 was obtained based on Eq. (3) and the information in Fig. \n2(a)–(c). Figure 2(d) shows the resulting values of 𝛼േ as a function of ∆𝑇. For ∆𝑇ଵ൏\n∆𝑇 ൏ ∆𝑇 ଶ, 𝛼ା is nearly constant, while 𝛼ି varies significantly. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 \n∆𝑇ଶ, on the other hand, 𝛼ି is almost constant, while 𝛼ା varies significantly. At ∆𝑇 ൌ ∆𝑇 ଵ \nand ∆𝑇 ൌ ∆𝑇 ଶ, the two solutions are equal, corresponding to the aforementio ned case when \nthe energy dissipation through the translational and angular mo tion of the DW are identical. \nThe proper damping solution can be selected by following the gu ideline obtained \nfrom the above analysis. For ∆𝑇ଵ൏∆ 𝑇൏∆ 𝑇 ଶ, which includes 𝑇, the energy dissipation \nshould be dominated by the translational motion, and thus 𝛼ା is a physical solution. Note \nalso that 𝛼ି becomes zero at 𝑇, which results in infinite DW mobility in contradiction with \nthe experimental observation. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 ∆𝑇 ଶ, where the energy dissipation is \ndominated by the angular motion of the DW, 𝛼ି is the physical solution. 9 \n Figure 3 shows the resultant Gil bert damping parameter in all t ested temperature \nranges. The Gilbert damping parameter was almost constant acros s 𝑇 with 𝛼ൌ7.2 × 10-3 \n(see the dotted line in Fig. 3). This result is in stark contra st to the previous prediction. In ref. \n[42], Stanciu et al. investigated the temperature dependence of the effective Gilb ert damping \nparameter based on a ferromagnet-based model and found that the damping diverged at 𝑇. \nBecause they analyzed the magnetic resonance in ferrimagnetic m aterials based on a \nferromagnet-based model, which cannot describe the antiferromag netic dynamics at 𝑇 a t \nwhich the angular momentum vanis hes, it exhibits unphysical res ults. However, our \ntheoretical analysis for field-driven ferromagnetic DW motion b ased on the collective \ncoordinate approach can properly describe both the antiferromag netic dynamics in the \nvicinity of 𝑇 and the ferromagnetic dynamics away from 𝑇 [38]. Therefore, the \nunphysical divergence of the Gilbert damping parameter at 𝑇 is absent in our analysis. \nOur results, namely the insensitivity of damping to the compens ation condition and \nits low value, have important implications not only for fundame ntal physics but also for \ntechnological applications. From the viewpoint of fundamental p hysics, nearly constant \ndamping across 𝑇 indicates that the damping is almost independent of the total angular \nmomentum and is mostly determined by electron spin scattering n ear the Fermi level. \nSpecifically, our results suggest that the 4f electrons of RE e lements, which lie in a band far \nbelow the Fermi level, do not play an important role in the mag netic damping of RE-TM \nferrimagnets, whereas the 3d and 4s bands of TM elements have a governing role in magnetic \ndamping. This result is consistent with the recently reported t heoretical and experimental \nresults in FeCo alloys [20]. From the viewpoint of practical ap plication, we note that the \nestimated damping of 𝛼ൌ7.2 × 10-3 is the upper limit, as the damping estimated from DW 10 \n dynamics is usually overestimated due to disorders [53]. The ob tained value of the Gilbert \ndamping parameter is consistent with our preliminary ferromagne t i c r e s o n a n c e ( F M R ) \nmeasurements. The experimental results from FMR measurements an d the corresponding \ntheoretical analysis will be publ ished elsewhere. This low valu e of the Gilbert damping \nparameter suggests that ferrimagne ts can serve as versatile pla t f o r m s f o r l o w - d i s s i p a t i o n \nhigh-speed magnetic devices such as spin-transfer-torque magnet ic random-access memory \nand terahertz magnetic oscillators. \nIn conclusion, we investigated the field-driven magnetic DW mot ion in ferrimagnetic \nG d F e C o a l l o y s o v e r a w i d e r a n g e o f t e m p e r a t u r e s a c r o s s 𝑇 and extracted the Gilbert \ndamping parameter from the DW mobility. The estimated Gilbert d amping parameter was as \nlow as 7.2 × 10-3 and almost constant over the temperature range including 𝑇, which is in \nstark contrast to the previous prediction in that the Gilbert d amping parameter would diverge \nat 𝑇 due to the vanishing total angular momentum. 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Shinjo, Science 284, 468 \n(1999). \n[49] In this Letter , the parameters such as the spin angular mo mentum density 𝑠 r e p r e s e n t \nthe magnitudes of the quantities. Their directions are separate ly handled through the signs in \nthe equations of motion. \n[50] C. Kittel, Phys. Rev. 76, 743 (1949). \n[51] G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). \n[52] B. I. Min and Y.-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). \n[53] H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, Phys. Rev. Lett. \n104, 217201 (2010). \n 16 \n Figure Captions \nFigure 1(a) Schematic illustration of the GdFeCo microwire devi ce. (b) The averaged DW \nvelocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures \n𝑇∗ (202, 222, 242, 262, and 282 K). The dots indicate the best li n e a r f i t s . ( c ) T h e D W \nmobility 𝜇 as a function of 𝑇∗ at several current densities ( |𝐽|ൌ1.3, 1.7, and 2.0 ×1010 \nA/m2). \nFigure 2 The temperature-dependent (a) DW mobility 𝜇, (b) sub-magnetic moment 𝑀, and \n(c) sub-angular momentum 𝑠. Here, we use the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ\n𝑇. (d) The Gilbert damping parameter 𝛼േ as a function of ∆𝑇. Here, we use 𝜆ൌ15 nm for \nproper solutions of Eq. (3). \nFigure 3 The resultant Gil bert damping parameter 𝛼 in all tested temperature ranges. \n 17 \n Acknowledgements \nThis work was supported by the JSPS KAKENHI (Grant Numbers 15H0 5702, 26103002, and \n26103004), Collaborative Research Program of the Institute for Chemical Research, Kyoto \nUniversity, and R & D project for ICT Key Technology of MEXT fr om the Japan Society for \nthe Promotion of Science (JSPS). This work was partly supported by The Cooperative \nResearch Project Program of the Research Institute of Electrica l Communication, Tohoku \nUniversity. D.H.K. was supported as an Overseas Researcher unde r the Postdoctoral \nFellowship of JSPS (Grant Number P16314). S.H.O. and K.J.L. wer e supported by the \nNational Research Foundation of Korea (NRF-2015M3D1A1070465, 20 17R1A2B2006119) \nand the KIST Institutional Program (Project No. 2V05750). S.K.K . was supported by the \nArmy Research Office under Contract No. W911NF-14-1-0016. K.J.K . was supported by the \nNational Research Foundation of Korea (NRF) grant funded by the Korea Government \n(MSIP) (No. 2017R1C1B2009686). \nCompeting financial interests \nThe authors declare no competing financial interests. 200 225 250 275 3000.00.51.01.52.0\n 1.3\n1.7\n2.0\n [104 m/sT]\nT* [K]J [1010 A/m2]0 50 100 1500.00.51.01.5\n 202\n 222\n 242\n 262\n 282 [km/s]\n0H [mT]T* [K]\nFigure 1b\nca\nWriting line\n\tܫ\nܸ\nߤܪ\ny xz-60 -40 -20 0 20 40 60 801.52.02.53.0 s1\n s2s [10-6 Js/m3]\nT [K]-60 -40 -20 0 20 40 60 800.00.51.01.52.0\n [104 m/sT]\nT [K]\n-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \n +\n -\nT [K]T1T2-60 -40 -20 0 20 40 60 800.30.40.50.6 M1\n M2M [MA/m]\nT [K]a\nb\nc\nd\nFigure 2Figure 3-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \nT [K]" }, { "title": "1602.02239v1.Lieb_Mattis_ferrimagnetism_in_diluted_magnetic_semiconductors.pdf", "content": "Lieb-Mattis ferrimagnetism in diluted magnetic semiconductors\nR.O. Kuzian,1, 2J. Richter,3M. D. Kuz'min,4and R. Hayn4\n1Institute for Problems of Materials Science NASU, Krzhizhanovskogo 3, 03180 Kiev, Ukraine\n2Donostia International Physics Center (DIPC), ES-20018 Donostia-SanSebastian, Spain\n3Institut f ur Theoretische Physik, Otto-von-Guericke-Universit at Magdeburg,\nPF 4120, D - 39016 Magdeburg, Germany\n4Aix-Marseille Universit\u0013 e, IM2NP-CNRS UMR 7334,\nCampus St. J\u0013 er^ ome, Case 142, 13397 Marseille, France\n(Dated: 11.10.15)\nWe show the possibility of long-range ferrimagnetic ordering with a saturation magnetisation of\n\u00181\u0016Bper spin for arbitrarily low concentration of magnetic impurities in semiconductors, provided\nthat the impurities form a superstructure satisfying the conditions of the Lieb-Mattis theorem.\nExplicit examples of such superstructures are given for the wurtzite lattice, and the temperature\nof ferrimagnetic transition is estimated from a high-temperature expansion. Exact diagonaliza-\ntion studies show that small fragments of the structure exhibit enhanced magnetic response and\nisotropic superparamagnetism at low temperatures. A quantum transition in a high magnetic \feld\nis considered and similar superstructures in cubic semiconductors are discussed as well.\nPACS numbers: 75.10.-b, 75.20.-g, 75.50.Gg, 75.50.Pp\nIn order to launch the engineering of a new generation\nof electronic devices, one needs new materials with spe-\ncial properties. For instance, spintronics has a need for\nroom-temperature ferromagnetic semiconductors1. Since\nthe discovery of high- TCferromagnetism in GaAs:Mn2\nand the prediction of room-temperature ferromagnetism\ninp-doped ZnO:Co,Mn systems3, a lot of attempts have\nbeen made to obtain ferromagnetism in transition metal\ndoped ZnO, GaN and in other oxides and nitrides. The\np-type carriers doping is necessary for the p-dZener ferro-\nmagnetic long-range interaction4. Up to now all attempts\nto obtain ZnO with p-type current carriers have failed.\nNevertheless, several reports of \\ferromagnetic\" room\ntemperature behavior have been published5{7. \\Perhaps\nthe most surprising development of the past decade in\nthe science of magnetic materials is the abundant ob-\nservations of spontaneous magnetization persisting to\nabove room temperature in semiconductors and oxides,\nin which no ferromagnetism was expected at any temper-\nature, particularly in the p-dZener model\"5.\nIn the absence of p-type current carriers, the inter-\naction between magnetic impurities is governed by the\nsuperexchange mechanism. Superexchange is often re-\ngarded as an obstacle in the way towards magnetic semi-\nconductors as it has antiferromagnetic (AFM) charac-\nter and tends to anti-align the interacting spins, leading\nto a cancellation of the net magnetization. In fact, the\nAFM interaction does notpreclude spontaneous mag-\nnetization. In a seminal paper8, E. Lieb and D. Mattis\nshowed that the ground state of an AFM system depends\non the topology of the interacting bonds and, under cer-\ntain conditions, it is ferrimagnetic rather than AFM. The\nLieb-Mattis theorem applies if there is no magnetic frus-\ntration in the spin system.\nIn this communication we study various structures\nformed by the interacting magnetic impurities in wurtzite\nsemiconductors. We take antiferromagnetic nearestneighbor interaction into account and consider diluted\nlattices without frustration, in order to remain within\nthe Lieb-Mattis scheme. First we construct several \fnite\nclusters that show an enhanced magnetic response at low\ntemperatures. Not alone do they possess a net magnetic\nmoment, they all share a further interesting peculiarity:\nbelow a certain temperature their magnetic susceptibility\nexceeds that of non-interacting spins. We call it isotropic\nsuperparamagnetic response9,10. Next we construct ex-\ntended lattices of these clusters, which undergo a fer-\nrimagnetic ordering transition at a \fnite temperature.\nThe average ground-state spin per magnetic ion of spin\nStends to a \fnite value (of about S=3) despite the low\nconcentration of magnetic ions. The extension of our idea\nto other lattices and the in\ruence of frustration will be\nbrie\ry discussed at the end of the communication.\nWe take the interaction in the form\n^H=1\n2X\nR;rJr^SR^SR+r; (1)\ni.e., we adopt the notation Jrfor the interaction be-\ntween one pair of spins11. We assume that only the\nnearest-neighbor (in the metal sublattice) interaction\nis nonzero. This assumption is relevant to magnetic\nsemiconductors, where the nearest-neighbor exchange\ndominates12{14. Two kinds of nearest neighborships are\npresent in wurtzites: those where both ions lie in the\nsame plane and those where they lie in two adjacent\nplanes. The corresponding exchange integrals, J1(in-\nplane) and J2(out-of-plane), are di\u000berent15{17.\nThe magnetic response of a system is characterized by\nits magnetic susceptibility. Talking about a compound\nA1\u0000xMxX (where X is a ligand of V or VI group, A is a\nmetal of IIId or IId group, and M is a transition metal),\nwe shall attribute all the magnetic moment to transition\nmetal ions (TMIs) only. We now introduce the magneticarXiv:1602.02239v1 [cond-mat.mtrl-sci] 6 Feb 20162\nsusceptibility per one spin,\n\u001f\u0011\u0016M\nH; (2)\nwhere\u0016Mis the average magnetic moment of one TMI.\nFor non-interacting spins, the susceptibility obeys the\nCurie law\u001fC= [(g\u0016B)2S(S+1)]=(3kBT);whereSis the\nspin of the TMI and gis its gyromagnetic ratio. Besides\nisolated spins, TMI impurities may form pairs, trimers,\ntetramers, and more complex structures (see Fig. 1). The\n1\n3\n4 4’3’a)2\nb)\nJ12J\nc)\nFIG. 1. (Color online) a): Complexes formed by transi-\ntion metal impurities (arrows): isolated ions (1), dimers (2),\ntrimers (3,30), tetramers (4,40). Black solid line segments de-\npict the nearest-neighbor interaction J1bonds. One wurtzite\nabplane is shown, blue circles denote non-magnetic host metal\nions, ligands are not shown. b), c) : More complex Lieb-\nMattis systems with ferrimagnetic ground state: linear chains\nof impurities in the abplane \"decorated\" by spins in adjacent\nplanes (gold arrows); pink line segments depict J2bonds.\nantiferromagneitc interaction depresses the magnetic re-\nsponse at high temperatures. For T\u001dJmaxS(S+ 1)\u0011\nTs, the susceptibility of an interacting system obeys the\nCurie-Weiss law \u001fCW= [(g\u0016B)2S(S+1)]=[3kB(T\u0000\u0012)]<\n\u001fC, with\u0000\u0012= [S(S+ 1)]=(3kBN)P\nR;r(R)Jr(R). Here\nNis the number of spins and Jmaxis the strongest ex-\nchange interaction in the system, Rruns over all spins of\nthe lattice, and rruns over all nearest neighbors of each\nspin.\nAt temperatures T.Ts, the response of the system\ndepends on its geometry. Analytic expressions for the\nsusceptibility can be obtained for small systems18. Fig.\n2a shows the results for the simplest S=1/2 case. We see\nthat atT\u0018Tsthe response of three spins arranged lin-\nearly30is larger than that of a triangular arrangement of\nthe same spins 3. For 4-spin systems we see the striking\ndi\u000berence between the response of a star arrangement 40\nand that of a rhombus 4.\nEven more interesting is the response of the complexes\nshown in Fig. 1b,c. Each one of these systems can be\ndecomposed into two sublattices A and B (denoted by\narrows \\up\" and \\down\"), the interaction being nonzero\nonly between sites that belong to di\u000berent sublattices.\nSuch a system satis\fes the requirements of the Lieb-\nMattis theorem8, and possesses a ferrimagnetic ground\nstate with total spin Sg=SjNA\u0000NBj. In this case, the\nterm \\ferrimagnetic\" refers to correlations of the spins\n 0 2 4 6 8 10\n 0 0.5 1 1.5 2 2.5 3g2µB2/χ\nkBT/|J|S(S+1)a)\n1 \n2 \n3’\n3 \n4’\n4 \n 0 1 2 3 4 5 6\n 0 0.5 1 1.5g2µB2/χ\nkBT/J1S(S+1)b)S=1/2\nS=1 \n 0 0.2 0.4 0.6 0.8 1 1.2\n 0 0.1 0.2 0.3 0.4g2µB2/χ\nkBT/J1S(S+1)c)J2/J1=1\n0.75\n0.33\n 0 0.2 0.4 0.6 0.8 1\n 0 0.1 0.2 0.3g2µB2/χ\nkBT/J1S(S+1)d)J2/J1=1\n0.75\n0.33FIG. 2. (Color online) Inverse susceptibility (per spin) \u001f\u00001\nfor the complexes shown in Fig. 1. Straight solid red line\nshows the Curie law \u001f\u00001\nC; straight dashed lines show the low-\ntemperature asymptotics: \\super\" -paramagnetic Curie laws\n(g\u0016B)2=\u001fg= 3NkBT=[Sg(Sg+ 1)] for Lieb-Mattis systems.\na): clusters shown in Fig. 1a with S= 1=2;b): the com-\nplex shown in Fig. 1b with two di\u000berent values of spin S;\nthe straight dash-dotted red line is the high- TCurie-Weiss\nasymptote. c)the same complex with S= 1 and various val-\nues ofJ2=J1;d)the complex shown in Fig. 1c with S= 1=2\nand various values of J2=J1.\nin the ground state, in the absence of a long-range mag-\nnetic order19. We have performed full exact diagonaliza-\ntion studies (ED) of thermodynamic properties of clus-\nters shown in Fig. 1b,c using J. Schulenburg's spinpack\nprogram20,21. The susceptibility \u001f(T) is calculated as the\nratio of the induced magnetization Mto the \"vanishing\"\nmagnetic \feld H= 10\u00005J1=g\u0016B. One observes in Figure\n2b,c,d that the response of the systems shown in Fig. 1b,c\nexceeds the response of non-interacting spins at low tem-\nperature. Thus, an antiferromagnetic interaction may\nresult in an enhancement of magnetic response if the ge-\nometry of spin arrangement favors the formation of a fer-\nrimagnetic ground state. Then for temperatures T\u001cTs\nthe susceptibility per spin shows superparamagnetic re-\nsponse\u001fg= [(g\u0016B)2Sg(Sg+ 1)]=[3kBT(NA+NB)]. Evi-\ndently, the enhancement of the low-temperature response\ntakes place, if\nK\u0011\u001fg\n\u001fC=jNA\u0000NBj(jNA\u0000NBjS+ 1)\n(NA+NB)(S+ 1)>1:(3)\nNot every system satisfying the requirements of the Lieb-\nMattis theorem and having a ferrimagnetic ground state\nhas an enhanced susceptibility. Thus, the clusters 30\n(NA= 1;NB= 2) and 40(NA= 1;NB= 3) both have\nK < 1, i.e. their response is weaker than that of the\nsame number of non-interacting spins.\nThe \"S\"-shape form of the T-dependence of the inverse3\nsusceptibility (Figure 2b) was previously reported for\nsmall fragments of ferrimagnetic superstructure in double\nperovskites10,22. It interpolates between the Curie-Weiss\nlaw\u001fCWatT\u001dTs, and the \"super\"-spin Curie law\n\u001fg=K\u001fCatT\u001cTs.\nIf impurity spins arrange themselves in a periodic su-\nperstructure having two (or more) non-equivalent spin\npositions, a ferrimagnetic ground state is possible for this\nsuperstructure. Let us denote the number of spins in the\nsuperstructure unit cell nA+nB, where A and B refer to\nthe non-equivalent positions. If the spins of the sublattice\nA interact (antiferromagnetically) only with the spins of\nthe sublattice B (absence of frustration), and nA6=nB,\nthe ground-state spin of the unit cell is Sc=SjnA\u0000nBj8.\nFor a fragment of such a ferrimagnetic superstructure\na) b)\nc)\nd)\nFIG. 3. (Color online) Examples of ferrimagnetic superstruc-\ntures a), b) : \rat and three-dimensional two-leg honey-\ncombs,L= 1; c): four-leg honeycomb, L= 2; d): a unit\ncell of a square network, it may be also regarded as a face of\ncubic unit cell. The notations is the same as in Fig. 1. The\ncyan rhombi show the unit cells.\ncontainingNccells, the ground-state spin is Sg=NcSc=\nNcjnA\u0000nBjS, and the enhancement ratio equals K=\njnA\u0000nBj(NcjnA\u0000nBjS+ 1)=[(nA+nB)(S+ 1)]. It is\nclear that for a su\u000eciently large number of cells Ncthe\nratioKwill be not only greater than 1, but can reach\nvery large values. Fig. 3a shows a honeycomb superstruc-\nture that may be formed by TMIs in the abplane of the\nwurtzite structure. The hexagon edge length is ah= 2a,\nabeing the lattice parameter of the wurtzite. It is easy\nto imagine superstructures with ah= 2La,L= 1;2:::, all\nof them being ferrimagnetic.\nFlat superstructures like those shown in Fig. 3a can be\nlinked together by some bridging spins to form a three-\ndimensional ferrimagnetic superstructure, which will un-dergo a ferrimagnetic phase transition, provided that the\nnumber of cells is macroscopically large. Figure 3b,c\n 0 2 4 6 8 10 12 14 16 18\n 0 1 2 3 4 5(gµB)2/χ\nkBT/J1S(S+1)a)\n1/2\n1 \n3/2\n2 \n5/2\n 0 2 4 6 8 10\n 0 0.5 1 1.5 2(gµB)2/χ\nkBT/J1S(S+1)b)\n1 \n0.75\n0.33\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0.4 0.5 0.6 0.7(gµB)2/χ\nkBT/J1S(S+1)c) [4,4]\n[5,5]\n[5,6]\n[6,5]\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0.4 0.5 0.6 0.7(gµB)2/χ\nkBT/J1S(S+1)d) 1\n 2\n 3\n 4\n5\n6\nFIG. 4. (Color online) Temperature dependence of inverse\nsusceptibility given by [5,5] Pad\u0013 e approximants for tenth-\norder high-temperature expansion (HTE) for ferrimagnetic\nsuperstructures: a) two-leg honeycomb ( L= 1), vari-\nous spin values are shown, solid (dash-dotted) straight red\nline shows Curie (Curie-Weiss) law; b)four-leg honeycomb\n(L= 2),S= 5=2 variousJ2=J1values are shown; c)four-leg\nsystem,L= 2,S= 2 various Pad\u0013 e approximants for eighth-\norder ([4,4])23,24, tenth-order([4,6], [5,5], [6,4])25, and eleven-\norder ([5,6], [6,5])26HTE; d)the vicinity of TCfor various\nhoneycomb superstructures with size parameter L= 1;2:::6,\nS= 5=2,J2=J1.\nshows examples of the structures. It is clear that this\nmotif may be repeated in an in\fnite number of varia-\ntions. Like the host wurtzite lattice, the unit cell of the\nsuperstructure contains metal ions in two planes. The\nmagnetic ions in one plane (green \\down\" and brown\n\\up\" arrows) form a honeycomb lattice with the hexagon\nedge 2La. In the second plane, the magnetic ions (gold\n\\up\" arrows) occupy the positions nearest to the green\n\\down\" arrows. The interaction between the ions in\nthe \frst plane is J1, whereas the interaction between\nthe ions in two adjacent planes is J2. We note that\nthe complexes shown in Fig. 1b,c are building blocks\nof the honeycombs. It will be demonstrated below that\nmany other Lieb-Mattis networks can be built of such\nblocks. The number of magnetic ions in the unit cell is\nnA+nB= 9L\u00001 , the ground state spin of the cell being\nSc=SjnA\u0000nBj=S(3L\u00001). Now the total number\nof ions in the cell is nc= 24L2. Thus, the concentra-\ntion of magnetic ions equals x= (9L\u00001)=(24L2), and\ncan be made very small for su\u000eciently large L. At the\nsame time, the average ground-state spin per magnetic\nion,hSRi=Sc=(nA+nB) =S(3L\u00001)=(9L\u00001), tends\nto a \fnite value, S=3, asL!1 .\nThe inverse magnetic susceptibility \u001f\u00001of such super-4\nstructures is presented in Fig. 4 as a function of nor-\nmalized temperature T=Ts. It was calculated using a\nprogram25based on the tenth-order high-temperature ex-\npansion (HTE)27. The program computes the exact coef-\n\fcients of the HTE as well as its Pad\u0013 e approximants (ra-\ntios of two polynomials), \u001f(T)\u0019[m;n] =Pm(T)=Pn(T).\nThe Pad\u0013 e approximants allow to extend the region of va-\nlidity of the HTE down to T\u00180:5Ts25(Fig. 4c). This\nextension sometimes fails if an approximant has a pole\nin the temperature region of interest. Our experience\nshows that the [5,5] approximant works well in almost\nall cases. Sometimes di\u000eculties arise for S= 1=2, and\nfor smallJ2=J1ratios, i.e., for the extreme quantum case.\nNevertheless, due to the weak dependence of the shape\nof the curve \u001f\u00001(T=Ts) on the spin value S(Fig. 4a),\nit can still be analyzed. At T&3Ts, the inverse sus-\nceptibility follows the Curie-Weiss asymptotic law with\n\u0012=\u0000[S(S+ 1)=3kB]12L(J1+J2)=(9L\u00001). ForT.Ts\nit sharply deviates from the asymptotic behavior and\nchanges sign at T=TC. This is the temperature of\nferrimagnetic ordering | the Curie temperature.\nThe precision of the determination of critical temper-\natures from the zero of \u001f\u00001(Fig. 4c) was estimated to\nbe about 10%25. Figure 4b shows that TCdecreases as\nthe ratio of out-of-plane to in-plane couplings, J2=J1, is\nreduced. At J2= 0 the system becomes a stack of non-\ninteracting two-dimensional planes, and TCshould van-\nish. This limit lies outside the range of applicability of\nthe HTE, and we postpone its study to future works.\nHere we mention only that magnetic anisotropy, which is\nneglected in our study, should act in the opposite direc-\ntion, i.e., it should enhance the TCas it depresses spin\n\ructuations.\nFigure 4d shows that the ordering temperature de-\ncreases very slowly as Lis increased. Note that the\nsuperstructure parameter values L= 1;2;3;4;5;6 corre-\nspond to the following concentrations of magnetic ions:\nx= 0:33;0:18;0:12;0:09;0:07;0:06. To get a closer rela-\ntion to experiments, we may consider, e.g., ZnO:Mn,Co,\nwhere the in-plain superexchange values are J1=kB\u0018\n50 K11,13,14andTs=J1S(S+ 1)=kB\u0018438(188) K for\nS= 5=2(3=2). For other Co-doped semiconductors 66 K\n.J1=kB.100 K12,17,28(and references therein), i.e., Ts\nlies within the interval 248 K .Ts.375 K. The Mn-\ndoped semiconductors have 12 K .J1=kB.32 K12,29,\nand 105 K .Ts.280 K.\nThus, a very diluted system may have an appreciable\nordering temperature ( TC&100 K) provided that the\nmagnetic ions are arranged in a Lieb-Mattis ferrimagnetic\nsuperstructure.\nIn many aspects, the behavior of a ferrimagnet in its\nordered state is similar to that of a ferromagnet with the\nsame value of spontaneous magnetization Ms. But in\na high magnetic \feld the ferrimagnet exhibits a transi-\ntion accompanied by reorientation of its sublattices30{32.\nAtT= 0 the magnetization per spin has a constant\nvalue,\u0016M;s=g\u0016BSjnA\u0000nBj=(nA+nB), up to a cer-\ntain critical \feld, Hc;1; then it grows up linearly to thesaturation value, \u0016M;max =g\u0016BS, which is reached at a\nsecond critical \feld, Hc;2. For a two-sublattice ferrimag-\nnet having the structure shown in Fig. 3a ( L= 1) and\nJ1=J2=Jwe \fndg\u0016BHc;1=JS, andHc;2= 5Hc;1.\nForJ=kB\u001820 K this gives Hc;1\u001837 T,Hc;2\u0018185 T.\nThe complexes shown in Fig. 1b,c may be arranged\nin many kinds of networks, to form Lieb-Mattis ferri-\nmagnetic superstructures in various host semiconductors.\nFigure 3d shows an example of a 2D square superstruc-\nture unit cell with L= 2, which is possible in a cubic\nhost. It has nA= 1 + 2(L\u00001) andnB= 4(L\u00001) + 2L.\nOne can also imagine a 3D cubic network; then Fig. 3d\ncorresponds to a face of the cubic unit cell having nA=\n1+3(L\u00001),nB= 3L+12(L\u00001), and the concentration of\nmagnetic ions x= (nA+nB)=nc= (9L\u00007)=(4L3). For-\nmation of such superstructures is possible in perovskite\nsolid solutions, like KMn xMg1\u0000xF333,34, or in solutions of\nmultiferroics PbFe 1=2Nb1=2O3or PbFe 1=2Ta1=2O3with\nferroelectric perovskites35{39.\nWe conclude that Lieb-Mattis ferrimagnetism is a\npossible route to obtaining long-range magnetic order\nin semiconductors containing transition metal ions as\nsubstitutional impurities, which requires no additional\ncharge carriers. A precursor of the ordering transition\nis the enhanced magnetic response of \fnite cluster show-\ning isotropic superparamagnetism. Our results for the\ninverse susceptibility show a characteristic \"S\"-like form\nof the curves, which could be used to identify the present\nmechanism. Adding the magnetic anisotropy to our the-\nory, we expect also other ingredients of superparamag-\nnetism, namely a \fnite blocking temperature and hys-\nteresis.\nThese superparamagnetic clusters serve as building\nblocks to create in\fnite sublattices of the wurtzite struc-\nture that obey the Lieb-Mattis rules. As we have al-\nready noted, there is an enormous wealth of such Lieb-\nMattis sublattices, our proposals (Fig. 3) may only serve\nas examples. We expect a \fnite transition tempera-\nture for all these lattices and we have shown it explic-\nitly for the subclass that we considered. Of course,\na question arises, whether frustration in a realistic di-\nluted semiconductor can in\ruence the above discussed\nscenario. First we argue that there are several numer-\nical studies showing that the Lieb-Mattis theorem, al-\nthough not rigorously valid, applies to many frustrated\nspin systems, see, e.g., Ref. 40. Furthermore, we know\nthat there are various frustrated 2D lattices with antifer-\nromagnetic nearest-neighbor exchange, such as the tri-\nangular or the Shastry-Sutherland lattices, which show\nground-state magnetic LRO41,42. Last but not least, the\nstability of the ferrimagnetic ground state against frus-\ntration has been demonstrated for several speci\fc ferri-\nmagnetic models, see, e.g., Refs. 43{45. Consequently,\nthere is ample evidence that the above sketched mech-\nanism should be robust against frustration. The \fnal\nproof that the here proposed mechanism can, indeed, be\nrealized in a real material demands further studies, in\nclose collaboration between experiment and theory.5\nIn this communication, we have considered only semi-\nconductors doped by one kind of magnetic ions, where\nferrimagnetism can appear due to the topology of inter-\nacting bonds. Another option is the co-doping with two\nkinds of ions having di\u000berent spin values. 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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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B 97,220403(R) (2018).\n12" }, { "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 \n����MSBVW 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. Hillebrands, J. Phys. D. Appl. Phys. 43, 264002 \n(2010). \n3) A. 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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": "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": "2312.01630v2.Ground_State_Phase_Diagram_of__1_2_1_2_1__Mixed_Diamond_Chains.pdf", "content": "arXiv:2312.01630v2 [cond-mat.str-el] 2 Mar 2024Journal of the Physical Society of Japan FULL PAPERS\nGround-State Phase Diagram of (1/2,1/2,1) Mixed Diamond Ch ains\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 two apical\nspins in each unit cell are coupled by an exchange coupling λ. The vertex spins are coupled with the top\nand bottom apical spins by exchange couplings 1 + δand 1−δ, respectively. Although this model has\nan infinite number of local conservation laws for δ= 0, they are lost for finite δ. The ground-state phase\ndiagram is determined using the numerical exact diagonaliz ation and DMRG method in addition to the\nanalytical approximations in various limiting cases. The p hase diagram consists of a nonmagnetic phase\nand several kinds of ferrimagnetic phases. We find two differe nt ferrimagnetic phases without spontaneous\ntranslational symmetry breakdown. It is also found that the quantized ferrimagnetic phases with large\nspatial periodicities present for δ= 0 are easily destroyed by small δand replaced by a partial ferrimagnetic\nphase. The nonmagnetic phase is considered to be a gapless To monaga-Luttinger liquid phase based on the\nrecently extended Lieb-Schultz-Mattis theorem to the site -reflection invariant spin chains and numerical\ndiagonalization results.\n1. Introduction\nIn low-dimensional frustrated quantum magnets, the\ninterplay of quantum fluctuation and frustration leads\nto the emergence of various exotic quantum phases.1,2)\nThe conventional diamond chain3,4)is known as one of\nthe simplest examples in which an interplay of quan-\ntum fluctuation and frustration leads to a wide variety of\nground-state phases. Remarkably, this model has an in-\nfinite number of local conservation laws, and the ground\nstates can be classified by the corresponding quantum\nnumbers. If the two apical spins have equal magnitudes,\nthe pair of apical spins in each unit cell can form a non-\nmagnetic singlet dimer and the ground state is a di-\nrect product of the cluster ground states separated by\nsinglet dimers.3,4)Nevertheless, in addition to the spin\ncluster ground states, various ferrimagnetic states and\nstrongly correlated nonmagnetic states such as the Hal-\ndane state are also found when the apical spins form\nmagnetic dimers. In these cases, all the spins collectively\nform a correlated ground state over the whole chain.\nIn the presence of various types of distortion, the spin\ncluster ground states also turn into highly correlated\nground states. Extensive experimental studies have been\nalso carried out on the magnetic properties of the nat-\nural mineral azurite which is regarded as an example of\ndistorted spin-1/2 diamond chains.5,6)\nOn the other hand, the cases with unequal apical spins\narelessstudied. In this case,the apicalspins cannotform\na singlet dimer. Hence, all spins in the chain inevitably\nform a many-body correlated state. As a simple example\n∗E-mail address: hida@mail.saitama-u.ac.jpof such cases, we investigated the mixed diamond chain\nwith apical spins of magnitude 1 and 1/2, and vertex\nspins, 1/2 in Ref. 7 assuming that the exchange inter-\nactions between the vertex spins and two apical spins\nare equal to each other. In the absence of coupling λbe-\ntween two apical spins, we found a quantized ferrimag-\nnetic (QF) phase with the spontaneous magnetization\nper unit cell mspquantized to unity as expected from\nthe Lieb-Mattis (LM) theorem.8)With the increase of λ,\nwe found an infinite series of QF phases with msp= 1/p,\nwherepis a positive integer (1 ≤p <∞) that increases\nwithλ. Finally, the nonmagnetic Tomonaga-Luttinger\nliquid (TLL) phase sets in at a critical value of λ=λc.\nThe width and spontaneous magnetization of each QF\nphase tend to infinitesimal as λapproaches λc.\nIf the two apical spins have different magnitudes, how-\never, it is natural to assume that the exchange interac-\ntions between these two kinds ofapicalspins and the ver-\ntex spins are also different. We examine this case in the\npresent work.Two QF phaseswithout translationalsym-\nmetry breakdown are found. The QF phases with large\npare replaced by a partial ferrimagnetic (PF) phase in\nwhich the magnetization varies continuouslywith the ex-\nchange parameter. With the help of numerical diagonal-\nization results, the nonmagnetic phase is considered to\nbe a TLL phase consistent with the Lieb-Schultz-Mattis\n(LSM) theorem9–13)that is recently extended to site-\nreflection invariant spin chains.11–13)\nThispaperisorganizedasfollows.InSect.2,themodel\nHamiltonian is presented. In Sect. 3, various limiting\ncases are examined analytically. In Sect. 4, the classical\nground state is analytically determined. In Sect. 5, the\n1J. Phys. Soc. Jpn. FULL PAPERS\nground-statephasediagramdeterminedbythenumerical\ncalculation is presented and the properties of each phase\nare discussed. The last section is devoted to a summary\nand discussion.\n2. Model\nWe investigate the ground-state phases of mixed dia-\nmond chains described by the following Hamiltonian:\nH=L/summationdisplay\nl=1/bracketleftBig\n(1+δ)Sl(τ(1)\nl+τ(1)\nl−1)\n+(1−δ)Sl(τ(2)\nl+τ(2)\nl−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, respectively. The\nnumber of unit cells is denoted by L, and the total num-\nber of sites is 3 L. The lattice structure is depicted in Fig.\n1. We consider the region λ≥0 and 1 ≥δ≥ −1. For\nSl1+δ\n1−δ1+δ\n1−δτl(1)\nSl+1\nτl(2)S=τ(1)=1/2\nτ(2)=1λ\nFig. 1. Structure of the diamond chain investigated in this work.\nδ= 0, (τ(1)\nl+τ(2)\nl)2commutes with the Hamiltonian (1)\nfor alll. In Ref. 7, we made use of this property to de-\ntermine the ground-state phase diagram. In the present\nwork, we examine the general case of δ/negationslash= 0.\n3. Analytical Results\n3.1 Lieb-Mattis theorem\nWe start with several limiting cases where we can ex-\namine the ground state analytically. Before going into\nthe discussion of specific cases, we briefly introduce the\nLieb-Mattis theorem which is useful to determine the\nspontaneous magnetization in the absence of frustration.\nLet us consider a Heisenberg model\nHH=/summationdisplay\ni,jJijSiSj (2)\ndefined on a lattice consisting of two sublattices A and\nB. Here, Siis the spin operator on the i-th site with\narbitrary magnitude. It is assumed that the magnetic\ninteractions Jijsatisfy the following condition:\n(1) Iftwospinsareondifferentsublattices,then Jij≥0.\n(2) If twospins areon the same sublattice, then Jij≤0.\n(3) All spins are connected by magnetic interaction.\nThese assumptions imply the absence of frustration.\nThen, the spontaneous magnetization Mspof the ground(c) TLL(a) LM1\n(b) LM2\n(d) p=2(b’) LM21+δ\nλ\n1−δ\n1+δλ\n1−δ1/2⇑⇑ ⇑1+δ\nλ\n1−δ\n1+δ\nλ\n1−δ⇑ ⇑1+δλ\n1−δ1/2⇑⇑ ⇑\nFig. 2. Schematic spin configurations in the ground-state\nphases. The ovals in (b’), (c), and (d) mean that the two spins\nin an oval form a spin-doublet state. The open arrows express the\ntotal spins Tlin ovals. In (c), the antiferromagnetic long-range or-\nder depicted in the figure melts due to the quantum fluctuation\nresulting in the nonmagnetic TLL.\nstate is given by Msp=|SA−SB|, whereSAandSB\nare the sums of the magnitudes of the spins on the A-\nsublattice and B-sublattice, respectively.8)\n3.2λ= 0\nIf−1< δ <1, the system is unfrustrated and the\nground state is the QF phase with msp= 1 according\nto the LM theorem.8)In this case, the sublattice A con-\nsists of the sites occupied by Sl, and the sublattice B,\nthose occupied by τ(1)\nlandτ(2)\nl. Hence, SA=L/2 and\nSB= 3L/2 which gives Msp=Landmsp=Msp/L= 1.\nThe numerical analysis in Sect. 5 shows that this phase\nsurvives even in the weakly frustrated regime of small λ.\nHereafter, this phase is called the LM1 phase. Schematic\nspin configuration is presented in Fig. 2(a).\n3.3δ= 1\nIfλ >0, the ground state is the QF phase with\nmsp= 1 according to the LM theorem. In this case, the\nsublattice A consists of the sites occupied by τ(1)\nl, and\nthe sublattice B, those occupied by Slandτ(2)\nl. Hence,\nSA=L/2 andSB= 3L/2 which gives Msp=Land\nmsp=Msp/L= 1. The numerical analysis in Sect. 5\nshows that this phase survives even in the weakly frus-\ntrated regime of small 1 −δ. Hereafter, this phase is\ncalled the LM2 phase. The schematic spin configuration\npresented in Fig. 2(b) demonstrates that this phase is\ndistinct from the LM1 phase. This is also numerically\nconfirmed in Sect. 5.\n2J. Phys. Soc. Jpn. FULL PAPERS\n3.4λ≫1+δ,1−δ\nIn the limit λ→ ∞, all pairs of τ(1)\nlandτ(2)\nlform\ndoublet states ||Tl,Tz\nl/angbracketrightwithTl= 1/2 andTz\nl=±1/2\nwhereTl≡τ(1)\nl+τ(2)\nl. They are expressed using the\nbasis/vextendsingle/vextendsingle/vextendsingleτ(1)z\nl,τ(2)z\nl/angbracketrightBig\nas\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2,1\n2/angbracketrightbigg\n=/radicalbigg\n1\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,1\n2/angbracketrightbigg\n−/radicalbigg\n2\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,−1\n2/angbracketrightbigg\n,(3)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2,−1\n2/angbracketrightbigg\n=/radicalbigg\n1\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,−1\n2/angbracketrightbigg\n−/radicalbigg\n2\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1,1\n2/angbracketrightbigg\n.(4)\nFor large but finite λ,TlandSl′(l′=lorl+1) are cou-\npled by an effective Heisenberg interaction Jeff. IfJeff\nis antiferromagnetic, the ground state is the nonmag-\nnetic TLL state as schematically shown in Fig. 2(c). If it\nis ferromagnetic, the ground state is the QF state with\nmsp= 1 as shown in Fig. 2(b’). This ground state con-\nfiguration is continuously deformed to that of the LM2\nphase depicted in Fig. 2(b) by reducing the coefficient\nof|0,1/2/angbracketrightin Eq. (3) continuously. Hence, this QF state\nalso belongs to the LM2 phase. We estimate Jeffwithin\nthe first order perturbation calculation with respect to\n(1±δ)/λas\nJeff=1\n3(3−5δ). (5)\nHence, we find the phase boundary between the TLL\nphase and the LM2 phase is given by δ= 0.6 for large\nenoughλ.\n3.5δ=−1\nThe system is unfrustrated and the ground state is the\nnonmagnetic phase with msp= 0 according to the LM\ntheorem. In this case, the sublattice A consists of the\nsites occupied by τ(2)\nl, and the sublattice B, those occu-\npied bySlandτ(1)\nl. Hence, SA=LandSB=Lwhich\ngivesMsp= 0 and msp=Msp/L= 0. According to the\nnumerical calculation of Sect. 5, this phase continues to\nthe TLL phase discussed in Sect. 3.4.\n3.61+δ≃0andλ≃0\nFor 1 +δ=λ= 0,Slandτ(2)\nlform a ferrimagnetic\nchain with msp= 1/2, andτ(1)\nlare free spins with mag-\nnitude 1/2. We take the zeroth order Hamiltonian as\nH0=L/summationdisplay\nl=1/bracketleftBig\n(1−δ)Sl(τ(2)\nl+τ(2)\nl−1)/bracketrightBig\n,(6)\nand the total spin of H0is defined by\nStot\n0=L/summationdisplay\nl=1(Sl+τ(2)\nl). (7)The perturbation part of the Hamiltonian is rewritten\nas,\nH1=L/summationdisplay\nl=1/bracketleftBig\n(1+δ)(Sl+Sl+1)+λτ(2)\nl/bracketrightBig\nτ(1)\nl.(8)\nWithin the subspace of the ferrimagnetic ground states\nofH0withStot\n0=L/2, each ground state is specified by\nStot\n0z=−L/2,...,L/2. From rotational symmetry, the\ntwo-spin effective Hamiltonian Heff\nlforStot\n0andτ(1)\nlis\nwritten down by their inner product as\nHeff\nl=J′\neff\n(L/2)Stot\n0τ(1)\nl. (9)\nIt should be noted that the terms of higher powers of\nStot\n0τ(1)\nlareunnecessarysince Stot\n0τ(1)\nlcantakeonlytwo\nvaluesStot\n0/2 and−(Stot\n0+1)/2. Then, the total effective\nHamiltonian Heffis given by\nHeff=L/summationdisplay\nl=1Heff\nl. (10)\nTo determine J′\neff, we estimate the expectation values of\nH1andHeffin the state |F/angbracketright=|Stot\n0=L/2;Stotz\n0=L/2/angbracketright\nto find\n/angbracketleftF|Heff|F/angbracketright=L/summationdisplay\nl=12J′\neff\nL/angbracketleftF|Stotz\n0|F/angbracketrightτ(1)z\nl\n=J′\neffL/summationdisplay\nl=1τ(1)z\nl, (11)\n/angbracketleftF|H1|F/angbracketright=L/summationdisplay\nl=1/bracketleftBig\n2(1+δ)/angbracketleftF|Sz\nl|F/angbracketright\n+λ/angbracketleftF|τ(2)z\nl|F/angbracketright/bracketrightBig\nτ(1)z\nl.(12)\nComparing (11) and (12), we find\nJ′\neff= 2(1+δ)/angbracketleftF|Sz\nl|F/angbracketright+λ/angbracketleftF|τ(2)z\nl|F/angbracketright.(13)\nThe expectation values /angbracketleftF|Sz\nl|F/angbracketrightand/angbracketleftF|τ(2)z\nl|F/angbracketright\nare calculated by the numerical diagonalization for L=\n2,4,6,12 and 14. After two steps of Shanks transforma-\ntion,14)we find\n/angbracketleftF|Sz\nl|F/angbracketright ≃0.7924871, (14)\n/angbracketleftF|τ(2)z\nl|F/angbracketright ≃ −0.2924871. (15)\nThus, we can determine the phase boundary between\nthe LM1phaseandthe nonmagneticphaseas λ/(1+δ)≃\n0.73815 where the sign of J′\neffchanges.\n4. Classical Phase Diagram\nBefore the description ofthe numericalphase diagram,\nwe examine the classical limit. We regard all spins as\nclassical vectors with fixed magnitudes. The magnitudes\nofSlandτ(1)\nlare denoted by s, and that of τ(2)\nl, byαs.\n3J. Phys. Soc. Jpn. FULL PAPERS\nWe assume a uniform ground-state spin configuration in\nthe form,\nSl= (0,ssinϕ,scosϕ),\nτ(1)\nl= (0,ssinθ,scosθ),\nτ(2)\nl= (0,0,αs),(16)\nas depicted in Fig. 3. We take the direction of τ(2)\nlasz-\ndirection. The nonuniform configurationssuch as period-\n1+δ\nλ\n1−δ\nτ(2)\nlτ(1)\nl\nSl+1 Slθ\nϕ ϕ\nFig. 3. Definition (16) of spin angles θandϕ.\ndoubledstatesandspiralstatesarealsoconsidered.How-\never, they turned out to have higher energies.\nThe ground-state energy per unit cell is given by\nE= 2s2(1+δ)cos(θ−ϕ)\n+2αs2(1−δ)cosϕ+αs2λcosθ.(17)\nMinimizing Ewith respect to θandϕ, we have\n∂E\n∂θ=−2s2(1+δ)sin(θ−ϕ)−αs2λsinθ= 0,(18)\n∂E\n∂ϕ= 2s2(1+δ)sin(θ−ϕ)\n−2αs2(1−δ)sinϕ= 0. (19)\nLet us start with trivial solutions.\n(1)θ=ϕ= 0\nSl= (0,0,s),\nτ(1)\nl= (0,0,s),\nτ(2)\nl= (0,0,αs).(20)\nThis is a ferromagnetic phase with spontaneous\nmagnetization msp= (2 +α)sper unit cell. This\nphase is not realized in the parameter regime con-\nsidered ( λ >0 and−1≤δ≤1).(2)θ=ϕ=π:\nSl= (0,0,−s),\nτ(1)\nl= (0,0,−s),\nτ(2)\nl= (0,0,αs).(21)\nForα= 2, this is a N´ eel-type ground state with\nlong-range antiferromagnetic order. However, once\nthe quantum fluctuation is switched on, it is ex-\npected that this state turns into the nonmagnetic\nphasewith msp= 0owingtotheone-dimensionality.\nHence, this corresponds to the classical counterpart\nof the nonmagnetic phase.\n(3)θ= 0,ϕ=π:\nSl= (0,0,−s),\nτ(1)\nl= (0,0,s),\nτ(2)\nl= (0,0,αs).(22)\nThis is the classical counterpart of the LM1 phase\nwith spontaneous magnetization msp=αsper unit\ncell.\n(4)θ=π,ϕ= 0 :\nSl= (0,0,s),\nτ(1)\nl= (0,0,−s),\nτ(2)\nl= (0,0,αs).(23)\nThis is the classical counterpart of the LM2 phase\nwith spontaneous magnetization msp=αsper unit\ncell.\n(5) Nontrivial solution :\nAssuming sin θ/negationslash= 0 and sin ϕ/negationslash= 0, we find\ncosϕ=λ(1+δ)\n4α(1−δ)2−αλ\n4(1+δ)−(1+δ)\nαλ,(24)\ncosθ=2(1−δ2)\nαλ2−(1+δ)\n2α(1−δ)−α(1−δ)\n2(1+δ),(25)\nafter some elementary manipulations from (18) and\n(19). The spontaneous magnetization mspper unit\ncell is given by\nm2\nsp= (αs)2+2αs2(cosθ+cosϕ)\n+2s2/parenleftbigg\n1+cosϕcosθ−λ\n2(1−δ)(1−cos2θ)/parenrightbigg\n.\n(26)\nThis state corresponds to the PF phase.\nThe phase boundary between the PF phase and other\nphases can be obtained in the following way,\n(1) PF-LM1 ( θ= 0,ϕ=π) phase boundary :\n4J. Phys. Soc. Jpn. FULL PAPERS\nSetting cos θ= 1 in (25), we have λ=λc1where\nλc1=2(1−δ2)\nα(1−δ)+(1+δ). (27)\nThe value of cos ϕatλ=λc1is obtained by substi-\ntuting (27) into (24) as\ncosϕ=−1. (28)\nThis implies that λc1corresponds to the PF-LM1\nphase boundary.\n(2) PF-LM2 ( θ=π,ϕ= 0) and PF-nonmagnetic ( θ=\nϕ=π) phase boundary :\nSetting cos θ=−1 in (25), we have λ=±λc2where\nλc2=2(1−δ2)\n(1+δ)−α(1−δ). (29)\nThe value of cos ϕatλ=±λc2is obtained by sub-\nsstituting (29) into (24) as\ncosϕ=±1. (30)\nThis implies that λc2and−λc2correspond to the\nPF-LM2 and PF-nonmagnetic phase boundary, re-\nspectively.\nThe classical phase diagram obtained in this section is\nshown in Fig. 4 for αs= 1 and s= 1/2.\n−1 0 1012\nPFnonmagneticLM2\nLM1λ\nδ\nFig. 4. Ground-state phase diagram in the classical limit with\nαs= 1 and s= 1/2.\n5. Numerical Results\nWe have carried out the numerical exact diagonaliza-\ntion forL= 4,6 and 8 with the periodic boundary con-\ndition to determine the phase boundary from the values\nof the spontaneous magnetization. The extrapolation of\nthe transition point to the thermodynamic limit is car-\nried out using the Shanks transform.14)If the data for\nlarger systems are necessary, the DMRG calculation for−1 0 1012\nLM2 TLLλ\nδLM1PFp=2PFPF\nFig. 5. Ground-state phase diagram. The open circles are the\nphase boundaries estimated from the numerical exact diagon aliza-\ntion data extrapolated to the thermodynamic limit from L= 4,6\nand 8 by the Shanks transform.14)The double circles are the phase\nboundaries estimated from the DMRG data for L= 48. The trian-\ngles atδ= 0 are the phase boundaries between the infinite series\nof QF phases determined in Ref. 7. The QF phases with p >2\nfor finite δare not shown, since they survive only for invisibly\nsmallδin the present scale. The deviation from the scaling rela-\ntion ∆E∼1/LforL= 18 and 24 is significant in the shaded area.\nThe curves are guides for the eye.\nL= 48 is carried out with the open boundary condition.\nThe obtained phase diagram is shown in Fig. 5.\n5.1 Ferrimagnetic phases with msp= 1\nAs in the classical case, the two QF phases (LM1,\nLM2) with msp= 1 do not form a single phase but are\nseparatedbythePFphase,theTLLphase,andthe p= 2\nQF phase. The δ-dependence of the spontaneous magne-\ntizationmspforL= 48calculatedbythe DMRG method\nis shown in Fig. 6 for λ= 0.6. This behavior shows that\na PF phase with msp<1 intervenes between the two QF\nphases with msp= 1. The corresponding behavior in the\nclassical limit is also shown in Fig. 7. The angles θand\nϕvary byπacross the PF phase. This behavior explic-\nitly shows that the LM1 and LM2 phases are different\nphases.\n5.2 The fate of the infinite series of QF phases\nFigure 8 shows the λ-dependence of the spontaneous\nmagnetization for (a) δ= 0.1 and (b) 0.02. The corre-\nsponding figure for δ= 0 taken from Ref. 7 that shows\nthe presence of an infinite series of QF phases is also\nshown as Fig. 8(c) for comparison. For δ= 0.02, the\nQF phases with msp= 1 (p= 1) and msp= 1/2 (p= 2)\nremain finite and the structures survive around the mag-\nnetizations corresponding to p= 3 and 4. For δ= 0.1,\nonly the QF phases with msp= 1 (p= 1) and msp= 1/2\n(p= 2) remain finite, while those with p≥3 are smeared\nout and replaced by a PF phase. These results suggest\n5J. Phys. Soc. Jpn. FULL PAPERS\n0.4 0.6 0.8 100.51msp\nλ=0.6\nδp=1\nL=48\nFig. 6. δ-dependence of mspin the ground state for λ= 0.6\ncalculated by the DMRG method for open chains with L= 48.\n0.4 0.6 0.8 100.51\nmsp\nθ/π ϕ/πλ=0.6\nδ\nFig. 7. δ-dependence of mspin the classical ground state for\nλ= 0.6. The angles θandϕare also plotted.\nthat the QF phases become more fragile with increas-\ningp. The corresponding curves in the classical limit are\nshown in Fig. 9. The QF phases vanish in the classical\ncase showing that these are essentially quantum effects.\nThe fragility of the QF phase with large pcan be un-\nderstood in the following way: If the spontaneous break-\ndown of the translational invariance were absent, the QF\ngroundstates with p≥2 areabsent and the groundstate\nis a PF state that is regarded as a magnetized TLL.15)\nThe low energy effective Hamiltonian in this phase is\ngiven by the U(1) compactified boson field theory with\nthe TLL parameter Kas follows:\nH(0)\nB=1\n2π/integraldisplay\ndx/bracketleftbigg\nK(πΠ)2+/parenleftbigg1\nK/parenrightbigg\n(∂xφ)2/bracketrightbigg\n,(31)\nwhereφis a boson field defined on a circle φ∈[0,√\n2π),\nand Πis the momentum density field conjugate to φ. The\nspin wave velocity is set equal to unity. Extending the0.6 0.7 0.8 0.900.51msp\nδ=0.1\nλp=1\np=2\np=3\np=4L=48(a)\n0.6 0.7 0.8 0.900.51msp\nδ=0.02\nλp=1\np=2\np=3\np=4L=48(b)\n0.6 0.7 0.800.51\nλmsp\np≤38λc(∞)−~0.807p=2p=1\np=3\np=4(c)\nFig. 8. λ-dependence of the spontaneous magnetization in the\nground state for (a) δ= 0.1 and (b) δ= 0.02 calculated by the\nDMRGmethod foropen chains with L= 48. (c)The corresponding\nfigure for δ= 0 is taken from Ref. 7.\nbosonization procedure of Ref. 16 to the case of mixed\nspin chains, a translation by one unit cell results in the\nshift of the boson field as\nφ→φ+√\n2π(Suc−msp), (32)\nwhereSucis the sum of the spin magnitudes in a unit\ncell. In the present case, Suc= 2. We consider the case\n6J. Phys. Soc. Jpn. FULL PAPERS\n0 1 2 301msp\nλδ=0\nδ=0.02\nδ=0.1\nFig. 9. λ-dependence of the spontaneous magnetization in the\nground state for δ= 0, 0.02 and 0.1 in the classical limit.\nmsp= 1/pthatcorrespondstothevalueof mspintheQF\nphase with period p. Then, taking the compactification\ncondition into account, the shift (32) is rewritten as\nφ→φ−√\n2π/p. (33)\nThe leading perturbation invariant under the shift (33)\nis given by\nH(1)\nB=c/integraldisplay∞\n−∞dxcos(√\n2pφ), (34)\nwherecis a constant. Although this operator is transla-\ntionally invariant, if it is relevant, the phase φis pinned\ntooneofthe minimaof ccos(√\n2pφ)andthe translational\ninvariance is spontaneously broken. Since the scaling di-\nmension of the operator(34) is xp=p2K/2,it is relevant\nifxp<2. Although the p-dependence of Kis unknown,\nassuming that it is moderate, the main pdependence of\nxpcomes from the factor of p2. This explains why the\nQF phases with large pare more fragile than those with\nsmallp.\n5.3 Nonmagnetic phase\nFor larger λ, the nonmagnetic phase appears and it\ncontinues to the TLL phase for large λdiscussed in Sect.\n3.4. Since the sum of the spin magnitudes in a unit cell is\nan integer, the conventional LSM theorem9,10)does not\nexclude the unique gapped phase. However, our model\n(1) is invariant under the site-reflection about the vertex\nspinSlwhose magnitude is 1/2. Hence, our model sat-\nisfies the condition to exclude the unique gapped phase\nin the recent extension of the LSM theorem to the site-\nreflection invariant spin chains.11–13)Taking the continu-\nity to the TLL phase in the limit λ→ ∞into account,\nthewholenonmagneticphaseisconsideredtobetheTLL\nphase. This is confirmed by the numerical diagonaliza-\ntion calculation of the singlet-triplet energy gap ∆ E. It\nis checkedthat ∆ Eapproximatelyscaleswith the system0 1 2024\nλL∆Eδ=−0.3\nL=4\nL=6\nL=8(a)\n0 1 200.51\nλL∆Eδ=0.3\nL=4\nL=6\nL=8(b)\n0 1 2 3 400.10.20.3\nλL∆Eδ=0.5\nL=4\nL=6\nL=8(c)\nStot>1 Stot=1\nFig. 10. λ-dependence of the scaled gap L∆Eof the lowest ex-\ncitations for (a) δ=−0.3, (b) 0.3 and (c) 0.5 with L= 4,6 and\n8. The open and filled symbols are excitations with Stot= 1 and\nStot>1, respectively.\nsizeLas ∆E∼1/Las shown in Fig.10(a) for δ=−0.3\nand (b) for 0.3. Similar analyses are also carried out for\nseveral other values of δ. In the vicinity of the PF phase\nindicated by the shaded areaofFig. 5, however,the devi-\nationfromthescalingrelation∆ E∼1/Lissignificantas\nshown in Fig. 10(c). Nevertheless, this area shrinks with\nthesystemsize.Hence,itislikelythatthewholenonmag-\n7J. Phys. Soc. Jpn. FULL PAPERS\nnetic phase is a TLL phase. It should be also remarked\nthat the nonmagnetic ground state for δ= 0 is also a\nTLL phase7)since it is exactly mapped onto the ground\nstate of the spin-1/2antiferromagneticHeisenbergchain.\nForλ/greaterorsimilarλc, whereλcis the nonmagnetic-ferrimagnetic\ntransition point, the ferrimagnetic state with total spin\nStot>1 comes down resulting in the level-crossing with\nthe nonmagnetic state at λc. Their energies measured\nfrom the ground state are plotted by the filled symbols\nin Fig. 10(c). These ferrimagnetic states are macroscop-\nically different from the nonmagnetic state in the ther-\nmodynamic limit and cannot be regarded as elementary\nexcitations in the nonmagnetic state, even though they\nhave the next-lowest energy for finite-size systems. Un-\nfortunately, in the region where this type of state has the\nnext-lowest energy, it is difficult to identify the singlet-\ntriplet gap, since we can calculate only several lowest\neigenvalues by the Lanczos method we employ in this\nwork.\n6. Summary and Discussion\nThe ground-state phases of mixed diamond chains (1)\nare investigated numerically and analytically. For com-\nparison, the ground-state phase diagram of the corre-\nsponding classical model is calculated analytically. In\nthe quantum case, the ground-state phase diagram is de-\ntermined using the numerical exact diagonalization and\nDMRG method in addition to the perturbation analysis\nforvariouslimitingcases.Thegroundstateofthepresent\nmodel has a rich variety of phases such as the two kinds\nof QF phases with msp= 1, the QF phase with sponta-\nneous translational symmetry breakdown, the PF phase,\nand the nonmagnetic TLL phase.\nThefate oftheinfinite seriesofQFphasesobservedfor\nδ= 0 is also investigated numerically. It turned out that\nthe QF phases with largespatial periodicities pare easily\ndestroyed by small δand replaced by the PF phase. The\ninterpretation of this behavior is also discussed using the\nbosonization argument.\nIn the nonmagnetic phase, the unique gapped ground\nstate is excluded based on the recently extended LSM\ntheorem.11–13)Combined with the numerical calculation\nofthe energygap, this regionis consideredto be the TLL\nphase.Sofar,theexperimentalmaterialscorrespondingtothe\npresent mixed diamond chain arenot available.However,\nconsidering the rich variety of ground-state phases, the\nexperimental realizationofthe present model is expected\nto produce a fruitful field of quantum magnetism. With\nthe recent progress in the synthesis of mixed spin com-\npounds,17)we expect the realization of related materials\nin the near future.\nThenumericaldiagonalizationprogramisbasedonthe\nTITPACK ver.2 coded by H. Nishimori. 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B102, 060408(R) (2020) and references therein.\n8" }, { "title": "1703.08263v1.Anomalous_current_induced_spin_torques_in_ferrimagnets_near_compensation.pdf", "content": "1 \n Anomalous current -induced spin torques in ferrimagnets near \ncompensation \nRahul Mishra1, Jiawei Yu1, Xuepeng Qiu2, M. Motapothula3, T. 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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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": "1502.01071v1.Instability_of_a_ferrimagnetic_state_of_a_frustrated_S_1_2_Heisenberg_antiferromagnet_in_two_dimensions.pdf", "content": "arXiv:1502.01071v1 [cond-mat.mtrl-sci] 4 Feb 2015Japanese Journal of Applied Physics RAPID COMMUNICATION\nInstability of a ferrimagnetic state of a frustrated S= 1/2\nHeisenberg antiferromagnet in two dimensions\nHiroki Nakano1∗and Toru Sakai1,2\n1Graduate School of Material Science, University of Hyogo, K amigori, Hyogo 678-1297, Japan\n2Japan Atomic Energy Agency, SPring-8, Sayo, Hyogo 679-5148 , Japan\nTo clarify the instability of the ferrimagnetism which is th e fundamental magnetism of ferrite, numerical-\ndiagonalization study is carried out for the two-dimension alS= 1/2Heisenberg antiferromagnet with frus-\ntration. We find that the ferrimagnetic ground state has the s pontaneous magnetizationin small frustration;\ndue to a frustrating interaction above a specific strength, t he spontaneous magnetization discontinuously\nvanishes so that the ferrimagnetic state appears only under some magnetic fields. We also find that, when\nthe interaction is increased further, the ferrimagnetism d isappears even under magnetic field.\nFerrite is a magnetic material that is indispensable in modern society. It is be-\ncause this material is used in various industrial products including mo tors, generators,\nspeakers, powder for magnetic recording, and magnetic heads et c. It is widely known\nthat fundamental magnetism of the ferrite is ferrimagnetism.1–4)The ferrimagnetism is\nan important phenomenon that has both ferromagnetic nature an d antiferromagnetic\nnature at the same time. The occurrence of ferrimagnetism is unde rstood as a mathe-\nmatical issue within the Marshall-Lieb-Mattis (MLM) theorem5,6)concerning quantum\nspin systems. A typical case showing ferrimagnetism is when a syste m includes spins\nof two types that antiferromagnetically interact between two spin s of different types in\neach neighboring pair, for example, an ( S,s)=(1, 1/2) antiferromagnetic mixed spin\nchain, in which two different spins are arranged alternately in a line and coupled by the\nnearest-neighbor antiferromagnetic interaction. The ferrimagn etic state like the above\ncase, in which the spontaneous magnetization is fixed to be a simple fr action of the\nsaturated magnetization determined by the number of up spins and that of down spins\nin the state, is called the Lieb-Mattis (LM) type ferrimagnetism. Ano ther example of\nferrimagnetism is a system including single-type spins that are more t han one in a unit\ncell, although the ferrimagnetism can appear even in a frustrating s ystem including\n∗E-mail: hnakano@sci.u-hyogo.ac.jp\n1/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n\tB\n \tC\n \nβα\nα\b\nFig. 1. (Color) Network of antiferromagnetic interactions studied in this p aper. The black and red\nbonds represent J1andJ2interactions. Green squares denote finite-size clusters of 24 and 30 sites in\n(a) and (b), respectively. Note that the two-dimensional networ k composed only of the black bonds is\ncalled the Lieb lattice.\nonly a single spin within a unit cell.7,8)\nThe antiferromagnet on the Lieb lattice illustrated in Fig. 1 correspo nds the second\ncase, in which there are three spins in a unit cell. The MLM theorem hold s in the Lieb-\nlattice antiferromagnet. If antiferromagnetic interactions are a dded to this Lieb lattice\nso that magnetic frustrations occur, however, the MLM theorem no longer holds. In this\nsituation, the ferrimagnetic state is expected to become unstable . The problem of how\nthe ferrimagnetism collapses owing to such frustrating antiferrom agnetic interactions is\nan important issue to understand the ferrimagnetism well and to ma ke ferrimagnetic\nmaterials more useful in various products. This problem was studied in theS= 1/2\nHeisenberg antiferromagnet on the spatially anisotropic kagome lat tice,9,10)where the\nexistence of an intermediate phase with weak spontaneous magnet ization is clarified\nbetween the LM type ferrimagnetic phase and the nonmagnetic pha se including the\nisotropic kagome-lattice antiferromagnet. We are then faced with a question: is there\nany other different behavior of the collapse of the ferrimagnetism?\nUnder circumstances, the purpose of this study is to demonstrat e the existence of\na different behavior of collapsing ferrimagnetism in the case of an S= 1/2 Heisenberg\nantiferromagnet on the lattice shown in Fig. 1 to answer the above q uestion. When the\nantiferromagnetic interactions denoted by the red bonds vanish, the system is unfrus-\ntrated and thus it certainly shows ferrimagnetism in the ground sta te. In this study, we\nexamine the case when the red-bond interactions are switched on.\n2/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\nThe model Hamiltonian examined inthis study is given by H=H0+HZeeman, where\nH0=/summationdisplay\ni∈α,j∈βJ1Si·Sj+/summationdisplay\ni∈α′,j∈βJ1Si·Sj\n+/summationdisplay\ni∈α,j∈α′J2Si·Sj, (1)\nHZeeman=−h/summationdisplay\njSz\nj. (2)\nHereSidenotes an S= 1/2 spin operator at site i. Sublattices α,α′, andβand the\nnetwork of antiferromagnetic interactions J1andJ2are depicted in Fig. 1. Here, we\nconsider the case of isotropic interactions. The system size is deno ted byNs. Energies\nare measured in units of J1; thus, we take J1= 1 hereafter. We examine the properties\nof this model in the range of J2/J1>0. Note that, in the case of J2= 0, sublattices α\nandα′are combined into a single sublattice; the system satisfies the above conditions\nof the MLM theorem. Thus, ferrimagnetism of the LM type is exactly realized in this\ncase. In the limit of J2/J1→ ∞, on the other hand, the lattice of the system is reduced\nto a trivial system composed of isolated S= 1/2 spins and isolated dimers of two spins.\nIts ground state is clearly different from the state of the LM-type ferrimagnetism in the\ncase ofJ2= 0. One thus finds that while J2becomes larger, the ground state of this\nsystem will change from the ferrimagnetic one in the case of J2= 0 to another state,\nwhich we survey here.\nNext, we discuss the method we use here, which is numerical diagona lization based\non the Lanczos algorithm.11)It is known that this method is nonbiased beyond any\napproximations and reliable for many-body problems, which are not o nly localized spin\nsystems such as the Heisenberg model12,13)treated in th present study but also strongly\ncorrelatedelectronsystemsincludingtheHubbardmodel14–16)andthet-Jmodel.14,17,18)\nA disadvantage of this method is that the available system sizes are lim ited to being\nsmall. Actually, the available sizes in this method are much smaller than t hose of the\nquantum Monte Carlo simulation19,20)and the density matrix renormalization group\ncalculation;21)however, it is difficult to apply both methods to a two-dimensional (2D )\nfrustrated system like the present model. This disadvantage come s from the fact that\nthe dimension of the matrix grows exponentially with respect to the s ystem size. In\nthis study, we treat the finite-size clusters depicted in Fig. 1 when t he system sizes\nareNs= 24 and 30 under the periodic boundary condition. Note that each o f these\nclusters forms a regular square although cluster (b) is tilted from a ny directions along\n3/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n–2 0\nh/J 1–1 01M/M sJ2/J 1 = 0.55(b)0 5 10 \nM–10–5 E /J 1Ns = 30\nJ2/J 1 = 0.55(a)\n2\nFig. 2. (Color) Results for J2/J1= 0.55. Lowest energy in each subspace of Mfor the system of\nNs= 30 is shown in panel (a). The magnetization process is depicted in pa nel (b); red and black lines\nrepresent results for Ns= 24 and 30, respectively.\ninteraction bonds.\nWecalculatethelowestenergyof H0inthesubspacecharacterizedby/summationtext\njSz\nj=Mby\nnumerical diagonalizations based on the Lanczos algorithm and/or t he Householder al-\ngorithm. The energy is represented by E(Ns,M), whereMtakes every integer up to the\nsaturation value Ms(=SNs). We here use the normalized magnetization m=M/Ms.\nSome of Lanczos diagonalizations have been carried out using the MP I-parallelized\ncode, which was originally developed in the study of Haldane gaps.22)Note here that\nour program was effectively used in large-scale parallelized calculation s.23–25)\nTo obtain the magnetization process for a finite-size system, one fi nds the magneti-\nzation increase from MtoM+1 at the field\nh=E(Ns,M+1)−E(Ns,M), (3)\nunder the condition that the lowest-energy state with the magnet izationMand that\nwithM+1 become the ground state in specific magnetic fields. Note here th at it often\nhappens that the lowest-energy state with the magnetization Mdoes not become the\n4/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n–0.2 0\nh/J 1–0.4 –0.2 00.20.4M/M sJ2/J 1 = 0.64(b)\n0010 2 4 6\nM–13–12–11E /J 1Ns = 30\nJ2/J 1 = 0.64(a)\n0.2 3\nFig. 3. (Color) Results for J2/J1= 0.64. Lowest energy in each subspace of Mfor the system of\nNs= 30 is shown in panel (a). The magnetization process is depicted in pa nel (b); red and black lines\nrepresent results for Ns= 24 and 30, respectively. Main panel is a zoomed-in view of its inset wit h a\nwide range. The broken lines represent the results before the Max well construction is carried out.\nground state in any field. The magnetization process in this case is de termined around\nthe magnetization Mby the Maxwell construction.26,27)\nNow, we observe the case of J2/J1= 0.55; results are shown in Fig. 2. Figure 2(a)\ndepicts the lowest energy level in the subspace belonging to MforNs= 30. The levels\nforM= 0 toM= 5 are identical within the numerical accuracy. For M >5, the\nenergies increase with M. This behavior indicates that the spontaneous magnetization\nisM= 5. In Fig. 2(b), we draw the magnetization process determined by eq. (3) in the\nfull range from the negative to the positive saturations. The spon taneous magnetization\nm= 1/3 appears and the state at m= 1/3 shows the plateau with a large width. It is\nobserved that, above m= 1/3, the magnetization grows continuously. These behaviors\nare common with those of the LM ferrimagnetism at the unfrustrat ed case of J2= 0.\nNext, let us examine the case of J2/J1= 0.64; results are shown in Fig. 3. The\nMdependence of the lowest energy belonging to Mis different in M <3 from the\n5/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\ncase ofJ2/J1= 0.55. This difference affects with the disappearance of the spontane ous\nmagnetization, which is shown in Fig. 3(b). This discontinuous disappe arance occurs\natJ2/J1∼0.59 forNs= 24 and at J2/J1∼0.63 forNs= 30. An important point\nis that an intermediate state with smaller but nonzero spontaneous magnetizations is\nabsent between the m= 1/3 state and the nonmagnetic state. This behavior is clearly\ndifferent from the presence of such an intermediate state in the sp atially anisotropic\nkagome lattice.9,10)We speculate that this difference comes from the point that the\ncompeting interaction in the present model has a strong quantum n ature localized\nat pairs of dimerized spins. The discovery of the future third case o f the collapsing\nferrimagnetism would contribute to confirm our speculation. Note a lso that the plateau\natm= 1/3 shows a large width. This suggests that the ferrimagnetic state is realized\nif external magnetic fields are added.\nTo examine the properties of the m= 1/3 states in a more detailed way, we evaluate\nthe local magnetization defined as\nmξ\nLM=1\nNξ/summationdisplay\nj∈ξ/angbracketleftSz\nj/angbracketright, (4)\nwhereξtakesα,α′andβ. Here, the symbol /angbracketleftO/angbracketrightdenotes the expectation value of the\noperator Owith respect to the lowest-energy state within the subspace with a fixed\nMof interest. Recall here that the case of interest in this paper is M=Ms/3. Here\nNξdenotes the number of ξsites. Results are shown in Fig. 4. In the region of small\nJ2/J1,αandα′spins are up and βspin is down, although each of magnetizations is\nslightly deviated from the full moment due to a quantum effect. This s pin arrangement\nis a typical behavior of ferrimagnetism. On the other hand, in the re gion of large J2/J1,\nthe magnetizations at αandα′spins are vanishing and βspin shows almost a full\nmoment up. This marked change in the local magnetizations occurs a tJ2/J1∼1.38 for\nNs= 24 and at J2/J1∼1.40 forNs= 30, which suggests the occurrence of the phase\ntransition around at J2/J1∼1.4. Therefore one finds that, for J2/J1larger than this\ntransition point, the ferrimagnetic state cannot be realized even u nder magnetic fields.\nItisunfortunatelydifficulttodeterminethetransitionpointintheth ermodynamiclimit\nprecisely only from the present two samples of small clusters. For t he determination,\ncalculations of larger clusters are required in future studies. Note here that similar\nobservations of the local magnetizations were reported in Refs. 2 8 and 29, which treated\ntheHeisenberg antiferromagnet ontheCairo-pentagonlattice,30)a2Dnetwork obtained\nby the tiling of single-kind inequilateral pentagons. The same behavio r ofmξ\nLMis also\n6/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\nobserved when the kagome-lattice antiferromagnet31–37)is distorted in the√\n3×√\n3\ntype.25,38)The relationship between these models should be examined in future s tudies.\nNote also that, in the present model, the change around the trans ition point seems\ncontinuous irrespective of whether the system size is Ns= 24 or 30. This aspect is\ndifferentfromtheobservationintheCairo-pentagon-latticeantif erromagnet,28,29)where\nthe change around the transition point seems continuous for Ns= 24 but discontinuous\nforNs= 30. We speculate that whether the change is continuous or discon tinuous in\nfinite-size data is related to whether the number of unit cells in finite- size clusters is an\neven integer or an odd integer. To confirm this speculation, furthe r investigations are\nrequired in future. It will be anunresolved question whether the tr ansitionis continuous\nordiscontinuousinthethermodynamiclimit.Figure5depictsthemagn etizationprocess\natJ2/J1∼1.39. No jumps seem to appear in the process at J2/J1corresponding to\nthe transition point. It is unclear whether the width at m= 1/3 survives or vanishes\nalthough this m= 1/3 width at J2/J1∼1.39 is smaller than those in Figs. 2(b) and\n3(b). Future studies would clarify how themagnetization process b ehaves in the vicinity\nof the transition point.\nIn summary, we have investigated how the ferrimagnetic state of t heS= 1/2\nHeisenberg antiferromagnet onthe 2Dlatticecollapses owing to mag netic frustrationby\nnumerical-diagonalization method. We capture a discontinuous vanis hing of the sponta-\nneous magnetization without intermediate phase showing spontane ous magnetizations\nthat are smaller than that of the Lieb-Mattis ferrimagnetic state w hen a frustrating\ninteraction is increased. We also observe the disappearance of the ferrimagnetic state\nunder magnetic fields for even larger interaction showing frustrat ion. It is known that\norganicmolecularmagnetscanrealizeferrimagnetism.39,40)Sincevarietyoflatticestruc-\ntureleadingtoaninteractionnetworkisavailableinsuchorganicmolec ularmagnets,the\nexperimental confirmation might be done in these magnets more eas ily than metallic-\nelement compounds. Further studies concerning instability of the f errimagnetism would\ncontribute much for our development of more stable ferrimagnetic materials.\nAcknowledgments This work was partly supported by JSPS KAKENHI Grant Numbers 23 340109\nand 24540348. Nonhybrid thread-parallel calculations in numerical diagonalizations were based on\nTITPACK version 2 coded by H. Nishimori. Part of calculations in this st udy were carried out as an\nactivity of a cooperative study in Center for Cooperative Work on C omputational Science, University\nof Hyogo. Some of the computations were also performed using fac ilities of the Department of\nSimulation Science, National Institute for Fusion Science; Center f or Computational Materials\nScience, Institute for Materials Research, Tohoku University; Su percomputer Center, Institute for\n7/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n0.5 1 1.5\nJ2/J 1–0.4 –0.2 00.20.4L ocal magnetization1.38 1.4 1.4200.5\n2\nFig. 4. (Color) Behavior of local magnetizations vs. the ratio of interactio nsJ2/J1together with a\nzoomed-in view near the transition point in inset. Closed circles and clo sed diamonds denote results\nforαandβforNs= 24, respectively. Results for α′forNs= 24 are identical those for αwithin the\nnumerical accuracy because αandα′are symmetric in the Ns= 24 cluster. Open circle, open\ntriangle, and open squares represent results for α,α′, andβforNs= 30, respectively. Due to the\ntilting for Ns= 30,αandα′are not symmetric, although results of αandα′forNs= 30 are slightly\ndifferent but very similar. To avoid invisibility from overlapping of symbo ls, results of α′are shown\nonly in inset.\n0 1 2 \nh/J 100.51M/M sJ2/J 1 = 1.39\n3\nFig. 5. (Color) Magnetization process for J2/J1= 1.39. Red and black lines represent results for\nNs= 24 and 30, respectively.\nSolid State Physics, The University of Tokyo; and Supercomputing D ivision, Information Technology\nCenter, The University of Tokyo. This work was partly supported b y the Strategic Programs for\nInnovative Research; the Ministry of Education, Culture, Sports , Science and Technology of Japan;\nand the Computational Materials Science Initiative, Japan. 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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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V Nagar, Kalapet, Pondicherry\n-\n605014, India.\n \nCondensed \nM\natter \nP\nhysics \nD\nivi\ns\nion, Saha Institute of Nuclear Physics, 1/AF \nBidhannagar, Kolkata\n-\n700064\n \n*\nCorresponding author: Tel.: +91\n-\n9944064547; E\n-\nmail: rnbhowmik.phy@pondiuni.edu.in\n \nAbstract\n \nA\n \ncomparative \nstudy of the non\n-\nequilibrium magnetic phenomena (magnetic blocking, memory, \nexchange bias and aging effect) has been presented for \nferrimagnetic \nCo\n1.75\nFe\n1.25\nO\n4\n \n(CFO) and \nits composite with \nnon\n-\nmagnetic \nBaTiO\n3\n \n(BTO). \nS\nynchrotron X\n-\nRay diffraction \npatterns h\nave \nconfirmed \ncoexistence \nof \nCFO and BTO structures \nin composite\n, but \nmagnetic spin dynamics \nhave \nbeen \nremarkabl\ny\n \nmodifi\ned\n. The blocking \nphenomenon \nof ferrimagnetic \ndomains below \nthe \nroom temperature \nhas been \nstudied \nby \ndifferent \nmodes of \n(\nzero field coole\nd and field cooled\n)\n \nmagnetic \nmeasurements \nin \ncollaboration with \nmagnetic fields\n \nON and OFF modes and time \ndependent magnetization\n. \nThe \napplications of \nunconventional pr\notocols \nduring \ntime dependent \nmagneti\nzation\n \nmeasurement \nat \ndifferent stages of \nthe \ntempe\nrature and field dependence of \nthe \nmagnetization curves\n \nhave been useful to \nreveal\n \nt\nhe non\n-\nequilibrium dynamics of magnetic spin \norder\n. \nThe \napplying\n \nof\n \noff\n-\nfield relaxation experiments\n \nhas made possible to tune \nthe \nmagnetic \nstate and coercivity of the \nsyst\nems\n.\n \nThe role of interfacial coupling between magnetic and non\n-\nmagnetic particles has been understood on different\n \nmagnetic \nphenomena\n \n(\nmeta\n-\nstable magnetic \nstate, exchange bias\n \nand \nmemory effect\n)\n \nby comparing the experimental results of \nCo\n1.75\nFe\n1.25\nO\n4\n \nspin\nel oxide \nand \nit’s\n \ncomposite with \nBaTiO\n3\n \nparticles\n.\n \nKeywords\n:\n \nSpinel\n \nferrite, \nBaTiO\n3\n, \nComposite magnet, Exchange bias\n, \nMemory \nand aging \neffect\n.\n 2\n \n \n1. \nIntroduction\n \nThe \nnon\n-\nequilibrium spin dynamics \nin magnetic materials strongly \ndepend\nent\n \non \nspin \ndisorder \nand \nm\nanifested by \nmany unusual \nmagnetic \nphenomena, e.g., spin glass, super\n-\nspin glass\n \n/\ncluster spin glass, superparamagnetic blocking, exchange bias, domain wall pinning, memory \nand training \neffect [\n1\n-\n7\n].\n \nEach of these phenomena has their own characteristics. \nT\nhe s\npin glass\nes\n \nare \ndefined by \na typical \ncompet\nit\nion \nbetween \nferromagnetic (FM) and antiferromagnetic (AFM) \nexchange \ninteractions \nand \nfrustrat\nion\n \nof \nthe \nspins\n \nin lattice structure\n. \nThe spin dynamics below \na \ncharacteristic \nfreezing\n \ntemperature\n \nbecomes slow \ndue to \nincreasing \ninter\n-\nspin interaction\ns\n. \nThe \nsuperparamagnetic blocking\n \nof non\n-\ninteracting \nmagnetic \nparticles (group of spins) \noccurs below \na typical temperature\n \ndue to relaxation of the particles along \ntheir \nlocal anisotropy\n \naxes. \nTaking \ninto account th\ne existence of strong inter\n-\nparticle interactions, the freezing of \nnanoparticles \nassembly\n \nis defined as super\n-\nspin glass or cluster\n-\nspin glass\n \n[\n3,7\n-\n8\n]\n. \nIt is practically difficult to \ndistinguish the features of super\n-\nspin glass from superparamagnetic block\ning in magnetic \nnanoparticles, having finite inter\n-\nparticle interactions, and distribution in size and anisotropy. In \nsuch systems, t\nhe aging effect \n(relaxation phenomenon) \nplays an important role in determining \nspin dynamics below the\nir\n \nfreezing/blocking \ntemperature. \nThe \nmagnetic exchange bias effect \nwas primarily modeled for \nFM and AFM \nbi\n-\nlayers\n \n[\n9\n], \nbut \nit has been found in many particulate \nsystems where interfacial exchange coupling between FM (core) and weak FM/AFM (shell) \nstructure control \nthe \nshape o\nf magnetic hysteresis loop \n[\n2\n, \n1\n0\n-\n1\n1\n]. \nThe \nmemory \neffect is another \nform\n \nof non\n-\nequilibrium\n \nspin dynamics, where \nnew \nspin configuration/\nmeta\n-\nstable state \nachieved \nduring\n \nintermediate stops \nof \nzero field or field cool\ned \nmagnetization curves \ncan be retrieved\n \nduring re\n-\nheating\n \nprocess\n \n[\n1\n-\n3\n]\n.\n \nThe memory effect \nhas been \nobserved \nin a wide range of \nmaterials, irrespective of \nstrong\nly\n \ninteracti\nng \n[\n12\n-\n13\n]\n \nand non\n-\ninteracting \nspin sy\ns\ntems\n \n[\n14\n-\n15\n]. 3\n \n \nThe artificially designed \nferrimagnetic\n-\nferroelectric composite \nand h\netero\n-\nstructured \nspin\n \nsystems \nalso \nshow\ned\n \nexchange bias and memory effect \n[\n10\n-\n11, 16\n-\n18\n].\n \nThe\n \nexchange bias effect \ndominates \nat lower temperatures \nand \nmemory \neffect \ndominates at higher temperatures\n \n[1\n0\n, 1\n3\n, \n1\n9\n], and both are \nnot free from spin glass\n \nfreezi\nng\n, superparamagnetic blocking, anisotropy and \ndomain wall pinning effect. \nThe training effect, on the other hand, \nis related to \nan irreversible \nchange in spin structure pinned at domain walls or at the interfaces of FM\n-\nAFM structure or at \nthe interfaces o\nf ferromagnetic and ferroelectric systems [\n20\n-\n21\n]. \nThe disorder induced by \ncoexisting crystalline phases\n \nalso played \nrole\n \non \nspin \ndependent electronic conductivity \n[\n22\n]\n. \nApart from basic understanding, t\nhe \nstudy \nof non\n-\nequilibrium \nspin dynamics is\n \nuseful f\nor \napplications of \nstrongly interacting electronic spin systems\n,\n \nsuch as random alloy [\n3, 7\n], \nperovskite [\n2\n,\n \n6\n,\n \n15\n], and spinel ferrite [\n3\n-\n5\n], \nin spin valves, spins filter, read\n-\nwriting devices, \nmagneto\n-\nresistive random access memories, \nsensors and magneti\nc switches [23\n-\n24\n]. This \nrequires an effective strategy \nfor \ntun\ning\n \nthe \nferro/ferri\nmagnetic parameters\n \nby controlling \nthe \neffects of \nspin disorder\n \ninside the domains or at interfaces \nof \nthe composite \nmaterials\n.\n \n \nThe present work focuses on s\npinel ferrites\n, \nwhich \nare defined by \na \ngeneral \nformula \nunit \nAB\n2\nO\n4\n,\n \nwhere \ncat\nions occupy \nthe \ntetrahedral (A) and octahedral (B) coordinated \nlattice sites \nwith\n \nanion\ns\n \n(\nO\n2\n-\n) \nat \nfcc \npositions\n \nof\n \nthe \nlattice \nstructure\n. \nIn long range ferrimagnetic \n(FiM) \nspinel \nferrite\n, antiferr\nomagnetic (AFM) superexchange interactions between A and B site moments \n(J(A\n-\nO\n-\nB) are expected to be strong in comparison to intra\n-\nsublattice interactions (J(B\n-\nO\n-\nB) and \n(J(A\n-\nO\n-\nA))\n \n[2\n5]\n. \nIn this work, we will study \nthe effects of intrinsic disorder in ferri\nmagnetic \nCo\n1.75\nFe\n1.25\nO\n4 \nparticles\n \n[2\n6\n] and extrinsic \nspin \ndisorder \n(interfacial effect) \nin its composite with \nnon\n-\nmagnetic BaTiO\n3 \n[2\n7\n]\n \nto control the non\n-\nequilibrium magnetic phenomena, e.g., exchange \nbias, memory and aging effect\n.\n \n 4\n \n \n2. \nExperimental\n \n2.1. Ma\nterial Preparation\n \nT\nhe \nmaterial preparation and characterization of the \nCo\n1.75\nFe\n1.25\nO\n4\n \n(CFO) ferrite and its \ncomposite with BaTiO\n3\n \n(BTO) were \ndescribed\n \nin \nearlier works [\n2\n6\n-\n2\n7\n]. \nThe \nferrite \npowder \nwas \nprepared by \nchemical \nco\n-\nprecipitation route and \nthermal\n \nanneal\ning \nat \n8\n00\n \n0\nC\n \n(CF80) and 9\n00 \n0\nC\n \n(CF90) \nfor 2 hrs\n. \nT\nhe CF80 sample formed bi\n-\nphased cubic spinel structure\n, unlike single phase \nstructure in \nCF90 \nsample. \nThe composite \nsample \nCF80_BTO was prepared by mixing \nof \nCF\n80\n \nferrite and \nBTO\n \npowders \nwith mass r\natio 50:50\n, and final \nheat\n \ntreatment was performed\n \nat 1000 \n0\nC for 4 hrs.\n \nS\nynchrotron X\n-\nray diffraction \npattern\n \nconfirmed the \ncoexist\nence of \ncubic spinel \nstructure\n \nof\n \nCFO and \ntetragonal phase\n \nof\n \nBTO\n \nin \nthe composite CF80_BTO sample\n \nwithout any \nintermediate \nphase formation\n.\n \nInterestingly, \nbi\n-\nphased nature of CF80 sample (as seen from \nsplit\n \nof \nX\n-\nray diffraction peaks of \ncubic spinel phase) disappeared in \nCF80_BTO composite\n. \nThe \nspin \nstructure in \nCF90 ferrite and CF80_BTO composite samples \nare schematically mod\neled in Fig. \n1(a\n-\nb) \nand origin of the \nspin disorder for non\n-\nequilibrium spin dynamics \nare summarized below.\n \nThe single phase ferrite sample CF90 is \nmodeled as \nconsisting of average \nparticle \nsize \n\n \n40 nm \nand each magnetic particle is assumed to be consistin\ng of core\n-\nshell spin structure [1\n0\n, 1\n9\n]. The \ncore (interior) part is consisting of more than one domain (multi\n-\ndomain structure)\n. The\n \nspins \ninside each domain are ferrimagnetically (\n\n\n\n\n)\n \nordered \nand disordered or pinned at the \ndomain\n-\nwalls. \nEffectively, t\nhe shell (outer) part of a particle spreads over few lattice parameters \nwhose length is more than domain\n-\nwall thickness and spins \ntherein \nare more disordered than the \ncore\n \nspins\n. \nThe magnetic exchange interactions inside the core are stronger than \nthe \nshel\nl\n \nand \nparticles are strongly interacted in \nC\nF90\n. In case of \nCF\nO\n_BTO\n \ncomposite, \nthe ferrite particle \nof \nsize\n \n\n \n90 nm\n \nare dispersed in matrix of BTO of \nmicron size\nd\n \nparticle\ns\n. \nThe presence non\n-5\n \n \nmagnetic (NM) \nBTO \nparticle dilutes the \nmagnetic exchange interact\nions \nbetween two CFO \n(FiM) \nparticles and it \nincrease\nd\n \nferrimagnetic softness in composite sample\n \n[2\n7\n]\n. \nThe interfacial \nexchange interactions are affected by \npossibl\ne\n \nmagneto\n-\nelectric coupling [21, 28\n] \nand\n \nhidden \nexchange coupling [\n29\n]\n \nbetween FiM\n \nand \nferro\nelectric (FE) \nBTO particles\n. \nAlthough\n, both CF90 \nand CF\nO\n_BTO are \nhetero\n-\nstructured spin systems\n, but the nature of interfacial \nspin disorder \nis \ndifferent. \nIn case of hetero\n-\nstructured spin systems, the time evolution of \nspin \nvector \ninside an \nordered magnet\nic domain \ncan be re\n-\nwritten as \n, where \n \n= \n \n[\n30\n]. \nA competition between the free spin torque under external field (first \nterm) and \nintrinsic \ndamping torque (second term) under internal field \nand meta\n-\nstable states \ndete\nrmines the relaxation/orientation of spin vectors towards its nearest \nnew \nmagnetic state. \nThe \ninternal field \nis controlled by spin disorder, frustration and inter\n-\nparticle interactions. In case of \nCF90 sample, the spin disorder is contributed by intrinsic \ndisorder at core (ordered FiM) and \nextrinsic disorder at shell (disordered FiM). \nI\nn the spinel \nferrite\n \nCo\n1.75\nFe\n1.25\nO\n4\n,\n \nintrinsic spin \ndisorder is expected due to \ndistribution of magnetic moment and magneto\n-\ncrystalline anisotropy \nof the cations among A and \nB sites of the spinel structure (\nFe\n3+\n \nions at A and B sites in \nhigh spin \nstate\n \nand low anisotropic, \nCo\n2+\n \nions at A and B sites in high spin state and highly anisotropic,\n \nand Co\n3+\n \nions \nat B sites\n \nare \nnon\n-\nmagnetic \nand isotropic)\n \n[\n25\n]. \nIn case of CF\nO\n_BTO comp\nosite, \nadditional \nextrinsic \nspin disorder \nis introduced at the interfaces of \nshell (disordered FiM\n \nof CFO\n) \nand shell (non\n-\nmagnetic and ferroelectric\n-\nBTO).\n \nThis\n \nis produced \ndue \nto \nstructural and magnetic \nmismatch at the interfaces of two \nphases [\n31\n].\n \nHence,\n \nthe change of both external magnetic field \n(ON\n/\nOFF) and internal field control the \ntime response of magnetization in the temperature and \nfield dependence of magnetization curves.\n \nThe basic difference is that \nthe \nmagnetization will be 6\n \n \nwell below of the sat\nuration level in case of the temperature dependence of the magnetization \ncurves, where as \nthe \nmagnetization will be close to the saturation \nlevel \n(high magnetic state) \nin \ncase of the field dependence of magnetization curves at the starting of relaxation\n \npr\nocess\n.\n \n2.2. Measurement protocols\n \nPhysical\n \nproperty measurement system (PPMS\n-\nEC2\n, Quantum\n \nDesign\n, USA\n)\n \nwas used \nfor magnetic measurements\n.\n \nThe \ntemperature dependence of \nmagneti\nzation \nwas \nrecorded using \nzero field cool\ned \n(ZFC) and field cool\ned \n(FC) mode\ns\n \nwi\nth \nconventional and \nunconventional\n \nprotocols\n \n(PCs)\n. \nThe \nPC1\n \nis a c\nonventional \nZFC mode\n \n(Fig. 1(\nc\n))\n, where t\nhe \nsample \nwa\ns \ncooled \nfrom \n33\n0\n \nK \nto \n10\n \nK in the absence of external \nmagnetic \nfield or \napplying \na \nsmall field to \nmaintain \nthe \nresidual magnetization \ncl\nose\n \nto zero\n \nduring cooling\n. \nThis \nwa\ns followed by \nmagnetic \nmeasurement at set \n(constant) \nmagnetic field while \ntemperature of \nthe sample \ni\ns warming up to \n300 K/3\n3\n0 K.\n \nThe \nPC2\n \nis the \nconventional \nFC mode\n \n(Fig. 1(\nd\n))\n, where t\nhe sample \nwa\ns \ncooled \nfrom 300\n \nK\n/3\n3\n0\n \nK\n \nto 10 K\n \nin the presence of constant magnetic field\n.\n \nThe \nmagnetization \nwa\ns \nrecorded during \nfield \ncooling \n(MFCC(T)) \nof the sample \nfrom higher temperature \nor \nwarming \n(MFCW(T)) of \nthe \nsample \nfrom 10 K to 300 K\n/3\n3\n0 K\n \nwithout changing the field that was \nappli\ned during pre\n-\ncooling down to 10 K\n. \nThe conventional (ZFC and FC) measurement \nprotocols \nprovide\n \ngeneral features (magnetic blocking and anisotropy effect) of the \nmagnetic \nparticles. \nThe \nnon\n-\nequilibrium spin dynamics \n(memory and aging effect) \nwe\nre examined \nby \nadopting \nfew \nunconventional \nprotocols \nto \nrecord\n \ntime\n \ndependent magnetization during \nintermediate stop on \ntemperature and field dependence of\n \nmagneti\nzation\n \n[\nM(T, H\n, t\nw\n)\n]\n \ncurves \n[\n2, \n5\n, \n14\n-\n15\n]\n.\n \nWe followed \nFC protocol (PC3) (Fig. 1(\ne\n)) for studying \nthe \nmem\nory effects. In FC\n-\nPC3\n, \nMFCC(T) \ncurve was recorded \nwith \nintermediate \nstop\ns\n \nat \n250 K, 150 K and 50 K\n \nby \nswitching off the cooling field for time t\nw\n. T\nhe M(t\nw\n) data \nat the stopping temperature \nwere\n 7\n \n \nrecorded before \nswitching \nthe cooling field again ON\n \nand \nres\numing the \nMFCC(T) \nmeasurement\n \non lowering the temperature\n \ndown to 10 K\n.\n \nAfter reaching the temperature 10 K, the MFCW(T) \ncurve was recorded from 10 K to 300 K without changing the cooling field and without \nintermediate stops. \nThe \nPC\n4\n \nis the\n \nconventional fi\neld dependence of magnetiz\nation (M(H)) \nmeasurement (Fig. 1(\nf\n))\n, \npre\n-\ncooled under ZFC and FC modes\n \nfrom 300 K\n \nto \nthe \nset \ntemperature\n. The shift of FC\n-\nM(H) loop with respect to ZFC\n-\nM(H) loop \ncan be used \nto \nstudy \nexchange bias effect. \nThe \nprotocol PC5 \nin \nFig\n.\n \n1(\ng\n)\n \nis the super\nposition of PC4 \nwith \nM(t\nw\n) \nmeasurement\n, where \nt\nhe\n \nM (T = constant, H, t\nw\n) curve\n \nwa\ns \nrecorded by varying the \nmagnetic \nfield \nwith \nan \nintermediate \nstop\n \nfor waiting time (t\nw\n) \nat \nmagnetic \nf\nield to\n \nzero or \nbefore \ncoercive \nfield point in\n \nnegativ\ne \nfield \naxis \nand M(t\nw\n) data \nwe\nre \nrecorded\n. \nAfter M(t\nw\n) measurement, the \nM(H) measurement is continued\n \nin negative field side\n. The steps of \nM(H) measurement\ns\n \nare\n \nrepeated \nwith different \nt\nw\n \nvalues.\n \nThe protocol PC6 in Fig. 1(h) is \nsimilar to \nthe protocol \nPC5\n. \nThe \nonly exception is that M(t\nw\n) \nmeasurements were\n \ncarried out at \nmultiple \npoints \n(at zero field \nor points close to \ncoercive fields \nboth \non negative \nand \npositive field axes\n) \nof \nthe \nM(H) curve\ns\n. \n \n3. Result\ns\n \nand discussion\n \n3.1. \nTemperature and field depend\nen\nt\n \nmagnetization [M(T,H\n, t\nw\n)]\n \nfor CF90 sample\n \n \nFirst, we\n \nshow basic properties \nof \nthe temperature dependence of magnetization \nin \nCF90 \nsample. \nT\nhe \nMZFC(T) and MFC\nW\n(T) curves\n \nat +500 Oe\n \n(\nFig. 2(a)\n) were\n \nmeasured\n \nusing \nPC1 \nand \nPC\n2\n. The MZFC\n(T)\n \ncurve exhibit\ns\n \nmagnetic blocking temperature (T\nm\n) at \n\n \n300 K. A wide \nbifurcation between MZFC\n(T)\n \nand \nMFCW(T) \ncurves \nbelow T\nm\n, where \nMZFC curve \ndecreased \nrapidly \nbelow \nT\nm\n \nand \nbecomes \nnearly temperature independen\nce\n \nbelow \n150 K\n, and \nMFC curve \nslowly increased\n. \nThe behavio\nr \ns\nhow\ns high anisotropic \neffect at low temperatures\n.\n \nTo overcome \nthe anisotropic effect, we measured\n \nMZFC(T) curve\ns by increasing the \nmagnetic fields \nup to \n\n \n2 8\n \n \nkOe\n \n(\nFig. 2(b)\n).\n \nThe \nMZFC(T) curves \nshow\ned\n \nmore or less symmetric \nrespons\ne \nof the spin\n-\nclusters \nunder field reversal\n. \nM\nagnitude \nof \nthe \nMZFC(T) curve\ns\n \nincrease\nd\n \nwith a shift \nof peak \nposition \nto low temperature \non \nincreasing \nthe \nmagnetic \nfield\n. \nIn highly anisotropic sample, the \nenergy density in ferrimagnetic state \nin \nthe presence of magnetic field is\n \nE = \nK\nH\n \n+\n \nK\nA\n, where \nK\nH\n \n(\n= \n\n \nM\nsat \nH cos(θ\n–\n \nφ)) is the \nZeeman energy \nand \nK\nA\n \n(\n= K\neff \n(T) sin\n2\nθ) is the crystalline \nanisotropy \nenergy\n \n[\n20\n]. \nThe\n \nMZFC(T) curves below 150 K are \nalso \nnot significantly affected \nwithin \n\n \n2 \nkOe\n, except some\n \nminor difference\ns\n. It indicates that \nZeeman energy \nis not \neno\nugh \nin this field \nrange \nto overcome the anisotropy\n \nenergy\n \nand\n \ndomain\n-\nwall pinning effect \ncontrols the shape of \nmagnetization curves \n[\n32\n]\n. \nOn the other hand, \nbroad\n \npeak \nin MZFC(T) \ncurve\ns\n \ndescribes \na \ndi\nstribution of anisotropy in the system and it \ncan\n \nbe \nquantified from \nfirst order derivative of \nthe \nMZFC(T) curves (Fig. 2(c))\n. \nThe \npeak profile in \nthe \ndM/dT vs. T \ncurves\n \n(\nFig. 2(d\n-\ne)\n)\n \nw\nas\n \nfit\nted\n \nwith Lorenzian \nshape\n \nto determine the peak parameters\n. \nThe intercept of the dM/dT curve on \ntemperature axis (> T\np\n)\n \ndefines the blocking temperature (T\nm\n).\n \nThe peak \ntemperature \n(T\np\n) of \ndM/dT vs. T curve corresponds to the inflection point of the MZFC(T) curve below T\nm\n. \nFig. 2(f) \nshows \na symmetrically decrease of \nt\nhe \npeak \nparameters (T\np\n, width, \nT\nm\n) \nabout the zero point\n \no\nf \nmagnetic field axis\n \nwith the increase of field magnitude\n. \nThe increase of peak height\n, along with \ndecrease of peak width,\n \narises due to field induced clustering of \nsmall\n \nparticles [1\n3\n].\n \nThe T\nm\n(H) \ncurve is fitted with \na power law:\n \nT\nm\n(H) = \na\n-\nb\nH\nn\n \n(\na\n \nand \nb\n \nc\nonstants) \nwith \nexponent\n \nn\n \n\n \n0.25 and \n\n \n0.2\n1\n \nfor \npositive and \nnegative fields\n, respectively\n. \nThe exponent values for T\np\n(H) curve are \n\n \n0.29 and \n\n \n0.27 for positive and negative fields, respectively. \nThe\n \nvalues of \nn\n \nin CF90 sample are \nsuggestive of magnetic \nspin\n-\nclusters coexisting in ferri\nmagnetic \nstate\n \n[\n33\n]. \nThe \nM(T) curves \nshow \nbulk \nresponse of a ferrimagnet\n \nwithout \nmuch \ninformation of \nlocal \nspin dynamics\n. \n 9\n \n \n \nIn order to get information of local spin dynamics, t\nhe memory effect \nwa\ns tested using\n \nprotocol \nPC\n3\n.\n \nFig. 3(a)\n \nshows the \ncorresponding \nMFCC(T) curve \nat \ncooling field +200 Oe \nwith \nintermediate stops and subsequent\n \nMFCW(T) curve.\n \nThe appearance of kinks in the MFCW(T) \ncurve\n \nat \nthe previously \nintermittent stop\ns \n(\nfield off condition \nat 250 K, 150 K and 50 K\n \nduring \nMFCC(T) process\n) suggests a \nrecover\ny/memory of \nthe magnetic \nspin states\n \nthat w\nere\n \nimprinted \nthrough redistribution of energy barriers during the cooling process.\n \nThe\n \nmemory \nis \nreduce\nd\n \non \nlower\ning\n \nthe stopping \ntemperature\ns\n \nand \nneg\nl\nigible \nat 10 K\n. \nTh\ne magnetization that is recovered \non re\n-\napplying the cooling \nfield depends on \nthe response of spins in relaxed/quasi\n-\nrelaxed state\n. \nIn a \nstrongly interacted \nspin\n-\nsystem\n, \nan increasing slow down of the spin \ndynamics \non \ndecreasing the sample temperature belo\nw its spin \nfreezing\n/blocking \ntemperature\n \nhinder\n \nthe \nrecovery \nof initial magnetic state. It \nlead\ns\n \nto a large step in \nMFCC(T) curve immediately after \nswitching OFF and re\n-\napplying (ON) of the cooling field at the temperatures (e.g., 250 K in our \ncase). The s\ntep in MFCC(T) decreases as sample temperature decreases far away from its spin \nfreezing temperature. It is due to increasing inter\n-\nspins interactions in a strong spin\n-\npinning state \n(e.g., 50 K)\n. \nInterestingly, \nthe \nMFCW(T) curve overshoots the MFCC(T) curv\ne \nat \ntemperatures \nabove 300 K. It \nshow\ns \nin\n-\nfield growth of magnetization due to non\n-\nequilibrium spin state of the \nmagnetic particles below the\nir\n \ntrue blocking temperature\n \n(\nabove 300 K\n)\n. \nThe \nmeasurement\n \nof \nMFCC(T) and MFCW(T) \nat 200 Oe \nwithout \nfield\n-\noff\n \nat \nintermediate temperatures \nformed a \nthermal hysteresis loop\n \n(Fig. 3(b))\n. \nI\nt \nis a characteristic feature of first order magnetic \nphase \ntransition (short range spin order coexists in long range spin order) \nin the sample\n \n[\n34\n]\n.\n \nIn our \nsample, t\nhe \nin\n-\nfield \nMFCW(\nT) \ncurve \nstarts with \nthermal activated de\n-\npinning of the spins \nthat \nwere\n \nin \npinn\ning \nin intrinsically disordered ferrimagnetic state \nat 10 K after completing the \nMFCC(T) \nmeasurement\n. \nHowever, \na\n \ndifference between MFCW(T) and MFCC(T) curves (right 10\n \n \nY axis of \nFig. 3(b)) showed \na \nmaximum at about 210 K\n \nand it marked different spin dynamics at \nlower and higher temperatures\n. \nM\nagnitude of the difference \ndecreases \nat higher temperatures \ndue \nto approaching of spin system\n \ntowards blocking temperature (less \ninteracting\n/\npinning effect) and \nat low\n \ntemperatures\n \ndue to \napproaching towards \na \nstrongly \nspin\n-\npinning\n/interacting\n \nregime. \nOn \nincreasing the magnitude of cooling field to 500 Oe \n(Fig. 3(c))\n, the memory effect is observed \nonly at 250 K and \nsuppressed \nat low temperatur\nes (50 K and 150 K)\n. \nT\nh\nis \nis \ndue to \nclustering of \nsmaller \nmagnetic \nparticles\n \nand de\n-\npinning of\n \nthe spins \n(domain wall motion)\n \nat higher magnetic \nfield\n. In this process\n, the distribution of\n \nexchange interactions and anisotropy barriers related to \ncluster si\nze\n \ndistribution\n \nis \nnarrow\ned down\n. I\nt results in strongly spin\n-\ninteracting clusters\n \nduring \nfield cooling process and reduces the memory effect at 500 Oe\n.\n \nOn the other hand, \nspin system\n \nswitch\nes\n \nits magnetic state \nfrom high to \nlow \ni\nmmediately after switching\n \noff the \ncooling \nfield\n. \nThe \nspins in \nlow \nmagnetic state \n(non\n-\nzero remanent magnetization) \nrelax \nfor sample temperature \nin blocking state (T < T\nB\n)\n \n[\n5\n-\n6, 15\n]. \nThe \ntime \ndependence of \nFC\n-\nremanent magnetization (\nM\n \n(t, \nH= 0)\n)\n \ncurves\n \nhave been \nanalyzed \nby \nvarious\n \nequations, e.g., \nstretched exponential\n \nform [\n7\n], \na \ncomplicated form of equation that consists of essentially two power law terms [\n3\n]\n.\n \nIn our sample, \n \nM(t)\n \n(\nnormalized \nby initial value \nM(t\n0\n))\n \ncurves \nduring \nfield\n-\noff \ncondition\n \n(t = \nt\nw\n \n=\n \n1500 s\n)\n \n(\nFig. \n3(d\n-\ne\n)\n)\n \nare best fitted \nwith a function\n, consisting \nof \na constant and \ntwo exponential decay terms.\n \n \nM(t) = \n\n0\n \n\n \n\n1\nexp(\n-\nt/\n\n1\n) \n\n \n\n2\nexp(\n-\nt/\n\n2\n) \n \n \n(1)\n \nS\nign of \nthe \npre\n-\nfactors \n\n1 \nand \n\n2\n \nis \ntaken as\n \npositive and negative \nto\n \nrepresent \nthe magnetization \ndecay and growth, respectively. \nOut of the two exponential terms, one represents fast relaxation \n(initial \nprocess\n) and other one represents a slow relaxation \n(secondary process \nat higher time\ns)\n. \nSimilar \nspin relaxation \nproces\ns\ne\ns \nwe\nre found in \nmagnetic systems with \nheterogene\nous spin \nstructure \n[\n35\n]\n.\n \nThe \nfit of \nM(t) data at 50 K with \na \nlogarithmic decay M(t) = \n\n0\n \n–\nm\n*lnt\n \n(with \nm\n \n= \n 11\n \n \n0.0002 and 0.0160 at cooling fields \n200 Oe and 500 Oe, respectively) \nrepresents \na\nn extremely\n \nslow \nspi\nn systems \nand generally \nrepresent\ns\n \na distribution \nof \nactivation energy\n \nin spin glass state\n \n[\n1\n,\n3\n, \n36\n]\n. \nA comparative fit of the M(t) data \nduring \nOFF condition of \n500 Oe \nat 250 K \n(Fig. 3(f)) \nsuggest\ns\n \nthat logarithmic decay \nis s\natisfied \nfor \nlimited portion of\n \nthe \nM(t) curves\n, \nbut \n \nequation \n(1) \nwidely \nmatched \nto the\n \nM(t) curves. Hence, \nequation (1) \nis more acceptable in fitting the \nM(t) \ncurves \nduring \nfield\n-\noff condition of \nM(T) and M(H) measurements\n. \n \nThe \nnon\n-\nequilibrium spin dynamics \nduring \nZFC\n-\nM(H) loop \nmeasu\nrement \nwithin field \n\n \n70 kOe at 10 K (Fig. 4(a))\n \ncan be studied using PC\n4\n \nand \nPC\n5\n. \nThe \nM(H) loop\n \nunder zero field \ncooled mode \nwas recorded \nat 10 K \nusing PC\n4\n. Next, \nM(H) \nmeasurement \nbetween \n+70 kOe \nto \n-\n10 \nkOe \nwas repeated 6 times \nwith intermediate \nwait\n \nat 0\n \nOe\n \nto record \nthe \nM(t\nw\n) curve\ns\n \nfor \ndifferent \nt\nw\n. \nIn principle, spins in ferrimagnetic state is expected to relax \nduring waiting, irrespective of \nsweeping field ON or OFF conditions\n, if finite amount of disorder coexists in spin order\n, and it \ncould produce \nnew meta\n-\nstable state in the M(H) path\n. \nAs shown in\n \nFig. 4(b)\n, the \nM(H) curve\ns\n \nbetween 0 Oe and \n-\n10 kOe\n \nare\n \nextremely sensitive to spin relaxation\n \nduring \nt\nw\n \nat 0 Oe\n. T\nhe M(H) \ncurve\ns\n \nafter waiting at 0 Oe \nmove upward with the increase of \nt\nw\n \nwith reference t\no \nthe \nfirst \ncurve\n \n(default \nt\nw\n \n= 10 s)\n.\n \nThe \nM(\nt\nw\n) \ncurve\ns\n \nat 0 Oe (\nFig. 4(c)\n)\n \nslow\ned\n \ndown\n \nfor \nhigher \nt\nw\n \nand \nfollowed \n \nequation (1)\n. Fig. 4(d\n-\nf) shows \nthe \nwaiting time dependence \nof \nthe fit \nparameters (\nH\nC\n, \nM\n0\n,\n \n\n1\n, \n\n1\n,\n \n\n2\n, \n\n2\n)\n \nfrom M(H) curves (0 Oe to \n–\n \n10 kO\ne) and M(\nt\nw\n) curves at 0 Oe\n. \nC\noercivity \n(H\nC\n) \nof the \nCF90 \nsample\n \nsignificantly \nincreased \n(\n6628\n \nOe to \n6954\n \nOe) \nwith the increase of \nt\nw\n \nfrom \n100 s to \n7200 s \nat 0 Oe\n, unlike \na \ndecrease of the \nfit parameter M\n0\n \n(\n35.3473\n \nemu/g to \n35.304\n \nemu/g)\n. This\n \nis associated\n \nwith faster relaxation of initial process (increasing \n\n1\n \nand smaller \n\n1\n) and slower \nrelaxation of secondary process (\ndecreasing \n\n2\n \nand larger \n\n2\n)\n \nwith the \nincreas\ne of \nt\nw\n. \nA wide \ndifference\n \nbetween \n\n1\n \nand \n\n2 \nconfirms\n \nthe existence of \ntwo relaxation mechani\nsms\n \nin the sample\n. \n 12\n \n \nIn \norder to \nstudy the non\n-\nequilibrium spin dynamics in FC\n-\nM(H) \nloops\n, we have \nrecorded \n \nM(H) loop\ns\n \nat 10 K\n \nusing \nFC\n-\nPC\n4\n \nat cooling field\ns\n \n+70 kOe and \n-\n70 kOe\n. \nT\nhe \nM(H) curve\n \nstart\ned \nfrom \nfield \nsweeping\n \n+70 kOe to \n-\n70 kOe and back to +70\n \nkOe\n \nfor the \nFC loop (\ncooling \n@ \n+70 kOe)\n \nand in reverse way for the \nFC loop (\ncooling \n@ \n-\n70 kOe)\n. \nAs shown in \nFig. 5(a)\n, t\nhe \nFC \nloops\n \nexhibit widening and shifting along field and magnetization directions \nwith \nimproved \nsquare\nness\n \nin comparison to ZFC loop\n \na\nt 10 K\n.\n \nIt occurs due to exchange coupling of hetero\n-\nstructured spins at the interfaces \nor frozen in the system \nthat favor ordering along cooling field \ndirection and irreversible under reversal of the field \ndirection\n \n[Khur].\n \nT\nhe centers (H\nC0\n, M\nR0\n) and \ncoer\ncivity (H\nC\n) of the FC and ZFC loops \nhave been used \nto calculate the shift of coercivity (ΔH\nC \n= |H\nC\nFC\n \n-\n \nH\nC\nZFC\n|), exchange bias field (H\nEB \n= H\nC0\nFC\n \n–\n \nH\nC0\nZFC\n) and magnetization (ΔM\nR \n= M\nR\nFC\n \n-\n \nM\nR\nZFC\n)\n.\n \nT\nhe FC loop \n(@\n \n+70 kOe\n)\n \nis \nnearly symmetric with \nminor \nexchan\nge bias \nshift (\nH\nEB\n \n\n \n+ \n8 Oe)\n \nand\n \nits\n \nH\nC \n\n \n8295 Oe\n. \nHowever, \na \nlarge \npositive shift of \nmagnetization (ΔM\nR\n \n\n+ 1.55 \nemu/g) and coercivity (ΔH\nC\n \n\n \n+ 1\n5\n95\n \nOe)\n \nare noted \nwith respect to ZFC loop with H\nC \n\n \n6\n760\n \nOe\n. \nAs compared in the inset of Fig. 5(a), \nFC loop (@\n \n+\n70 kOe)\n \nand \nFC loop (@ \n-\n70 kOe)\n \nshowed similar features\n, \nbut \nFC loop (@ \n-\n70 kOe)\n \nshows \nhigher widening \n(H\nC \n\n \n8495 Oe, ΔH\nC\n \n\n \n+ \n1\n735\n \nOe, ΔM\nR\n \n\n \n+ 1.99 emu/g) \nand squareness\n.\n \nThis means spin dynamics is \nanisotropic to the\n \nreversal of high field cooling\n \nand \ni\nt could be \nrelated to spin pinning in ferrimagnetic domains \n[\n1\n1\n]\n. \nIn \norder to \nstudy\n \nthe \naging effect \nat intermediate point of the FC\n-\nM(H) \ncurves\n, \nwe repeated \nM(H) \nmeasurement \nwithin field range \n+ 70 kOe to \n-\n10 kOe \nfor\n \n6 times \nwith wait\ning\n \nat \n-\n2.5 kOe\n \nby ad\nopting PC5\n. Fig. 5(b) demonstrates that \nthe\n \nshap\ne (\nupward \nincrease\n)\n \nof \nthe \nM(H) curve \nin \nthe field range \n-\n2.5 kOe to \n-\n10 kOe\n \nis controlled by \nspin\n \nrelaxation at \n-\n2.5 kOe during \nt\nw\n \n(\n140 s \nto 7200 s\n)\n. \nAs shown in \nFig. 5(c)\n, the \nM(t\nw\n) curve\ns\n \nat \n-\n2.5 kOe \nalso \nfollowed \nequation (1)\n. \nThe \nincrease of \nt\nw\n \nat \n-\n2.5 kOe\n \nof the FC\n-\nM(H) loop (@+70 kOe) has \nincrease\nd the\n \nH\nC\n \n(Fig. 5(d)\n-\nleft 13\n \n \nY axis), the \noverall \nmagnetization after \n-\n2.5 kOe \nand \nfit parameter M\n0\n \n(Fig. 5(d)\n-\nright Y axis). \nThe\n \nfit parameters (Fig. 5(e\n-\nf))\n \nfor \nfast relaxation and slow relaxation\n \nprocesses showed similar \nfeatures as observed \nwith t\nw\n \nin case of ZFC\n-\nM(H) loop experiment \n(Fig. \n4\n(e\n-\nf\n)). \n \nNext, w\ne tested \nthe \nspin relaxation \non the M(H) curves \nat 150 K, \na temperature \njust above \nthe magnetization blocki\nng temperature \n\n \n125 K in MZFC(T) curve \nfor field \n50 kOe (Fig. 6(a)). \nAt this \ntemperature\n, \ndomain wall pinning is less effective\n \nbut magnetic clusters are not free from \nmutual interactions\n. T\nhe\n \nZFC\n-\nM(H) curve\n \nwas measured \nby sweeping\n \nfield \nfrom +70 kOe to \n-\n6\n \nkOe \nand \nintermediate waiting at \n-\n1 kOe\n. \nAfter measurement of \nthe \nfirst M(H) curve, the field \nwas made to zero and back to +70 kOe before starting the next curve and repeated \nit \n7 times. \nFig. \n6(b) \nshows\n \nall \nthe \nrelaxation regime of M(H) curves at \n-\n1 kOe \nduring waiting time \nand \nsubsequent \nfield dependent regime \n(\n-\n \n1 \nkOe to \n-\n5 kOe\n)\n. \nIt is noted (\nFig. 6(c)\n)\n \nth\nat\n \nmagnitude of \nthe \nM(H) curves \nfor H < \n-\n1 kOe \nis\n \nsystematically \nsuppressed\n \non increasing the \nwaiting at \n-\n1 kOe\n. \nThis trend is in contrast to the i\nncre\nasing \nincrement \nfor similar experiment \nat 10 K (Fig. 4(b)). \nThe M(t\nw\n) \ncurves\n \nin the relaxation regime \n(inset of Fig. 6(c))\n \nat 150 K also \nfollow equation (1)\n \nand \nfit parameters are shown in Fig. 6(d\n-\nf). \nThe \nH\nC\n \nhas shown \na small increment, whereas \nM\n0\n \ndecreas\ne\ns\n \nwith the increase of \nt\nw\n. \nThe\n \nvalues of \nthe \npre\n-\nfactors (\n\n1\n, \n\n2\n) and time constants (\n\n1\n, \n\n2\n) \nat 150 K are relatively larger than the values at 10 K. It indicates \na fast\ner\n \ndecay of magnetization \nat 150 K\n, \nwhere \nspin dynamics\n \nis still slow \ndue \nto strong in\ntra\n-\ncluster spin interactions\n. \n \n \n3.2 Temperature and field dependen\nt\n \nmagnetization [M\n \n(T,\n \nH\n, t\nw\n)] for CF80_BTO sample\n \n \nFig. 7(a)\n \nshows the \nfeatures of \nthe \nMZFC(T) \nand \nM\nFC\n(T)\n \ncurves at \n500 Oe\n \nin composite \nsample\n. \nIt is seen that \nbasic magnetic\n \nfeatures of \nt\nhe \nferrite particles,\n \ne.g., \nblocking temperature \n(T\nm\n) at about 300 K\n, \nwide \nmagnetic \nbifurcation at low temperatures\n, and \na weak temperature \ndependent MZFC\n(T) curve\n \nbelow 150 K, are retained \nin \nthe \nBTO matrix\n \n[2\n7\n]\n. \nMZFC\n(T)\n \ncurves 14\n \n \n(Fig. 7(b))\n \nalso \nshow\ned fie\nld induced magnetic changes, including \nincreasing \nmagnetization\n \nand \nshift of the \nbroad maximum \nto lower\n \ntemperature\ns\n.\n \nF\nirst order derivative of the MZFC\n(T)\n \ncurves \n(\n\nMZFC/\n\nT) at different magnetic fields\n \n(Fig. 7(c)) \nshowed an asymmetric shape \nabout \nthe peak\n \ntemperature (T\np\n)\n, which is the inflection point below the broad maximum of MZFC(T) curves\n. \nT\nhe \npeak profile\n \nof \n\nMZFC/\n\nT curves\n \nwere \nfitt\ned \nwith \nLorentzian\n \ncurve\n \nand the peak parameters \nare shown in Fig. 7(d)\n. \nThe peak temperature (T\np\n) decreases at higher \nfield by following a power \nlaw: T\np\n(H) = \na\n-\nb\nH\nn\n \nwith exponent (\nn\n) \n\n \n0.31\n, which is close to that obtained for CF90 sample\n. \nIt \n \nsuggests the retaining of the \nglass\ny\n \nbehavior \nof spin\n-\nclusters \nin composite system\n \n[\n33\n]\n. \nIt \nis\n \nnoted \nthat \npeak height of the \n\nMZFC/\n\nT curves initially increased for field up to 2.5 kOe, followed a \ngradual decrease at higher fields. This corresponds to \na \nminimum peak width at 2.5 kOe\n, along \nwith an increase of peak width both at low\ner\n \nand higher magnetic fields. \nThe features are \nconsis\ntent to \nfield induced \nnucleation\n \nof \nsmall particles \nby \nde\n-\npinning the spins \nat domain wall\n \nor \nat the interfaces of \nferrimagnetic and ferroelectric particles (via grain boundary)\n \nat low field \nregime\n. The \nincreas\ne of b\nroadness \nin the \nfirst order derivative c\nurves\n \nfor fields higher than 2\n.5 \nkOe \nis \nattributed \nto \nan \nincrease of \nintrinsic \ndisorder, arising from\n \na \ncompetition \nof \nanisotropy\n \nconstants\n \nand exchange interactions \ninside \nthe clusters\n, where as the\n \nreduced peak height\n \nis \nattributed to \nquasi\n-\nsaturation st\nates\n \nof magnetization curves at higher fields\n. \n \nThe \nretaining of \nmemory effect \nof the ferrite particles \nin composite sample \nis confirmed \nfrom \nMFCC(T) \nand \nMFCW(T) \ncurves\n \n(Fig. 8(a\n-\nd\n)), measured\n \nat \ncooling field\n \nrange \n200 Oe\n \nto \n10 kOe\n.\n \nThe MFCW(T) curves sho\nw\ned\n \nkinks\n \nat the temperatures \n(250 K, 150 K, 50 K) \nwhere \nfield was switched off during FCC mode\n. T\nhe \nkinks\n \nare more pronounced than the CF90 sample\n. \nThe \nmemory effect \nin \nCF80_BTO sample \nalso \nreduc\ned\n \nat low \ntemperature\ns\n \nand \nat higher \nmagnetic fields\n, simila\nr to the features in CF90 sample\n.\n \nIn Fig. 8 (e\n-\nf), we have compared the \n% 15\n \n \ndrop and relax\nation of remanent \nmagnetization \nfor \nboth \nthe samples \nduring field off condition \nof \nthe \nMF\nC\nC(T)\n \ncurves\n.\n \nT\nhe \n% \nof\n \ndrop represents fraction of \nthe \nreversible spins in the \nsystem\n \nimmediately after switching off the cooling field\n. It is 100 % for non\n-\ninteracting paramagnetic \nspins and less than 100 % for \nexistence of \nfinite interactions among \nthe \nspins or cluster of spins. \nIn case of interacting spins, the relaxation componen\nt is non\n-\nzero\n \nand i\nt represents the fraction of \nirreversible spins in the system that shows aging \neffect\n. \nT\nhe \n% \ndrop \nis \nseen to be \nhigh\ner\n \nthan the \nrelaxation \npart\n \nduring waiting time\n, as schematized in the inset of Fig. 8(e)\n. \nIt is seen that \n% of \ndrop and \nrelaxation both are drastically reduce on lowering the measurement temperatures from \n250 K to 50 K. \nHowever, \ndistinct differences \ncan be noted \nin the \noff\n-\nfield properties between \nCF90 and CF80_BTO \ncomposite sample\ns\n.\n \nFor example, t\nhe \n% of \ndrop\n \ndecreased on \nincreasing \nthe \ncooling field \nfrom 200 Oe to \n500 Oe in case of CF90 sample, whereas a monotonic increase \nnoted \nwith the increase of \ncooling field\ns\n \nfrom \n200 Oe \nto 10 kOe\n \nin \ncomposite sample\n. The higher \n% drop of \nmagneti\nzation\n \nindicates\n \nweak\nening of\n \nmagnetic \nspin \ninteraction\ns\n \nat the interfaces of \nCFO (ferrimagnetic) and BTO (non\n-\nmagnetic) particles\n. At the same time, \nincreas\ning \ndrop \nat \nhigher cooling field\ns\n \nis \nattributed to fast de\n-\nnucleation of \nlarge\nr\n \nclusters\n \ninto small\ner\n \nclusters \n(\ncomposed \nof magnetic ferri\nte and non\n-\nmagnetic BTO particles)\n \non switching off the cooling \nfield.\n \nThe \nde\n-\nnucleation of the large clusters is slow \nin CF90 sample \ndue to strong \ninter\n-\nparticle \ninteractions\n.\n \nIt gives rise to \nrelatively \nlow values of % drop and relaxation at all the meas\nurement \ntemperatures for CF90 sample. \nB\nased on the data for cooling field 500 Oe\n,\n \nit can be mentioned \nthat composite sample is consisting of approximately 2\n9\n% \nparamagnetic/non\n–\ninteracting spins \nand 3% of \nthe \ninteracting spins relaxed during waiting time 16\n00 s \nat \n250 K\n. I\nt \nis \nreduced to 1 % \n(paramagnetic spins) and 0.1\n4\n \n% (relaxation of interacting spins) for temperature 50 K. In case \nof CF90 sample, the paramagnetic spins (22%) \nand relaxation of \nthe \ninteracting spins\n \n(0.44 %) at 16\n \n \n250 K are reduced to 0.52 %\n \n(paramagnetic spins) and 0.03 % (relaxation of interacting spins) at \n50 K.\n \nThe\n \nM(t\nw\n) \ncurves\n \nduring \ncooling field off \ncondition of \nMFCC(T) measurement\ns\n \nwere fitted\n \nusing \nequation 1\n \n(Fig. \n9\n \n(a\n-\nd)) \nand \nthe \nfit parameters \nare shown in \nFig. \n9\n(e\n-\ni)\n.\n \nV\nariation o\nf \nthe \nparameter \nM(0) \nis \nconsistent to\n \nthe \ntemperature and field \ndependence of magnetization \ncurves\n. \nThe fit parameters \nassociated with relaxation processes \n(\n\n1\n, \n\n2\n, \n\n1 \nand \n\n2\n) \nare \nless sensitive to \nhigher magnetic fields (5 kOe and 10 kOe)\n \nand suggest quas\ni\n-\nequilibrium state due to nucleation \nof clusters\n.\n \nHowever, t\nhe \ntime constants (\n\n1\n, \n\n2\n) \nare relatively high at 50 K and further increased \nfor higher cooling fields. \nIt indicates slowing down of the spin dynamics at low temperature due \nto increasing interac\ntions among the spins/cluster of spins and intra\n-\ncluster interactions \n(domain \nwall motion) \ndominate at higher cooling fields. Most importantly, magnitude of the \ntime \nconstants in composite sample is nearly one order less than the values in CF90 sample. Thi\ns is an \nevidence of faster relaxation in composite sample due to less inter\n-\ncluster exchange interactions.\n \nThe \nvariation of \ncoercivity \nin \nthe \ncomposite s\nample\n \nas the function of\n \nin\n-\nfield \nwaiting \ntime \non \nFC\n-\nM(H) \ncurve\n \nhas been examined \nby \nusing PC5 \nat \n10 K \n(\nFig. 1\n0\n \n(\na\n)\n) and 150 K (\nFig. \n1\n0\n(b)\n)\n.\n \nT\nhe sample was first \nzero field cooled \nfrom 300 K to 10 K/150 K\n \nand \nM(H) curve \n(N = \n1) \nwas measured \nduring sweeping of \nthe \n \nfield from \n+\n70 kOe to \n-\n20 kOe\n \n(10 K) or complete loop \nwas measured at 150 K\n. \nAfter first round\n \nmeasurement,\n \nthe \napplied field was made \nto \nzero\n \nbefore \nincreasing the temperature to 300 K. \nNext\n, \nthe \nsample was \ncooled \nunder \n+\n70 kOe down to 10 K/\n \n150 K. After temperature\n \nstabilization\n, \nthe \nM(H) curve (N = 2) \nwas recorded \nfrom \n+\n70 kOe to \n-\n20 kOe with \nan\n \nintermediate \nwaiting\n \nat \n-\n \n7 kOe for 10 K and at \n-\n2150 Oe for 150 K\n.\n \nT\nhe M(t\nw\n) \ncurve \nwas recorded \nfor different in\n-\nfield \nwaiting time\ns by repeating the \nFC\n-\nM(H) curve \nin the \nfield range +70 kOe to \n-\n20 kOe\n. The (\nnormalized\n) in\n-\nfield\n \nM(t\nw\n) curves are shown fo\nr 10 K\n \n(Fig. \n1\n0\n(c))\n \nand for\n \n150 K\n \n(\nFig. 1\n0\n(d)\n)\n. \nT\nhe inset of Fig. 1\n0\n(a)\n \nshows that \nH\nC\n \nat 10 K is \nenhanced \nin 17\n \n \nFC\n-\nM(H)\n \ncurve\n \nand the \nH\nC\n \ni\ns further enhanced by repeating the\n \nFC loop \nwith higher \nwaiting \ntime at \n-\n \n7 kOe. \nThe t\nw\n \nat 10 K was made high enough to o\nbserve an appreciable relaxation. The \nin\n-\nfield M(t\nw\n) curves at 10 K \nwere \nfitted with logarithmic decay law M(t\nw\n) = \n\n0\n \n–\nm\n*lnt\nw\n. The \ninset of Fig. 1\n0\n(c) shows the decrease of both \n\n0\n \nand slope (\nm\n) with the increase of waiting time \nat \n-\n \n7 kOe of the FC\n-\nM(H) c\nurves. In contrast, in\n-\nfield M(t\nw\n) curves at 150 K \nwere \nfitted with \nexponential law (1). The fitted parameters are not shown in the \ngraph\n.\n \nOn th\ne\n \nother hand, \nin\n-\nfield \n(\n-\n21\n50 \nOe) magnetic relaxation \nof the FC curve \nat 150 K is fast\ner \n(over coming spin pinni\nng) \nthan the \nslow relaxation\n \n(\nstrong domain wall pinning\n) \nat 10 K\n. \nT\nhe \nin\n-\nfield \nmagnetization \nat 150 \nK \nswitche\nd\n \nfrom positive to negative \nfor\n \nt\nw\n \n> \n180 s\n \nand\n \nwait\n-\nin field \nH\nC\n \nvalue (\n-\n2150 Oe\n)\n \nbecomes \nsmaller than the \nH\nC\n \nof ZFC curve (\n-\n2533 Oe)\n. \nThis propert\ny can be used in magnetic \nswitching/sensor applications. \nThe \ndecrease of \nH\nC\n \nat 150 K with the increase of waiting time at \n-\n21\n50 \nOe\n \nis \ncharacteristically \nopposite with respect to the increment \nof \nH\nC\n \nat 10 K\n. \n \nWe \nused PC\n6\n \nto study aging effect on the ZFC\n-\nM(H\n) loop of the composite sample at 10 \nK with the field sequence +70 kOe to 0 Oe and M(t\nw\n) measurement \nfor \nt\nw\n \n= 3600 s was carried \nout \nat 0 Oe (P1). This is followed by resumption of \nthe \nM(H) measurement down to \n-\n7 kOe (P2), \nwhere the sample was waited to re\ncord the second M(t\nw\n) curve. Then, recording of M(H) curve \nwas continued down to field at \n-\n70 kOe\n \nand \nreversed back to +7 kOe (P3) where third M(t\nw\n) data \nwere recorded. Finally, M(H) curve \ncontinued for \nfield\ns\n \nup to +20 kOe. A similar M(H) \nmeasurement prot\nocols were used to record the M(t\nw\n) curves at 10 K after cooling the sample in \nthe presence of +70 kOe from 300 K. The same experiments were carried out at \nrelatively higher \ntemperature \n100 K\n \nat field points 0 Oe and \n\n \n4 kOe\n. Fig. 1\n1\n \n(a\n-\nb) shows the \nrecord\ned \nZFC and \nFC\n-\nM(H) curves\n \nat 10 K and 100 K\n. \nThe\n \nM(t\nw\n) \ncurves\n \nat poin\nts\n \nP2 and P3 started to relax \ntowards the magnetization state \nat zero field\n. The behavior is consistent to the \nreverse torque 18\n \n \nacting on the \nspins when the field is reduced to zero and rot\nate over a time toward the positive or \nnegative direction depending on the local easy\n-\naxis orientation\n \n[\n20, 37\n]\n. The FC loop at 10 K \nshows nearly symmetric widening along the field and magnetization axes. \nThis showed field \ncooled induced enhancement of coe\nrcivity and magnetization in the composite sample\n, which \nshowed small exchange bias shift \n\n \n+ 64 Oe at 10 K [2\n7\n]\n. \nThe \nfield cooled induced \nwidening \nand \nexchange bias shift are\n \nnegligible \nat 100 K. \nOn the other hand, \nM(t\nw\n) curves\n \nat 10 K and 100 K \n(\nFig. 1\n1\n(\nc\n-\nd)\n) followed equation (1) with \nM(0) values positive and negative for points P2 and P3, \nrespectively. The M(t\nw\n) curves\n,\n \nmeasured on \nZFC and FC\n-\nM(H) curves, do \nno\nt show much\n \ndifferences \nat \n10 K and 100 K for \npoint P1 (\nwait\ning\n \nat 0 Oe\n)\n. The normalized M(t\nw\n)\n \ncurves \nmeasured on \nFC\n-\nM(H) curves\n \nsignificantly enhanced for field in\n-\nwait \n\n \n7 kOe at 10 K and \n\n \n4 \nkOe at 100 K in comparison to the \nmeasurement on \nZFC\n-\nM(H) curves\n. The differences between \nM(t\nw\n) curves measured \non \nFC\n-\nM(H) curves \nsubstantially decreased at\n \n100 K\n, unlike the case on \nZFC\n-\nM(H) curves at 10 K\n. The M(t\nw\n) curves in positive field side (+7 kOe at 10 K and + 4 kOe \nat 100 K) are found to be higher than their counter parts in the negative field side. This indicates \nthe effect of cooling field induced\n \nunidirectional anisotropy \non interfacial spin ordering [\n37\n] and \nit is reflected in the variation of time constants. \nFig. 1\n1\n(e\n-\nf) shows the time constants (\n\n1\n, \n\n2\n) at \ndifferent fields in\n-\nwait in FC process are larger than that in ZFC process both at 10 K a\nnd 100 K. \nThis \nshows a \nslow\ning\n \ndown of \nspin \nrelaxation \ndue to cooling field induced nucleation of clusters. \n \n \nFinally, \nwe repeated the \nmeasurement of M(H) curves \nfrom +70 kOe to \n-\n15 kOe at 10 K \n(Fig. 1\n2\n(a)) \nand \n+\n70 kOe to \n-\n1 kOe at 300 K\n \n(Fig. 1\n2\n(b))\n \nto ex\namine \nthe \ntraining effect in \nthe \ncomposite sample\n.\n \nT\nhe sample was zero field cooled from \n300 K (\n330 K\n)\n \nbefore repeating the \nM(H) \nmeasurement\ns\n \nat 10 K (300 K)\n \nfor \ndifferent \nfield \nsweeping rate without further heating\n \nthe \nsample \nto higher temperature\n. \nThe M(\nH) curves at 10 K are practically independent of the field 19\n \n \nsweeping rate, wh\nereas \nM(\nH) curves at 300 K\n \nshowed \nmagnetic \nrelaxation\n \nat higher fields\n. \nM\nost \nof the systems with training effect\n \nhave shown decrease of\n \ncoercivity [1\n1\n, 3\n8\n]\n,\n \nbut coercivity\n \nof \nthe c\nomposite sample is independent of sweeping rate\n \nin the temperature range 10 K to 300 K\n. \n \n4. \nC\nonclusions\n \nThe \nferrimagnetic\n \nCo\n1.75\nFe\n1.25\nO\n4\n \nferrite \nand its composite with non\n-\nmagnetic BaTiO\n3\n \n(BTO) \nparticles \nare modeled in core\n-\nshell spin structure. \nThe \nexiste\nnce of \nintrinsic \nspin \ndisorder \ninside the \nferri\nmagnetic \nferrite particles\n \nis confirmed by \nexchange bias, \nmemory and relaxation \neffects. The magnetic memory effect \nin ferrite particles dominated \nat higher temperatures\n, unlike \nobservation of \nexchange bias ef\nfect at low temperature. \nThe magnetic exchange interactions \nbetween ferrimagnetic particles are diluted and modified due to \npresence \nof intermediating \nnon\n-\nmagnetic BTO particles. \nHowever, \nbasic magnetic \nproperties \n(\nblocking of ferrimagnetic \nclusters\n, \nnon\n-\ne\nquilibrium\n \nferrimagnetic \nstate, memory, exchange bias and aging) \nof the ferrite particles \nare retained in \nthe BTO matrix of composite sample\n.\n \nThe \nslow \nspin dynamics \nlow temperature \ndue to strong spin pinning/inter\n-\ncluster interactions becomes faster on inc\nreasing the \ntemperature \nclose to the \nspin freezing\n/blocking \ntemperature\n \nof the samples at \n\n \n300 K (due to increasing \nfraction of non\n–\ninteracting/paramagnetic spins). The fraction of paramagnetic spins in composite \nsample is found \nto be \nmore\n \n(\nshowing\n \npronou\nnced \nmemory effect\n \nand faster spin relaxation\n) \nthan \nthat in \nthe \nferrite sample that exhibited relatively \nsmall \nmemory dip and slow relaxation. \nThe \nrelaxation of magnetization during field off condition\ns\n \nof the temperature and field dependence \nof magnetizat\nion \ncurves \nconfirm\ned \ntwo relaxation mechanisms\n \nin both the samples\n.\n \nThe fast \nrelaxation process (initial stage of the relaxation) \nis attributed to \nloosely bound shell/interfacial \nspins, whereas the slow relaxation process (later stage of the relaxation) \nis\n \nattributed to \nstrongly \ninteracting core/interior spins in the systems. \nThe \nM(H) curves are not much affected by the 20\n \n \nvariation of field sweeping rate\n, but magnetic state and coercivity \nof the samples \nare strongly \ndependent on in\n-\nfield or off\n-\nfield waiting \ntime during M(H) measurement\ns\n. \nWe showed \nvarious \noptions for tuning the \nferri\nmagnetic state and parameters, irrespective of the magnetic \nferrite\n \nand \nit’s \ncomposite in non\n-\nmagnetic matrix. The \ntuning capability of ferrimagnetic parameters is\n \npromising for t\nechnological applications\n. It may\n \nbe generalized by applying similar protocols of \nmeasurements on different systems. \n \n \nAcknowledgment\n \nRNB acknowledges the Research support from \nUGC (\nF.\n \nNo. F.42\n-\n804/2013 (SR)\n), Gov\nt. \nof \nIndia \nand UGC\n-\nDAE\n-\n \nCSR (No M\n-\n252/2017\n/1022), Gov. of India \nfor \ncarrying out \nthis \nresearch \nwork\n.\n \nRNB also acknowledges the PPMS facility at CIF, Pondicherry University.\n \n \nReferences\n \n[\n1\n] K. Jonason, P. Nordblad, E. Vincent, J. Hammann, and J.\n-\nP. Bouchaud, Eur. Phys. J. B \n13\n, 99 \n(2000).\n \n[\n2\n] A.\n \nRos\ntamnejadi, M.\n \nVenkatesan, H.\n \nSalamati, K.\n \nAckland, H.\n \nGholizadeh, P.\n \nKameli, and J. \nM. D.\n \nCoey, J. Appl. Phys.\n121\n, 173902 (2017).\n \n[\n3\n] D. Parker, V. Dupuis, F. Ladieu, J.\n-\nP. Bouchaud, E. Dubois, R. Perzynski, and E. Vincent, \nPhys. Rev. 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@A B CD E F\nG H I J K\nL M NO P QR S T\nUV WX\nY\nZ[\\\n]\n^_` a b cd e\nf g h i j k l mn o p q r s tu v v w x y z { |} ~ ~ \n \n\n\n\n \n \n ¡¢£¡¢¤¥¦§¨©ª«¬ ® ¯\n° ± ² ³ ´ µ\n¶ ·\n¸ ¹º»¼½¾¿ÀÁ»Á¿Ã»\nÄ Å\nÆ Ç\nÈ ÉÊËÌÍÎÏÐÑÒËÑÏÓË\nÔ Õ Ö× ØÙÚÛÜÝÞßàÚÞßáÜâÝãäåÛæç è é ê ë ì\níî ï ð ñ ñ ò\nóô õ ö ÷ ÷ ø\nù ú û ü ýþ ÿ \u0000 \u0001 \u0002\n\u0003 \u0004 \u0005 \u0006 \u0007 \b \t \n \b \t \u000b\f \r \u000e \u000f \u0010\u0011 \u0012 \u0013 \u0014\u0015\n\u0016\u0017\u0018\u0019\u001a\u0016\u0017\u001b\u0019\u001c \u001d \u001e \u001f ! \"\n#$ %& ' ( ) * * + & , - . / & , 0 1 1 2 - 3 * 4 5 6 - 7 - . / * 4 4 5 6 8 7 9 ' 3 : ' . 3 ; <=; / ' - 3 ; 9 - 0 3 ' . ( > 3 ' = ; * : > 0 . 0 4 5? @ A B C D E A A F @ G H I J K A ? L I M L NF L D @ K A D J NF B A O\nP Q R S T U VW X Y Z[ \\\n] ^ _ ` ab c de f gh i jk l mn o pq r st u vw x yz { { | } ~ \n \n \n \n¡¢£¤¥¦§¨ © ª « ¬ \n® ¯ ° ± ² ³´ µ ¶ · ¸ ¹º » ¼ ½ ¾ ¿À Á Â Ã Ä ÅÆ Ç Ç È É Ê Ë" }, { "title": "1808.09240v1.Magnetic_field___temperature_phase_diagram_of_ferrimagnetic_alternating_chains__spin_wave_theory_from_a_fully_polarized_vacuum.pdf", "content": "arXiv:1808.09240v1 [cond-mat.str-el] 28 Aug 2018Magnetic field - temperature phase diagram of ferrimagnetic alternating chains:\nspin-wave theory from a fully polarized vacuum\nW. M. da Silva and R. R. Montenegro-Filho\nLaborat´ orio de F´ ısica Te´ orica e Computacional, Departa mento de F´ ısica,\nUniversidade Federal de Pernambuco, 50760-901 Recife-PE, Brasil\n(Dated: August 29, 2018)\nQuantum critical (QC) phenomena can be accessed by studying quantum magnets under an ap-\nplied magnetic field ( B). The QC points are located at the endpoints of magnetizatio n plateaus and\nseparate gapped and gapless phases. In one dimension, the lo w-energy excitations of the gapless\nphase form a Luttinger liquid (LL), and crossover lines boun d insulating (plateau) and LL regimes, as\nwell as the QC regime. Alternating ferrimagnetic chains hav e a spontaneous magnetization at T= 0\nand gapped excitations at zero field. Besides the plateau at t he fully polarized (FP) magnetization;\ndue to the gap, there is another magnetization plateau at the ferrimagnetic (FRI) magnetization.\nWe develop spin-wave theories to study the thermal properti es of these chains under an applied\nmagnetic field: one from the FRI classical state, and other fr om the FP state, comparing their\nresults with quantum Monte Carlo data. We deepen the theory f rom the FP state, obtaining the\ncrossover lines in the Tvs.Blow-Tphase diagram. In particular, from local extreme points in\nthe susceptibility and magnetization curves, we identify t he crossover between an LL regime formed\nby excitations from the FRI state to another built from excit ations of the FP state. These two LL\nregimes are bounded by an asymmetric dome-like crossover li ne, as observed in the phase diagram\nof other quantum magnets under an applied magnetic field.\nI. INTRODUCTION\nThe theory of quantum phase transitions1,2provides\na framework from which the low-temperature behav-\nior of many condensed-matter systems can be under-\nstood. The quantum critical point separates an insu-\nlating gapped phase and a gapless conducting phase.\nOf particular importance are magnetic insulators3,4, for\nwhich the quantum critical regime can be experimentally\naccessed through an applied magnetic field. In these sys-\ntems, the gapped phases are associated to magnetization\nplateaus in the magnetization curves.\nIn one dimension, magnetization plateaus can be un-\nderstood as a topological effect through the Oshikawa,\nYamanaka, and Affleck (OYA) argument5, which gener-\nalizes the Lieb-Schultz-Mattis theorem6. The OYA ar-\ngument asserts that a magnetization plateau is possible\nonly if (Su−mu) = integer, where muis the ground-state\nmagnetization and Suis the sum of the spins in a unit\nperiod of the ground state, respectively. If the ground\nstate does not present spontaneous translation symme-\ntry breaking, Suis equal to the fully polarized magneti-\nzation per unit cell, while muis the magnetization per\nunit cell of the system. The OYA argument was further\nextended7to models in higher dimensions and to charge\ndegrees of freedom.\nDue to the gap closing a magnon excitation, the end-\npoints of magnetization plateaus are quantum critical\npoints. In three-dimensional systems, this transition\nis in the same universality class of the Bose-Einstein\ncondensation4,8and was studied in a variety of magnetic\ninsulators3,4,9. In the magnetic system, the magnetiza-\ntion and the magnetic field play the role of the boson\ndensity and of the chemical potential, respectively, of\nthe bosonic model. In one dimension the mapping toa hard-core boson model or a spinless fermion system8\nimplies a square-root singularity in the magnetization\ncurve:m∼/radicalbig\n|B−Bc|asB→Bc; and, if three-\ndimensional couplings are present, the condensate can\nbe stabilized at temperatures below that of the three-\ndimensional ordering8.\nExactly at the quantum critical field, the magnons\nhave a classical dispersion relation, ω∼q2, whereqis\nthe lattice wave-vector. In one dimension, this quantum\ncriticalfieldseparatesagappedphasefromagaplessLut-\ntinger liquid (LL) phase10,11, with excitations showing\na linear dispersion relation, ω∼q. The predictions of\nthe Luttinger liquid theory in magnetic insulators with\na magnetic field, including the quantum critical regime,\nwere investigated in many materials12–14. For finite tem-\nperatures and B≈Bc, the quantum critical regime is\nobserved, and the crossover line15to the LL regime is\ngiven by T(B)∼a|B−Bc|, with a universal, model-\nindependent, coefficient a.\nOne-dimensional ferrimagnets16,17show spontaneous\nmagnetization at T= 0, as expected from the Lieb and\nMattistheorem18, andagapintheexcitationspectrumis\nresponsible for a magnetization plateau in their magneti-\nzation curves at the ground-state magnetization value.\nIn zero field, the critical properties in the vicinity of\nthe thermal critical point at T= 0 were studied in the\nisotropic19,20and anisotropic cases20. Interesting physics\nemerges through the introduction of destabilizing factors\nof the ferrimagnetic state, such as doping21–27or geo-\nmetric frustration28–36. The spin-wave theory37of fer-\nrimagnetic chains37–45was developed from the classical\nferrimagnetic groundstate, considering free and interact-\ningmagnons, withemphasisonzero-fieldproperties. The\nmagnetization curves of these systems under an applied\nmagnetic field were discussed mainly through numerical2\nmethods38,42,46–51.\nInthiswork,weinvestigatethespin-wavetheoryoffer-\nrimagnetic alternating chains at low temperatures and\nin the presence of a magnetic field. We compare some\nresults with quantum Monte Carlo (QMC) data, ob-\ntained using the stochastic series expansion method code\nfrom the Algorithms and Libraries for Physics Simula-\ntions (ALPS) project52, with 1×106Monte Carlo steps.\nWe considerspin-waveexcitations from the ferrimagnetic\nand fully polarized classical states. In the ferrimagnetic\ncase, we consider interacting spin-waves, while in the\nfully polarized, only free spin-waves are discussed. Con-\nsidering the whole values of magnetization, from zero to\nsaturation, the two approachespresent similar deviations\nfrom the QMC data. We deepen the theory from the fer-\nromagnetic ground state and obtain the crossover lines\nbounding the plateau and LL regimes. In particular,\nwe show that susceptibility and magnetization data can\nbe used to identify a crossover between two LL regimes,\none built from excitations of the ferrimagnetic magnetic\nstate, and the other from the fully polarized one.\nThis paper is organized as follows. In Sec. II we\npresent the Hamiltonian model and discuss the mag-\nnetization curves from QMC calculations. In Sec. III\nthe spin-wave theories from the FRI and FP classical\nstates are discussed, particularly the methodology used\nto obtain the respective magnetization curves with a fi-\nnite temperature, and make a comparison between their\nresults and QMC data. In Sec. IV, we study LL and\nplateau regimes at finite temperature through the free\nspin-wave (FSW) theory from the FP vacuum (FSW-\nFPv). Finally, in Sec. V we summarize our results and\nsketchthe T-Bphasediagramfromthe FSW-FPvtheory\nof the alternating (1/2,1) spin chain.\nII. MODEL HAMILTONIAN AND QMC\nMAGNETIZATION CURVES\nAn alternating spin ( s,S) chain has two kinds of spin,\nSands, alternating on a ring with antiferromagnetic su-\nperexchange coupling Jbetween nearest neighbors, and\ndescribed by the Hamiltonian\nH=JN/summationdisplay\nj=1/parenleftBig\nsj·Sj+sj·Sj+1/parenrightBig\n−BN/summationdisplay\nj(Sz\nj+sz\nj),(1)\nwhereBis the magnetic field and Ndenotes the number\nof unit cells. We assume S > sand consider equal g-\nfactors for all spins, defining gµB= 1, where µBis the\nBohr magneton. The magnetization per unit cell is given\nby\nm=N/summationdisplay\nj(Sz\nj+sz\nj). (2)\nIn Fig. 1 we show QMC results for m(B) for the (1/2,\n1) chain in the low- Tregime. At T= 0,m(B) presents\nFIG. 1. (color online). Magnetization plateaus at finite tem -\nperature, Luttinger liquid phase and crossovers: Quantum\nMonte Carlo (QMC) data. Magnetization per cell mand the\nsusceptibility χ=∂m/∂B as a function of magnetic field B\nfor an alternating ( s= 1/2,S= 1) chain with N= 256 unit\ncells and the indicated values of temperature T. The critical\nendpoint of the ferrimagnetic (FRI) and the fully polarized\n(FP) plateaus are Bc,FRI = 1.76JandBc,FP = 3J, respec-\ntively. The presence of the FRI and FP plateaus, and the\nregion dominated by Luttinger liquid (LL) regime is a com-\nmon feature for all values of sandS, withS > s . AsT→0,\nχ→ ∞ at the critical values of B; forT/greaterorsimilar0, local maxima\nin theχcurves marks the crossover from the LL regime to\nthe quantum critical regime. The local minimum in the χ\ncurve (dashed line) between Bc,FRI andBc,FP separates the\nLL regime into two regions: one with excitations from the\nFRI state, LL 1; the other with excitations from the FP state,\nLL2.\ntwo magnetization plateaus: the ferrimagnetic (FRI), at\nmFRI= (S−s), and the fully polarized (FP) one, at\nmFP=s+S. In particular, at T= 0,m=mFRIfor\nB= 0, with a gapless Goldstone mode. There are quan-\ntum phase transitions at the endpoint of the plateaus:\nB=Bc,FRIandB=Bc,FP, respectively; which have\nthe values Bc,FRI= 1.76JandBc,FP= 3.00Jfor the\n(1/2, 1) chain. At the critical fields, there is a tran-\nsition from a gapped plateau phase to a gapless Lut-\ntinger liquid (LL) phase, as B→Bc,FRIfrom magnetic\nfieldsB < B c,FRI, orB→Bc,FPfrom magnetic fields\nB > B c,FP. In the LL phase, the excitations have a\nlinear dispersion relation, ω∼q, and present critical\n(power-law) transverse spin correlations. Exactly at the\ncritical fields, the excitations have a classical dispersion\nrelationω∼q2andin thehighdiluted limitcanberepre-\nsented by a hard-core boson model or a spinless fermion\nmodel. Hence, the magnetization has a square-root be-\nhaviorm∼/radicalbig\n|B−Bc|and a diverging susceptibility\nχ=∂m/∂B∼1//radicalbig\n|B−Bc|asB→Bc.\nFor finite- T, butT→0, the magnetization m= 0 for3\nB= 0, since the system is one-dimensional. Gapped\nmagnetic excitations are thermally activated and the\nplateau widths reduce. The susceptibility shows local\nmaxima, with distinct amplitudes, at B≈Bc,FRIand\nB≈Bc,FPmarkingthecrossoverbetweentheLLregime,\nwhere the excitations have a linear behavior, ω∼q, to\nthe quantum critical regime, for which ω∼q2. We can\ndefine the local minimum in the χcurve, at B≡Bi,\nas a crossover between the region where the excitations\narepredominantly from the FRI state, denoted by LL 1in\nFig. 1, and that where the excitations are predominantly\nfrom the FP state, denoted by LL 2in Fig. 1. In partic-\nular, for B≈Bi, the magnetization curve has its more\nrobust value and behavior as the temperature increases,\nshowing that the LL phase is more robust for B≈Bi.\nIII. SPIN-WAVE THEORY\nThe ferrimagnetic arrangement of classical spins is a\nnatural choice of vacuum to study quantum ferrimagnets\nthrough free spin-wave (FSW) theory38, if we want to\nstudy excitations from the quantum ground state. Two\ntypes of magnon excitations are obtained, one ferromag-\nnetic, which decreases the ground state spin by one unit,\nand the other antiferromagnetic, increasing the ground\nstate spin by one unit. In particular, the antiferromag-\nnetic excitation has a finite gap ∆, which implies the\nexpected magnetization plateau at m=S−sandT= 0.\nHowever, at this linear approximation, quantum fluctua-\ntionsareunderestimated,givingpoorresultsforthevalue\nof antiferromagnetic gap, and other quantities, like the\naverage spin per site.\nWhen one-dimensional ferromagnets are studied\nthrough the linear spin-wave theory at finite tempera-\ntures, a divergingzero-field magnetization is obtained for\nany value of T53–55. Takahashi56,57modified the theory\nby imposing a constraint on the zero-field magnetization\nand an effective chemical potential in the thermal boson\ndistribution. This so-called modified spin-wave theory\ndescribes very well the low-temperaturethermodynamics\nof one-dimensional ferromagnets, and was further suc-\ncessfully adapted to other systems, including ferrimag-\nnetic chains40. In the case of ferrimagnets, the introduc-\ntion of the magnetization constraint in the bosonic dis-\ntribution, with the linear spin-wave dispersion relations\ngives an excellent description of the low- Tbehavior. The\ndescriptionoftheintermediate- Tregimecanbeimproved\nby changing the constraint37.\nIn this Section, we discuss interacting spin-wave the-\nory using a ferrimagnetic vacuum (ISW-FRIv) for B/negationslash= 0\nandT/negationslash= 0, with the modified spin-wave approach (Taka-\nhashi’sconstraint); andfreespin-wavetheoryfromafully\npolarizedvacuum (FSW-FPv), alsofor B/negationslash= 0 andT/negationslash= 0.\nFIG. 2. (color online). Interacting spin-wave (ISW) magnon\nbranches from the classical ferrimagnetic vacuum (FRIv) - c al-\nculating the thermodynamic properties. (a) The classical f er-\nrimagnetic vacuum of the ( s,S) chain. (b) Magnon dispersion\nrelations for the ( s= 1/2,S= 1) chain with B= 0. There are\nferromagnetic and antiferromagnetic magnons, carrying sp in\n∆Sz=−1 and ∆Sz= 1, respectively. The values of the criti-\ncal fields are B(ISW-FRIv )\nc,FRI = 1.68JandB(ISW-FRIv )\nc,FP = 2.74J. To\ncalculate the thermodynamic functions, the antiferromagn etic\n(ferromagnetic) magnons occupies their respective bands f ol-\nlowing the Fermi (Bose) distribution function. An effective\nchemical potential µis introduced in the Bose distribution\nto prevent particle condensation at the k= 0 mode for\nB= 0 and T→0. (c) For each value of T, we use a value\nofµsuch that m= 0 for B= 0. The inset shows that\nµ(T→0)→0 asT→0. In this limit, both bands are empty\nandm= (S−s) = 1/2, the FRI magnetization.\nA. Spin-wave theory - ferrimagnetic vacuum\nThe Holstein-Primakoff spin-wave theory is developed\nfrom the classical ground state illustrated in Fig. 2(a),\nwhich has the energy E(FRIv)\nclass=−2JNsS−B/parenleftbig\nS−s/parenrightbig\nN.\nThebosonicoperators aj(a†\nj)andbj(b†\nj), associatedto A\nandBsites, respectively, have the following relation with\nthe spin operators (Holstein-Primakoff transformation):\nS+\nj=√\n2S/parenleftBig\n1−a†\njaj\n2S/parenrightBig1/2\naj, andSz\nj=S−a†\njaj; (3)\ns+\nj=b†\nj√\n2s/parenleftBig\n1−b†\njbj\n2s/parenrightBig1/2\n, andsz\nj=b†\njbj−s.(4)\nPutting the Hamiltonian (1) in terms of these\nbosonic operators, expanding to quadratic order, Fourier\ntransforming and making the following Bogoliubov\ntransformation38:\nak=αkcoshθk−β†\nksinhθk,\nbk=βkcoshθk−α†\nksinhθk, (5)\ntanh2θk= 2√\nsS\ns+Scos/parenleftBigk\n2/parenrightBig\n, (6)4\nwherekis the lattice wave-vector, the non-interacting\nspin-wave Hamiltonian is given by\nH(FSW-FRIv)=E0+/summationdisplay\nk/bracketleftBig\nω(FRIv)\nk,−α†\nkαk+ω(FRIv)\nk,+β†\nkβk/bracketrightBig\n.(7)\nThe magnon branches obtained are:\nω(FRIv)\nk,σ=σJ/parenleftbig\nS−s/parenrightbig\n−σB+Jω(FRIv)\nk,(8)\nwithσ=±, and\nω(FRIv)\nk=/radicalbigg/parenleftbig\nS−s/parenrightbig2+4sSsin2/parenleftBigk\n2/parenrightBig\n,(9)\nwhile the ground-state energy is\nE0=J/summationdisplay\nk/bracketleftBig\nω(FRIv)\nk−/parenleftbig\nS+s/parenrightbig/bracketrightBig\n. (10)\nTheω(FRIv)\nk,−modes carry a spin ∆ Sz=−1, having a\nferromagneticspin-wavenature, and is gaplessfor B= 0;\nwhileω(FRIv)\nk,+modes carry a spin ∆ Sz= +1, having an\nantiferromagnetic spin-wave nature and has a gap ∆ =\n2J(S−s) atB= 0. For the ( s= 1/2,S= 1) chain38,\nfor example, ∆ = 1, although the exact value is 1 .76J;\nwhile/angbracketleftSz\na/angbracketright= 0.695 and /angbracketleftSz\nb/angbracketright=−0.195 atT= 0, with\nthe exact values38:/angbracketleftSz\na/angbracketright= 0.792 and /angbracketleftSz\nb/angbracketright=−0.292.\nThe dispersion relations can be improved if interac-\ntions between magnons are considered. The corrected\ndispersion relations described in Ref.43, shown in Fig.\n2(b), are:\n˜ω(FRIv)\nk,σ=ω(FRIv)\nk,σ−Jδω(FRIv)\nk,σ, (11)\nwhere\nδω(FRIv)\nk,σ= 2Γ1(S+s)\nω(FRIv)\nksin2(k/2)−Γ2√\nsS/bracketleftBig\nω(FRIv)\nk+σ(S−s)/bracketrightBig\n,\nwith\nΓ1=1\nN/summationdisplay\nksinh2θk,and (12)\nΓ2=1\nN/summationdisplay\nkcos(k/2)sinhθkcoshθk.(13)\nUp toO(S0), the Hamiltonian is\nH(ISW-FRIv)=Eg+/summationdisplay\nk/parenleftbig\n˜ω(FRIv)\nk,−α†\nkαk+ ˜ω(FRIv)\nk,+β†\nkβk/parenrightbig\n,(14)\nwhere\nEg=Eclass+E0+E1, (15)\nwith\nE1=−2JN/bracketleftBig\nΓ2\n1+Γ2\n2−/parenleftBig/radicalbig\nS/s+/radicalbig\ns/S/parenrightBig\nΓ1Γ2/bracketrightBig\n.(16)AtT= 0, the magnetization as a function of B, shown\nin Fig. 1 for the ( s= 1/2,S= 1) chain, can be under-\nstood from these ferromagnetic (∆ Sz=−1) and anti-\nferromagnetic (∆ Sz= +1) magnon modes. For B= 0\nthe two bands are empty and the magnetization is the\nferrimagnetic one. Increasing the magnetic field, the fer-\nromagnetic band acquires a gap which increases linearly\nwithB, while the gap to the antiferromagnetic band\ndecreases linearly with B. Notice, in particular, that\nthe ferromagnetic band is empty for all values of B. At\nB=B(ISW-FRIv )\nc,FRI/2 = ∆/2, thek= 0 mode of the an-\ntiferromagnetic band is the lower energy state, and at\nB=B(ISW-FRIv )\nc,FRI = ∆ the gap to this mode closes. The\nvalue ofB(ISW-FRIv )\nc,FRI is\nB(ISW-FRIv )\nc,FRI = ˜ω(FRIv)\n0,+= 2(S−s)/parenleftbigg\n1+1√\nsSΓ2/parenrightbigg\nJ.(17)\nIn particular, for the ( s= 1/2,S= 1) chain, with Γ 1=\n0.305and Γ 2= 0.478,B(ISW-FRIv )\nc,FRI= 1.68J, which is very\nclose to the exact value (1 .76J).\nThe magnetization for B >∆ is obtained by consider-\ning the antiferromagnetic magnons as hard-core bosons8,\nor spinless fermions. The magnetization increases with\nBas the antiferromagnetic band is filled, and saturates\nwhen the Fermi level reaches the band limit, at k=π.\nThe saturation field is\nB(ISW-FRIv )\nc,FP = ˜ω(FRIv)\nπ,+= 2/parenleftBigg\nS−Γ1+/radicalbigg\nS\nsΓ2/parenrightBigg\nJ,(18)\nwhich for the ( s= 1/2,S= 1) chain is B(ISW-FRIv )\nc,FP=\n2.74, departing from the exact value 3 J, but much better\nthan the free spin wave result: 2 J.\n1. Thermodynamics\nForT >0, ferromagneticand antiferromagneticmodes\nare occupied in accord to Bose-Einstein ( n(FRIv)\nk,−) and\nFermi-Dirac ( n(FRIv)\nk,+) distributions, respectively, as indi-\ncated in Fig. 2(a). The magnetization, for example, is\ngiven by\nm(T,B) = (S−s)+1\nN/summationdisplay\nk(n(FRIv)\nk,+−n(FRIv)\nk,−).(19)\nWe notice, however, that with T >0 andB= 0\nthe ferromagnetic band will be thermally activated and\nm→ −∞ asTincreases. This problem arises, also,\nin one-dimensional ferromagnetic chains, and was over-\ncomebyTakahashi56,58, in the low- Tregime, throughthe\nintroduction of an effective chemical potential µin the\nbosonic distribution, and a constraint m(B= 0,T) = 0.\nA similar strategy was applied to one-dimensional ferri-\nmagnetic systems40and good results were also obtained\nin the low- Tregime. The intermediate- Tregime, where5\nthe minimum in the Tχcurve of the ferrimagnets17are\nobserved, can be more accurately described if other con-\nstraints are used37,43,45.\nHere, for B= 0, we use the simplest constraint\nm(T,B= 0) = 0 , (20)\nsince we are interested in the low- Tregime, with\nn(FRIv)\nk,−=1\neβ[˜ω(FRIv)\nk,−−µ]−1, (21)\nn(FRIv)\nk,+=1\neβ˜ω(FRIv)\nk,++1. (22)\nIn Fig. 2(b), we present m(T,B= 0) for the indicated\nvalues of T. As discussed, m→ −∞atµ= 0 and the\nvalue of µfor which the constraint m(T,B= 0) = 0 is\nsatisfied, monotonically decreases with T, in this low- T\nregime. A finite µimplies an effective gap for the ferro-\nmagneticband, with anexponentialthermalactivationof\ntheir magnons. In particular, notice that µ(T→0) = 0,\nas expected. To calculate the thermodynamic functions\nforB/negationslash= 0, we consider the distributions in Eqs. (21) and\n(22) and use the same value of µfound in the case B= 0:\nµ(B,T) =µ(B= 0,T), for any value of B.\nThe magnetization as a function of BforT/negationslash= 0,\nshown in Fig. 1, can be qualitatively understood from\nthis theory. For B= 0, the magnetization m= 0,\ndue to the constraint. As Bincreases, in the region\n0< B < B c,FRI/2, the gap to the ferromagnetic band\nincreases, but this band is thermally activated and the\nmagnetization decreases from the m=S−svalue. This\neffect can also be seen from Fig. 2(b). If we move the\nZeeman term, + B, from the ferromagnetic dispersion re-\nlation to the chemical potential, ˜ ω(FRIv)\nk,−→˜ω(FRIv)\nk,−−Band\n−µ→ −(µ−B), in Eq. (21), the magnetization value\nis the one shown in Fig. 2(b) for µlower than that of\nB= 0, and m= 0. From Fig. 2(b), we see that increas-\ningB(decreasing µ) fromB= 0 [from µ(B= 0,T)],\nthe magnetization rises exponentially to the ferrimag-\nnetic value. For B=B(FSW, FPv)\nc,FRI/2, the lower energy\nband is the antiferromagnetic (∆ Sz= +1 magnons)\nfermionic band. This band is thermally activated for\n[B(FSW, FPv)\nc,FRI/2]< B < B c,FRI, and the magnetization is\nhigher than S−s. The magnetization increases through\nthe filling of this band, in accord to the Fermi distribu-\ntion, up to the saturation value m=s+S, which is\nexponentially reached.\nB. Spin-wave theory - fully polarized vacuum\nIn this section, westudy the freespin wavetheoryfrom\na fully polarized vacuum, illustrated in Fig. 3(a). We\nshow that this theory provides a good description of the\nlow-Tphysics, andisquantitativelymuchbetterthanthe\nfree spin wave description from the ferrimagnetic vac-\nuum. The critical saturation field has an exact value,\nFIG. 3. (color online). Free spin-wave magnon branches from\nthe classical ferromagnetic vacuum - calculating the therm o-\ndynamic properties. (a) The classical fully polarized vacu um\nof the (s,S) chain. (b) Free spin-wave (FSW) results for the\nmagnon energies relative to the fully polarized vacuum (FPv )\nforT/negationslash= 0 and B= 0 for the ( s= 1/2,S= 1) chain. In this\ncase, both branches are ferromagnetic with magnons carryin g\na spin ∆Sz=−1. To calculate the thermodynamic functions,\nthe lower (higher) magnon band is filled following the Fermi\n(Bose) distribution function. An effective chemical potent ial\nµis introduced in the Bose distribution to prevent particle\ncondensation at the k=πmode for B= 0 and T→0. The\ncritical fields are B(FSW, FPv )\nc,FRI = 2.00JandB(FSW, FPv )\nc,FP = 3.00J.\n(c) The chemical potential µis chosen such that m= 0 for\nB= 0. The inset shows that µ(T→0)→ − 1 asT→0. In\nthis limit only the lower energy band is occupied, implying\nthatm→(S−s) = 1/2, the ferrimagnetic magnetization, as\nT→0 andB→0.\nwhile the critical field at the end of the ferrimagnetic\nplateau is B(FSW, FPv)\nc,FRI= 2J.\nThe Holstein-Primakoff transformation in this case is\nS+\nj=√\n2S/parenleftBig\n1−a†\njaj\n2S/parenrightBig1/2\naj, andSz\nj=S−a†\njaj;(23)\ns+\nj=√\n2s/parenleftBig\n1−b†\njbj\n2s/parenrightBig1/2\nbj, andsz\nj=s−b†\njbj,(24)\nwith the two bosons lowering the site magnetization by\none unit. To quadratic order in these bosonic operators,\nthe Hamiltonian of the system, Eq. (1), is\nH(FSW-FPv)=E(FPv)\nclass+J/summationdisplay\nj/braceleftBigg\n−s/parenleftBig\na†\njaj+a†\nj+1aj+1/parenrightBig\n−2Sb†\njbj+√\nsS/bracketleftBigg/parenleftBig\naj+aj+1/parenrightBig\nb†\nj+/parenleftBig\na†\nj+a†\nj+1/parenrightBig\nbj/bracketrightBigg\n+B/summationdisplay\nj/parenleftBig\na†\njaj+b†\njbj/parenrightBig/bracerightBigg\n, (25)\nwithE(FPv)\nclass= 2JNsS−B/parenleftbig\nS+s/parenrightbig\nN. Fouriertransforming\nthe bosonic operatorsand using the Bogoliubov transfor-\nmation\na†\nk=α†\nkcosθk−β†\nksinθk; (26)\nb†\nk=β†\nkcosθk+α†\nksinθk, (27)6\nwith\ntan2θk= 2√\nsS\nS−scos/parenleftBigk\n2/parenrightBig\n, (28)\nthe Hamiltonian in Eq. 25 is written as\nH(FSW-FPv)=E(FPv)\nclass+/summationdisplay\nk/bracketleftBig\nω(FPv)\nk,1α†\nkαk+ω(FPv)\nk,0β†\nkβk/bracketrightBig\n,(29)\nwhere the dispersion relations42ω(FPv)\nk,ηare\nω(FPv)\nk,η= (−1)η+1/radicalbigg/parenleftbig\nS−s/parenrightbig2+4sScos2/parenleftBigk\n2/parenrightBig\n−/parenleftbig\nS+s/parenrightbig\n+B, (30)\nwithη= 0 or 1.\nTo discuss the T= 0 magnetization curve implied\nby these spin-wave modes, we present in Fig. 3(b) the\ndispersion relations ω(FPv)\nk,ηfor the ( s= 1/2,S= 1)\nchain and B=B(FSW, FPv)\nc,FP = 2J(s+S) = 3J. At\nB=B(FSW, FPv)\nc,FP=Bc,FP, both bands are empty, and the\nmagnetization is the fully polarized one. Decreasing B,\ntheη= 0 band is filled in accord to Fermi-Dirac statis-\ntics, and the magnetization decreases. The critical field\nat the end point of the ferrimagnetic plateau is obtained\nmakingω(FPv)\nπ,0= 0, which implies B(FSW, FPv)\nc,FRI= 2SJ, equal\nto 2Jfor the (s= 1/2,S= 1) chain. At this value of B,\ntheη= 0 band is totally filled and m= (s+S)−1, giving\n1/2 for the ( s= 1/2,S= 1) chain. There is a gap of\n2(S−s)Jbetween the η= 0 and η= 1 bands, at k=π;\nhence, the bosonic η= 1 band should start to be filled\natB=B(FSW, FPv)\nc,FRI−2(S−s)J, and the theory does not\nqualitatively reproduce the T→0 magnetization curve.\nThis problem is overcome by considering the finite tem-\nperature theory, with Takahashi’s constraint and effec-\ntive chemical potential. For finite T, the magnetization\nis given by\nm(T,B) = (S+s)−1\nN/summationdisplay\nk[n(FPv)\nk,0+n(FPv)\nk,1],(31)\nwhere\nn(FPv)\nk,0=1\neβω(FPv)\nk,0+1, (32)\nn(FPv)\nk,1=1\neβ[ω(FPv)\nk,1−µ]−1. (33)\nThe constraint, which is applied at B= 0, is\nm(T,B= 0) = 0 . (34)\nIn Fig. 3(c)wepresentthe magnetizationasafunctionof\nthe effective chemical µfor the indicated values of tem-\nperature. We note that m→ −∞as the temperature\nincreases, similarly to the spin-wave theory with the fer-\nrimagnetic vacuum. However, in this case µ→ −1 as\nT→0, as shown in Fig. 3(b). Hence, a finite chemicalpotential µ=−1 associated to the bosonic η= 1 band\nmust be considered in the T= 0 theory. With this chem-\nical potential, the η= 1 band stays empty at T= 0 for\nany value of B.\nThe thermodynamic functions arecalculated using Eq.\n33, with µ(T,B) =µ(T,B= 0). For finite T, the\nfermionic η= 0 band is completely filled and the oc-\ncupation of the η= 1 band is such that m= 0. Con-\nsidering the low- Tregime, as Bincreases, the energy\nof the two bands raises, lowering the total occupation\nof theη= 1 band, since ω(FPv)\nk,1−µlinearly increases\nwithBfor any k, andmincreases. The magnetiza-\ntion exponentially reaches its value at the ferrimagnetic\nplateau, m=S−s, asBincreases, since n(FPv)\nk,1→0\nfor anykand the η= 0 band is completely filled. For\n[B(FSW, FPv)\nc,FRI/2]< B < B(FSW, FPv)\nc,FRI, with [B(FSW, FPv)\nc,FRI/2] re-\nlated to the point B=Bc,FRI/2 in Fig. 1, the occupa-\ntion of the η= 0 band decreases from the T= 0 case:\nn(FPv)\nk,0= 1 for any k, and the magnetization is higher than\nS−s. The magnetizationincreaseswith B, and exponen-\ntially reaches the fully polarized value at B > B(FSW, FPv)\nc,FP,\nsince magnons at the η= 0 band are thermally excited.\n0 1 1.76 3\nB(J)00.511.5\nQMC\nFRIv, ISW\nFPv, FSW\nJχ / 4m\nFIG. 4. (color online). Comparison between results from\nquantum Monte Carlo (QMC) method, N= 256 unit cells,\nand the two spin-wave approaches for the magnetization per\ncellmand the susceptibility χ: (s= 1/2,S= 1) chain\nat temperature T= 0.02(J/kB). Results from the inter-\nacting spin-wave theory from a ferrimagnetic vacuum (ISW-\nFRIv) and free spin-wave theory from a ferromagnetic vacuum\n(FSW-FPv) compare well with QMC for B/lessorsimilarBc,FRI and\nB/greaterorsimilarBc,FP. The maximum in χrelated to Bc,FRI (Bc,FP) is\nbetter localized, compared to QMC, through the ISW-FRIv\n(FSW-FPv) approach.\nC. Comparison between QMC data and the two\nspin-wave approaches\nIn Fig. 4 we present magnetization and susceptibil-\nityχ=∂m/∂Bas a function of Bfrom ISW-FRIv and7\nFSW-FPv theories along with QMC data, at T= 0.1J.\nSince the ISW-FRIv gives a better result for Bc,FRI, this\ntheory is better in the vicinity of this critical field. Oth-\nerwise, the FSW-FPv approach is better in the vicinity\nofBc,FP. Further, the amplitudes of the two peaks in\nχ(B), which marks the crossover to the LL regime, have\nvalues lower than the ones given by QMC. The difference\nbetween the amplitudes of the spin-wave approaches and\nQMC data is related to limitations in the spin-wavetheo-\nries. Despite it, the description from both spin-wave the-\nories are qualitatively excellent, and quantitatively very\nacceptable in the low- Tregime.\nBelow we calculate the TvsBphase diagram in the\nlow-Tregime from the FSW-FPv theory. We study the\ncrossover lines between the LL regimes and the quantum\ncritical regimes; as well as the crossovers lines between\nthe plateau regimes and the quantum critical regimes.\nWe use the FSW-FPv approach since it has essentially\nthesameprecisionoftheISW-FRIvtheory, ifweconsider\na range of Bfrom 0 to the saturation field; also, the\ncritical point Bc,FPis exact in the FSW-FPv theory.\nIV. LUTTINGER LIQUID REGIME\nIn the LL phase, the dispersion relationcan be approx-\nimated by ±vF|k−kF|, wherevFis the Fermi velocity.\nFurther, in this regime the magnetization has the form15:\nm=m(T= 0)−π\n6v2\nF∂vF\n∂B(kBT)2+O(T3).(35)\nIn our case, the Fermi velocity along the η= 0 band\nisvF= [∂ω(FPv)\nk,0/∂k]k=kF, withkFcalculated from\nω(FPv)\nk,0|k=kF= 0.\nIn Fig. 5(a) we present vFas a function of Bfor the\n(1/2,1) chain. Near the critical fields, |∂vF/∂B|is large\nandvFlittle. For a fixed B/greaterorsimilarB(FSW, FPv)\nc,FRI, as shown in\nFig. 5(b), the magnetization presents a fast decay from\ntheT= 0 value as Tincreases. Also, for B/lessorsimilarB(FSW, FPv)\nc,FP,\nas shown in Figs. 5(c), mincreases from m(0). In both\ncases, the curvature of the m(T→0)-curve increases as\nBget closer to the critical fields. The crossover tem-\nperature T(B) of the LL regime at a fixed Bis defined\nas the point at which m(T) departs from the quadratic\nbehavior in Eq. (35). So, T(B) is taken to be at the\nminima ( B/greaterorsimilarB(FSW, FPv)\nc,FRI) and maxima ( B/lessorsimilarB(FSW, FPv)\nc,FP)\nof them(T) curve15. In particular, as B→Bcthe\ncrossover line separates the LL regime and the quan-\ntum critical regime, for which the excitations have a\nquadratic dispersion relation. In this case, a universal,\nmodel independent, straight line kBT(B) =a|B−Bc|,\nwitha= 0.76238, can be derived15.\nIn the inset of Fig. 5(a), we show that the minimum\nin theχ(B) =∂m/∂B curve is found at B=Bi, a\nvalue of Bat which |∂vF/∂B|= 0. This value of B\nmarks a crossover from the regime where excitations are2 2.2 2.4 2.6 2.8 3\nB(J)00.10.20.30.40.5vF(J)\n2 3B (J)0123Jχ∂vF__\n∂B > 0∂vF__\n∂B < 0\nB = Bi≈ 2.366 J(a)\nB = BiT = 0.01 J_\nkB\n00.050.10.150.2\nT(J/kB)0.84(b) (c) (d)\n00.050.10.150.2\nT(J/kB)1.21.3\n00.050.10.150.2\nT(J/kB)0.60.7m2 3\nB(J)01JχT = 0.085 (J/kB)\nB = Bi\nB = 2.05 JB = 2.10 JB = 2.20 J\nB = 2.366 JB = 2.95 J\nB = 2.90 J\nB = 2.80 JT = 0.085 (J/kB)\nFIG. 5. (color online). Results from the free spin-wave ap-\nproach with the fully polarized vacuum (FSW-FPv). (a)\nFermi velocity vFas a function of the magnetic field Band\n[(b), (c) and (d)] magnetization curves m(T). (a)∂vF/∂B→\n+∞andvF→0 asB→B(FSW, FPv )\nc,FRI = 2.00J, while\n∂vF/∂B→ −∞ andvF→0 asB→B(FSW, FPv )\nc,FP = 3.00J.\nAs shown in the inset, for B=Bi≈2.366J,∂vF/∂B = 0\nand the susceptibility χ(B) has a minimum at this value of\nB. (b)m(T) for the indicated values of Bin the vicinity of\nthe critical field B(FSW, FPv )\nc,FRI . (c)m(T) for values of Bin the\nvicinity of the critical field B(FSW, FPv )\nc,FP . (d)m(T) forB=Bi.\nThem(T) curves to order O(T2), Eq. (35), are shown as\ndashed lines in (b) and (c) for the corresponding values of B,\narrows indicate local extreme points in m(T), which are used\nas a criterium to identify the LL regime. The inset in (d)\nshows that the minimum in m(T) is associated to the local\nminimum in χ(B), which is found between the two critical\nfields.\npredominantly from the FRI critical state to the regime\nwhere they come from the FP critical state. At B=Bi,\nthe Fermi wave-vector is at the inflection point of the\ndispersion curve ( d2ω(FPv)\nk,0/dk2= 0), since\n∂vF\n∂B=/bracketleftBigg\nd2ω(FPv)\nk,0\ndk2/bracketrightBigg\nk=kF/parenleftbigg∂kF\n∂B/parenrightbigg\n, (36)\nandkFincreases monotonically with Bbetween the crit-\nical fields. If the value of kat the inflection point is ki,\nwe can calculate Bifrom the equation ω(FPv)\nki,0= 0. For the\n(1/2,1) chain, for example, Bi= 2.366Jand is indicated\nin Fig. 5(a).\nAtB=Bi,∂vF/∂B= 0 and the quadratic term in8\n00.050.10.150.2\nT(J/kB)0.860.87m00.050.10.150.2\nT(J/kB)1.21.3\nm\n00.050.10.150.2\nT(J/kB)0.60.7m\nB = 1.80 JB = 1.85 J\n1.51.762 2.5 3\nB(J)00.10.2\nT(J/kB)B = 1.95 J\nB = 2.25 JB = 2.95 J\nB = 2.90 J\nB = 2.80 J(a) (b)\n(c) (d)\nFSW-FPv minima\n(shifted)QMC minimaQMC maximaFSW-FPv\nmaxima\na|B - B c,FRI| a|B - B\n c,FP\n|B > ~Bc,FRIB < ~Bc,FP\nFIG. 6. (color online). Magnetization per cell m(T) with fixed\nB: calculating the crossover lines bounding the Luttinger li q-\nuid regime. Quantum Monte Carlo (QMC) results for the\nmagnetization curves m(T) and the crossover lines for a sys-\ntem with N= 128. (a) m(T) for values of Bin the vicinity of\nthe critical field Bc,FRI = 1.76J. (b)m(T) for values of Bin\nthe vicinity of the critical field Bc,FP = 3.00J. (c)m(T) for a\nvalue ofBsuch that ∂χ/∂B≈0 atT= 0 and inside the Lut-\ntinger liquid phase, dashed line in Fig. 1. (d) Local extreme\npoints of m(T) curves from QMC and free spin-wave from the\nfully polarized vacuum (FSW-FPv). In the case of the FSW-\nFPv local minima, we shift BbyBc,FRI−B(FSW, FPv )\nc,FRI ≈0.24J.\nThe exact crossover straight lines as T→0, extended in the\nfigure for better visualization: a|B−Bc,FRI|anda|B−Bc,FP|,\nwitha= 0.76238, are also shown. The error bars are defined\nas half the temperature step (∆ T= 0.008) used to calculate\nm(T).\nEq. (35) is absent. So, the more stable, against T, LL\nregion is found for B≈Bi. Since the crossover temper-\naturesT(B)→0 near the critical fields, the T(B) line\nhas anasymmetric dome-like profile, which is a conse-\nquence of the vFcurve, shown in Fig. 5(a) for the case\nof the (1/2,1) chain, and is also observed in other quan-\ntum magnets3.\nA minimum in the m(T) curveis alsoobservedfor B=\nBi, due to the O(T3) in Eq. (35), as shown in Fig. 5(d).\nIn this case, however, this extreme point is associated\nwith the minimum in the χ(B) curve, at B=Bi, as\nshown in the inset of Fig. 5(d).\nIn Fig. 6 we show m(T) curves for the (1/2,1) chain\ncalculated with QMC method to discuss the qualitatively\nagreement between these almost exact results and the\nconclusions from the spin-wave theory. In Figs. 6(a) and\n(b), weshowtheminimum(maximum)inthe m(T)curve\nforB/greaterorsimilarBc,FRI= 1.76J(B/lessorsimilarBc,FP= 3J). InFig. 6(c),\nwe calculate m(T) for a value of Bin the vicinity of the\nminimum in the χ(B) curve, B=Bi. Using the data\nin Fig. 1, it is located at Bi= (2.27±0.07)J, and is\nindicated as a dashed line in that figure. As shown in\nFig. 6(c), the m(T→0) curve is also flat, as in Fig.\n5(d), for B= 2.25J. The minimum in the m(T) curveappearsat T≈0.1J. As canbe observedin the T= 0.1J\nsusceptibility curve in Fig. 1, it is also associated with\nthe minimum in the χ(B) curve, at B≈Bi.\nIn Fig. 6(d), we compare the position of the local\nextreme points in the m(T) curves from QMC and FSW-\nFPv methods. The values of Bat the minima of m(T)\nweretranslatedby Bc,FRI−B(FSW, FPv)\nc,FRI≈0.24J. The lines\nfor the maxima in m(T) from both methods are in very\ngood agreement since the FSW-FPv is almost exact for\nT→0, due to the low density of excited magnons in this\ntemperature regime. Otherwise, the minima from both\nmethods do not compare well, except for T→0, which\nis dominated by the critical point.\n2 2.5 3\nB(J)0510C(kB) / T (J/kB)\nB = BiT(J/kB)\nT→0\nC ∝ TLL regime\nB = Bi\nLL phase (T = 0)0.005\n0.007\n0.010\n0.020\n0.030\n0.040\n0.05000.025 0.05\nT(J/kB)00.050.1C(kB)\nFIG. 7. (color online). Specific heat from the free spin-wave\ntheory from a fully polarized vacuum (FSW-FPv) for T→0.\nIn the Luttinger liquid (LL) regime, C∼TasT→0, and\nC/T is approximately constant for B≈Bi= 2.366J. The in-\nset shows this linear behavior of CatB=Bi. The crossover\nfrom the T= 0 insulating plateau regime to the gapless quan-\ntum critical regime, at local maxima, are indicated by arrow s.\nWe determine the crossover lines between the LL and\nplateau regimes through specific heat data, C(B). In\nFig. 7 we present FSW-FPv results for C(B) in the low-\nTregime. In the LL phase, at T= 0, the specific heat\nC∼TasT→0, andC/Tis approximately constant\nin the LL regime, as shown in Fig. 7. The range of\nBnearB=Biis the more robust for this regime, and\nwe present in the inset of Fig. 7 the linear behavior\nofCas a function of T. ForB/lessorsimilarB(FSW, FPv)\nc,FRI orB/greaterorsimilar\nB(FSW, FPv)\nc,FP,theexcitationsareexponentiallyactivatedand\nthe crossoverto the quantum criticalregimeis markedby\na local maximum in C(B). The points of these crossover\nlines,Tplateau(B)∼ |B−Bc|, are indicated by arrows in\nFig. 7. The quantum critical regime is bounded by this\ncrossover line and that of the LL regime, which points\nappears as a second local maximum near B(FSW, FPv)\nc,FRIand\nB(FSW, FPv)\nc,FPin Fig. 7.9\nV. SUMMARY AND DISCUSSIONS\nFIG. 8. (color online). Spin-wave T−Bphase diagram of\nthe (s= 1/2,S= 1) chain from the FPv. The quantum\ncritical points B(FSW, FPv )\nc,FRI = 2.00JandB(FSW, FPv )\nc,FP = 3.00J\nbound the FRI and FP plateau regions, respectively. Increas -\ning temperature, the plateau width decreases and the lines\nkBT=|B−B(FSW, FPv )\nc,FRI|andkBT=|B−B(FSW, FPv )\nc,FP|limit\nthe plateau regions for B/lessorsimilarBc[ferrimagnetic (FRI) plateau]\nandB/greaterorsimilarBc[fully polarized (FP) plateau]. The LL regime has\ncrossover lines given by a|B−B(FSW, FPv )\nc,FRI|anda|B−B(FSW, FPv )\nc,FP|,\nwitha= 0.76238, for B→Bc, as indicated by local maxima\nof the susceptibility χ(B) =∂m\n∂B,χ(B)max. Between these\nlocal maxima, there is a local minimum [ χ(B)min] separating\nthe regions under the influence of the B(FSW, FPv )\nc,FRI critical point\nand that of the B(FSW, FPv )\nc,FP one.\nWe have calculated the critical properties of alternat-\ning ferrimagnetic chains in the presence of a magnetic\nfield from two spin-wave theories. We determine the bet-\nter low-energy description of the excitations, considering\nthe level of approximation, comparing the results with\nquantum Monte Carlo data. These ferrimagnetic chains\npresent two magnetization ( m) plateaus, the ferrimag-\nnetic (FRI) plateau, for which m=S−sand the fully\npolarized(FP)one,at m=s+S. Thefirstspin-wavethe-\nory, is an interacting spin-wave (ISW) approach with the\nFRI classical vacuum, ISW-FRIv. The second method-\nology, is a free spin-wave (FSW) calculation from the FP\nstate, FSW-FPv. In both cases, two bands are obtained.To calculate the finite temperature ( T) properties of the\nsystem, one of the bands is considered as a bosonic band,\nwithaneffectivechemicalpotentialtopreventbosoncon-\ndensation at B= 0; while the other is considered as a\nhard-corebosonband, with a fermionic one-particlether-\nmal distribution. Near the endpoint of the FRI plateau,\nthe ISW-FRIv theory is a better option; while the FSW-\nFPv is exact for T→0 near the endpoint of the FP\nplateau. Since we are interested in describing the whole\nTvs.Bphase diagram of the system, we deepen the\nstudy onthe FSW-FPv, calculatingthe finite Tcrossover\nlines bounding the plateau and the Luttinger liquid (LL)\nregimes.\nIn Fig. 8 we summarize our results in a Tvs.Bphase\ndiagram, and show specific heat data C/Tas a function\nofBandT. In the FRI and FP plateau regions the exci-\ntations are gapped, and ( C/T)→0 asT→0. The gaps\nclose at the quantum critical (QC) fields B(FSW, FPv)\nc,FRI= 2J\nandB(FSW, FPv)\nc,FP= 3J, andlocalmaximaappearsintheval-\nues ofC/Tfor a fixed T. These local maxima indicate\nthe crossover between the plateau and the QC regimes,\nand between the QC and LL regimes. As T→0, the\ncrossover line between the plateau and the QC regimes\n(P-QC line) is a straight line kBT(B) =|B−Bc|, for\nBc=B(FSW, FPv)\nc,FRI andBc=B(FSW, FPv)\nc,FP; while a straight\nlinea|B−Bc|, with a model-independent constant a=\n0.76238,marksthecrossoverbetweenLLandQCregimes\n(LL-QC lines). The LL-QC line which contains the crit-\nical point B=B(FSW, FPv)\nc,FRI[B=B(FSW, FPv)\nc,FP] was also cal-\nculated from local minima (local maxima) in the m(T)\ncurves:m(T)min[m(T)max]. The LL-QC lines were also\ncalculated from local maxima in the susceptibility curve\nχ(B) at fixed T:χmax(B).\nThe Luttinger liquid regime can be divided into two\nregions, separated by the minimum in the χ(B) curve\nwith a fixed temperature, χmin(B). The value of the\nmagnetic field at which this minimum occurs at T= 0,\nBi, is at the inflection point of the magnon band and\nchanges little with T. The line m(T)minas a function\nofBmeets the line χmin(B) forB≈Bi. 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Kubo, Physical Review 87, 568 (1952).\n56M. Takahashi, Prog. Theor. Phys. Suppl. 87, 233 (1986).\n57M. Takahashi, Physical Review Letters 58, 168 (1987).\n58M. Takahashi, Physical Review B 40, 2494 (1989)." }, { "title": "0802.3302v1.Microwave_spectral_analysis_by_means_of_non_resonant_parametric_recovery_of_spin_wave_signals_in_a_thin_magnetic_film.pdf", "content": "arXiv:0802.3302v1 [cond-mat.mtrl-sci] 22 Feb 2008APS/123-QED\nMicrowave spectral analysis by means of non-resonant param etric recovery of\nspin-wave signals in a thin magnetic film\nS. Sch¨ afer,1A.V. Chumak,1,2A.A. Serga,1G.A. Melkov,2and B. Hillebrands1\n1Fachbereich Physik and FSP MINAS,\nTechnische Universit¨ at Kaiserslautern,\n67663 Kaiserslautern, Germany\n2Departement of Radiophysics,\nTaras Shevchenko National University of Kiev, Kiev, Ukrain e\n(Dated: December 1, 2018)\nWe report on the storage and non-resonant parametric recove ry of microwave signals carried by a\ndipolar surface spin-wave pulse in a thin ferrimagnetic film . The information about the intensity of\nthe spectral components of the signal within a narrow freque ncy band is saved due to the excitation\nof a dipolar-exchange standing spin-wave mode across the fil m thickness and is afterwards restored\nby means of parametric amplification of this mode. The intens ity of the restored signal measured for\nvarying shifts between the signal carrier frequency and hal f of the pumping frequency, which is equal\nto the frequency of the standing mode, reveals information a bout the entire frequency spectrum of\nthe input microwave signal.\nPACS numbers: 75.30.Ds, 76.50.+g, 85.70.Ge\nIn ourrecent experiments[1, 2] a microwavesignalcar-\nried by a packet of dipolar spin waves propagating in a\nferrimagnetic film was stored in the form of spatially lo-\ncalized spin-wave excitations and restored thereafter by\nmeans of amplification of these excitations by paramet-\nric pumping. For those experiments, parametrical inter-\naction was performed by applying a microwave pumping\nwith a frequency νp= 2·νswhich is twice the carrier fre-\nquencyνsof the input microwave signal. Here we report\non the non-resonant case of parametric restoration with\nνp∝negationslash= 2·νs. This technique enables access to the spec-\ntral characteristics of the processes that underly both\nthe storage and the restoration phenomena. Therefore\nthese studies are relevant for the basic understanding of\ninteractions between different groups of spin waves and\nespecially the influence of the thermal magnon bath on\nthe parametric interactions as well as for technical appli-\ncations, using the ability to store, retrieve and process\nspectral information of microwave signals.\nThe experimental setup includes a long and narrow\n(30×1.2 mm2) spin-wave waveguide cut out of a thin\n(5.94µm) single crystal YIG film which is placed upon\nboth input and output microstrip antennae (see Fig. 1).\nThe signalmicrowavepulse excitesa packetofspin waves\nwhich propagates along the waveguide from the input to\nthe output antenna. The signal picked up by the out-\nput antenna is observed with an oscilloscope after am-\nplification and detection. The long axis of the YIG film\nwaveguide and therefore the propagation direction of the\nspin wave is perpendicular to the static magnetic field\nofH0= 1780.5 Oe applied in the film plane. Thus the\nexperimental geometry generally corresponds to the case\nof dipolar-dominated magnetostatic surface spin waves\n(MSSW) [3]. In addition, due to the finite thickness\nof the magnetic medium, discrete exchange-dominated\nstanding spin-wave modes (SSW) perpendicular to theFIG. 1: (Color online) a) Experimental setup of experiments\nreported on, consisting of a single crystal YIG film, input an d\noutput antennae and a dielectric resonator for the applicat ion\nofthepumpingmicrowave field. Thewaveforms schematically\ndepict the form of the input, pumping and restored pulse as\ncan be seen on the oscilloscope. b) Section of the spin-wave\ndispersion spectrum for a thin magnetic film schematically\nshowing the hybridization of MSSW (dashed line) with SSW\nmodes (cont. lines).\nfilm’s surface exist [4]. In crossing regions of SSW and\nMSSW modes, both magnon groups are interacting and\na hybridization takes place, leading to a dispersion spec-\ntrum as shown in the inset of Fig. 1. Thus, an excitation\nofSSWmodes by a bypassingMSSW packetoccurs. Due\nto the almost negligible group velocity of SSW modes,\ninformation encoded in the traveling wave is locally con-\nservedinthe magneticfilm andthereforecanberetrieved\nbymeansofparametricamplificationofthe SSWevenaf-\nterthetravelingspin-wavepulsehaslefttheareabetween\nthe antennae.\nIn order to amplify the SSW modes we used the par-\nallel pumping method [5]. The pumping magnetic field,\nwhich is parallel to the bias magnetic field, is created2\nFIG. 2: (Color online) Experimental results. Power of the re -\nstored pulse is depicted against the carrier frequency νsof the\ninput signal pulse. The dots represent the further discusse d\nmeasurement with signal pulse length of τs= 100 ns. As\ncomparison, the squares show the results for a pulse duratio n\nofτs= 50 ns, revealing a broader spectral distribution as is\nexpected from equ. (1).\nby the dielectric resonator attached to the waveguide in\nthe middle between the antennae (see Fig. 1). The res-\nonator is excited by a microwave pulse at a fixed carrier\nfrequency of νp= 14.258 GHz. This frequency was cho-\nsen to match the doubled frequency of one SSW mode in\norder to obtain the maximum efficiency of its parametric\namplification. In order to avoid a direct amplification of\nthe traveling MSSW the pumping pulse is only applied\nafter the spin-wave packet is detected at the output an-\ntenna and therefore has left the area of parametric in-\nteraction. In the experiments reported on, the pumping\npulse with a power of4 W had a duration of 7 µs and was\ndelayed by 400 ns with respect to the input pulse. As a\nresult of the pumping an additional, so-called ”restored”,\nMSSW pulse was received by the output antenna long\nafter the original delayed MSSW pulse as is indicated in\nthe waveform insets of Fig. 1. In general the intensity,\ndelay time and width of the restored pulse depend on\nthe intensity and delay time of the pumping as well as\non the frequency and intensity of the input signal pulse.\nIn Fig. 2 our experimental results on the intensity of the\nrestored pulse vs. the carrier frequency of the input sig-\nnal are presented. The measurements were performed for\nthe input microwave pulse durations of τs= 50 ns and\nτs= 100 ns. The results were obtained by keeping the\noutput voltage of the microwave detector constant while\nvarying the attenuation of the signal before it enters the\nlow noise microwave amplifier and semiconductor detec-\ntor. This was done in order to operate all relevant mi-\ncrowave components and the detector in a linear regime\nof operation and exclude all possible nonlinear influences\nof the experimental setup. One can see the dependence\nof the restored pulse intensity on the carrier frequency\nνsof the input signal pulse. The global maximum at\nνs,0= 7.129 GHz corresponds to half the pumping fre-\nquency as expected for the pure resonant case when the\ncarrier frequency of the MSSW packet matches the fre-quency of the parametrically amplified SSW mode. The\npositions of auxiliary maxima (as well as minima) are\ndirectly correlated with the Fourier spectra of the input\nsignals. Forexample,thedoublingofthefrequencywidth\nfor a pulse with half the duration is clearly visible. The\nreason for that correlation is the fact that only the in-\ntensity of spectral components of the signal pulse within\nthe frequency width of the parametricallyamplified SSW\nmode is relevant for the recovery process and determines\nthus the intensity of the restored pulse. As a result, the\nfrequency resolution of the process depends on the fre-\nquencywidth ofthe SSW mode, which ourcaseissmaller\nthan 1 MHz.\nThe absolute difference of the peak intensities for dif-\nferent signal pulse durations is caused by two effects.\nFirst of all, a longer MSSW pulse excites the SSW modes\nmore effectively due to the longer interaction time. Sec-\nond, since the delay was measured with respect to the\nsignal pulse rising edge, the time interval between the\nsignal and the pumping pulses is smaller for the longer\nsignal pulse, giving the SSW modes less time to loose\nenergy before the pumping sets in.\nIn order to explain the experimental results in detail\nwe now concentrate on the data obtained with a pulse\nduration of τs= 100 ns. In Fig. 3 the normalized experi-\nmental values (circular dots) are presented together with\nthe Fourier spectrum of the input signal pulse (dashed\nline)\nPs∝/bracketleftBig\nsin/parenleftBig\n(2πνs−2πνs,0)·τs\n2/parenrightBig\n/(2πνs−2πνs,0)/bracketrightBig2\n(1)\nnormalized with respect to the experimental values. In\nspite of clear similarities between these spectra one can\nsee that the relative intensities of the spectral compo-\nnents of the restoredpulse are significantly different from\nthe frequency spectra of the input pulse, especially the\nrange of output power is drastically reduced for the ex-\nperimentalvaluesobtainedfromtherestoredpulse. Thus\nno linear connection between the intensities of the re-\nstored and input signals exists.\nWe interpret this deviation using a simple model based\nupon a general theory of parametric interaction of spin\nwaves [6]. Two aspects are of particular interest for the\nexplanation of our experiments. First, the hybridization\nof MSSW and SSW modes and the creation of so-called\ndipole-exchange gaps reported on in [2] is the fundamen-\ntal precondition for the transfer of energy and informa-\ntion from the traveling spin wave to the standing modes\nand backagain, which enables us to storethe signalpulse\ninformation for some time. The second important aspect\nis the process of parametric amplification which is re-\nsponsible for the restoration of the stored signal. During\nthis process, two effects are important: the amplification\nof the standing spin wave and the amplification of other\nspin-wave modes from the thermal floor. A microwave\nmagnetic pumping field hpparametrically generates and\namplifies spin waves from the thermal floor as well as the\nSSW modes pre-excited by the signal spin-wave pulse,3\nFIG. 3: (Color online) Arrangement of experimental results\nfor a pulse duration of τs= 100 ns from figure 2 (circles) and\ntheoretical curve according to equation (1) (dashed line) f or\nthe inputpulse signal power against the signal frequency. T he\ncontinuous line shows the restored pulse power expected fro m\nequation 5 with δ= 0.77.\nincreasing their amplitudes with time as exp( hpVk−Γk)t\nand exp( hpVs−Γs)t, respectively [5]. The parameters\nVk,Vs, Γkand Γ sdenote the coefficients of parametric\ncoupling with the pumping field and the relaxation fre-\nquencies for the thermal magnons and the standing spin\nwaves, respectively.\nAccordingtoL’vov[6] however,due to the competition\nbetween the magnon groups only one dominating spin-\nwave group with the maximum gain factor of parametric\namplification is finally excited. This process can be de-\nscribedbythe conceptofaninternalpumpingfield gener-\nated by the dominating magnongroupand compensating\nthe external pumping field acting on the other magnon\ngroups (see [2, 6]). In our case the amplification factor\nhpVk−Γkof exchange spin waves with k≈105rad/cm\nis always higher than the amplification hpVs−Γsof the\nSSW [2, 5]. Therefore, no generation of standing spin\nwaves in the dipole-exchange gaps and consequently no\nrestored MSSW pulse is observed as long as no input\nsignal pulse is applied to the system.\nWhen a signal MSSW is exciting standing spin waves\nhowever, the pumping field amplifies the standing spin\nwaves from the level As,0which is significantly higher\nthen the initial amplitude of the dominating group deter-\nmined by the thermal level AT. Thus during some tran-\nsition times the suppressing influence of the dominating\ngroupremainssmall and amplification ofthe SSW modes\niseffective. Assoonastheamplitudeofthosedominating\nspin waves exceeds a critical threshold, the amplification\nof the standing spin waves and therefore the amplitude\nof the recovered pulse starts to be suppressed. This mo-\nment is associated with the maximum of the recovered\npulse, having an amplitude\nAs,max=As,0·/parenleftbiggAk,cr\nAT/parenrightbiggδ\n, (2)\nwhereAk,cris a critical amplitude of the thermalmagnons causing the increased damping and δis a pa-\nrameter defined by\nδ=hpVs−Γs\nhpVk−Γk. (3)\nThis relation describes a saturation behavior and decay\nfor parametrically amplified SSW due to the competi-\ntion with thermal magnons. However, for the case of\nthe constant thermal level the described model results in\na linear relation between the intensities of the restored\nand input signals. In order to interpret the experimen-\ntal data we supposed that the MSSW pulse irradiated\nby the input antenna is able to significantly increase the\nlevel of thermal spin waves within its spectral width due\nto two-magnonscattering processes [5]. We can therefore\nassumethat the levelofthethermalmagnonsisincreased\nby the signal pulse according to\nAT=AT,0+β·As,0≈β·As,0 (4)\nassuming β·As,0≫AT,0, where AT,0is the thermal\nmagnon level before application of the input signal. The\ncoefficient βdescribes the influence of the signal spin\nwave on the thermal magnon gas. Since As,0=/radicalbig\nPs,0,\nthe power of the restored pulse can be depicted as\nPr=c·P1−δ\ns,0, (5)\nwith the constant c= (Ak,cr/β)2δ.\nA linear regress of the restored signal power against\nthe input signal power reveals a value of 0.77 for the pa-\nrameter δin Eq. (5). A look at the continuous line in\nFig. 3, representing the input power (dashed line) multi-\nplied with α= 1−δ= 0.23 reveals the excellent accor-\ndance between the experimental data (circles) and our\nmodel, which predicts a linear dependency between in-\nput and restored signal intensities within the logarithmic\nscale. The remarkable feature visible from that curve is\nthe drastic increase in dynamic range of a device based\nupon the presented effects. The accessible range of mi-\ncrowave power detectable will be increased by the coeffi-\ncientαusing the logarithmic dB-scale.\nIn conclusion, the non-resonant restoration of mi-\ncrowavesignals by means of parametric pumping was ob-\nserved, basing on the storage of spin-wave information in\nthe hybridizedspin-wavespectraofathin YIG film. This\nis a first step of a deeper understanding of the process\nof restoration of spin-wave pulses by parametric inter-\naction. The simple phenomenological theoretical model\nproposed describes the experimental results very well.\nFurthermore, we propose to use the described process\nfor a spectrum analyzer, taking advantage of the range\nof the output power which is compressed by Pr=c·Pα\ns,0\nand thus increases the accessible dynamic range as this\nis usually limited by the restricted range of the analyz-\ners’detectingcomponent. FinancialSupportbytheDFG\nwithin the SFB/TRR 49 is gratefully acknowledged.4\n[1] G.A. Melkov, Yu.V. Kobljanskyj, A.A. Serga, A.N. Slavin ,\nV.S. Tiberkevich, Phys. Rev. Lett., 86, 4918-4921 (2001).\n[2] A.A. Serga, A.V. Chumak, A. Andr, G.A. Melkov, A.N\nSlavin, S.O. Demokritov, B. Hillebrands, Phys. Rev. Lett.,\n99, 227202 (2007).\n[3] R.W. Damon, J.R. Eshbach, J. of Appl. Phys. 31, 104\n(1960).[4] B.A. Kailinikos, A.N. Slavin, J. Phys. C 19, 7013 (1986).\n[5] A.G. Gurevich, G.A. Melkov, Magnetization oszillations\nand waves , CRC Press, New York (1996).\n[6] V.S. L’vov, Wave turbulence under parametric excitation ,\nin:Applications to magnetics , Springer, (1994)." }, { "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": "2305.00295v1.Weyl_metallic_state_induced_by_helical_magnetic_order.pdf", "content": "Weyl metallic state induced by helical magnetic order\nJian-Rui Soh,1Iri´an S´anchez-Ram ´ırez,2Xupeng Yang,1Jinzhao Sun,3Ivica Zivkovic,1J. Alberto\nRodr´ıguez-Velamaz ´an,4Oscar Fabelo,4Anne Stunault,4Alessandro Bombardi,5Christian\nBalz,6Manh Duc Le,6Helen C. Walker,6J. Hugo Dil,7, 8Dharmalingam Prabhakaran,3\nHenrik M. Rønnow,1Fernando de Juan,2, 9Maia G. Vergniory,2, 10and Andrew T. Boothroyd3\n1Institute of Physics, Ecole Polytechnique F ´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\n2Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San, Sebastian, Spain\n3Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom\n4Institut Laue-Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France\n5Diamond Light Source, OX11 0DE, United Kingdom\n6ISIS, Rutherford Appleton Laboratory - STFC, OX11 0QX, United Kingdom\n7Institute of Physics, ´Ecole Polytechnique F ´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\n8Spectroscopy of Quantum Materials Group, Paul Scherrer Institute, Villigen, Switzerland\n9IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain\n10Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany\n(Dated: May 2, 2023)\nIn the rapidly expanding field of topological materials there is growing interest in systems whose\ntopological electronic band features can be induced or controlled by magnetism. Magnetic Weyl\nsemimetals, which contain linear band crossings near the Fermi level, are of particular interest ow-\ning to their exotic charge and spin transport properties. Up to now, the majority of magnetic Weyl\nsemimetals have been realized in ferro- or ferrimagnetically ordered compounds, but a disadvantage\nof these materials for practical use is their stray magnetic field which limits the minimum size of de-\nvices. Here we show that Weyl nodes can be induced by a helical spin configuration, in which the\nmagnetization is fully compensated. Using a combination of neutron diffraction and resonant elastic\nx-ray scattering, we find that EuCuAs develops a planar helical structure below TN= 14.5 K which\ninduces Weyl nodes along the \u0000–A high symmetry line in the Brillouin zone.\nINTRODUCTION\nWeyl semimetals (WSMs) are crystalline solids charac-\nterized by points in momentum space where singly de-\ngenerate bands cross. These points, called Weyl nodes,\nare extremely robust against perturbations due to their\nnon-trivial topology. In recent years, WSMs have gar-\nnered significant attention from both theoretical and ex-\nperimental perspectives due to their potential to host\nrelativistic charge carriers which mimic the behavior of\nmassless fermions, and the associated exotic transport\nproperties [1–4]\nThe realization of a WSM requires the breaking of ei-\nther (or both of) inversion symmetry ( P) or time-reversal\nsymmetry (T). The latter approach is of particular inter-\nest because it provides a route to greater control over\nthe topology of electronic bands via magnetism or mag-\nnetic fields [5, 6]. Ferromagnetic WSMs are an obvi-\nous choice of system to work with, but their instability\nat small scales due to stray fields limits their potential\nfor downscaled applications [7–10]. Antiferromagnetic\n(AFM) WSMs are more promising in terms of stability,\nbut most AFM structures do not lift the double degener-\nacy of the bands because although collinear AFM order\nbreaksT, it usually preserves the combination P\u0002T ,\nwhich prevents any Dirac points from splitting into pairs\nof Weyl nodes.\nAn alternative approach is to look for AFM systemswith non-collinear spin arrangements that do not have\nEu\nCu As\na\n1d\n1b\n1a\n1b\n1c\n1\nc\n1\nc\n1\nHelical\n12c\n1A-AFM\n1\nJ1\n1J2\n1J0\n1DP-AFM\n1\nFigure 1. aThe hexagonal unit cell of EuCuAs is described\nby the centrosymmetric P63=mmc space group, with Cu-As\nlayers hosting topological fermions sandwiched between mag-\nnetic Eu layers. b–dPossible Eu spin configurations in Eu-\nCuAs: bA-type AFM order, with a magnetic propagation vector\nofqm= (0;0;0);cdouble-period antiferromagnetic structure\n(DP-AFM); dhelical spin arrangement (right-handed chirality\nis shown). The latter two structures possess a magnetic propa-\ngation vector of qm= (0;0;1\n2), i.e. with a doubling of the unit\ncell along the crystal caxis.arXiv:2305.00295v1 [cond-mat.str-el] 29 Apr 20232\nP\u0002T symmetry. An example is the Mn 3Xcompounds\n(X=Sn, Ge) which have a type of chiral 120\u000eAFM struc-\nture that supports Weyl nodes [11], but whose electronic\nbands near the Fermi level are strongly broadened and\nrenormalized and therefore difficult to probe experimen-\ntally [12]. On the other hand, the non-collinear config-\nurations of Eu spins in centrosymmetric materials such\nas EuCo 2P2[13], EuNi 2As2[14], EuZnGe [15] and Eu-\nCuSb [16] may provide a way to lift the band degen-\neracy without introducing significant electronic correla-\ntions. Sizable band splittings of order 0.1 eV are possible\ndue to the large exchange coupling between localized Eu\n4fstates and the conduction electrons [17]. Calcula-\ntion of the electronic structure of these Eu compounds\nby density functional theory (DFT) is problematic due\nto the large magnetic supercells, and up to now calcula-\ntions have either not been attempted or do not include\nthe full non-collinear incommensurate magnetic struc-\nture. Hence, a simpler system with a commensurate non-\ncollinear AFM order is desired for further investigation.\nIn this study we consider EuCuAs, which belongs to the\ncentrosymmetric Eu TX(T=Cu, Ag, Au; X=P, As, Sb,\nBi) family of materials [18–27] whose unit cell is shown\nin Fig. 1 a. We establish that below the N ´eel temperature,\nTN= 14:5K, the Eu moments are aligned ferromagneti-\ncally in the basal plane and exhibit a helical spin arrange-\nment along the caxis, with a magnetic propagation vec-\ntorqm= (0;0;\u001c),\u001c'0:5. The magnetic interactions\nare found to be highly two-dimensional, and the helix is\nmost likely stabilized by frustration in the out-of-plane\ncouplings. Our ab initio DFT calculations demonstrate\nthat the helical spin structure in EuCuAs generates Weyl\nnodes.\nRESULTS\nBulk properties\nIn Figure 2 a, we present the temperature-dependent\nsusceptibility of EuCuAs measured with applied field par-\nallel and perpendicular to the caxis. The peak observed\nin both field directions at TN= 14:5K indicates the onset\nof AFM order of the Eu magnetic sublattice. The suscep-\ntibility curves and ordering temperature are consistent\nwith earlier measurements reported in [21]. Below TN,\nit can be seen that the magnetic susceptibility along the\ncaxis (\u001fkc) is greater than that perpendicular to the c\naxis (\u001f?c) at low temperatures, suggesting that the Eu\nmagnetic moments lie in the abplane.\nFigure 2 bdisplays field-dependent magnetization\ncurves measured at T= 2 K for the two field direc-\ntions. When the field is applied along the caxis ( Bkc),\nthe magnetization increases smoothly and reaches sat-\nuration above a field of Bsat:\nkc'2:5T. This behavior is\nconsistent with an increasing tilt of Eu magnetic mo-\nc\n1d\n1a\n1b\n1\n2K\n115 K\n130 K\n1\nT=2K\n1B= 0.1 T\n1B⊥c\n1B⊥c\n1Figure 2. Magnetic and transport properties of EuCuAs.\nFigure 1: aMagnetic susceptibility curves for EuCuAs in the\nB?candBjjcconfigurations show an anomaly at TN=14.5 K.\nbMagnetization curves as a function of applied field along\nand perpendicular to cat fixed temperature of T=2 K. cTem-\nperature dependence of resistivity ( \u001axx) at various fixed field\nstrengths (jBj= 0, 2.5, 5 T; B?c).dMagnetic field dependence\nof\u001axxat various fixed temperatures ( T=2\u000030K).\nments out of the plane until full polarization is reached\nalong the field direction. The saturation magnetization,\nMsat= 7:0(1)\u0016Bf.u.\u00001, agrees well with the expected\nvaluegJJ\u0016Bfor a single divalent Eu2+ion ( 4f7,L= 0,\nS= 7=2andgJ= 2) per formula unit.\nIn the B?cfield configuration we observe a kink in the\nmagnetization at around Bt\u00180:3T, consistent with pre-\nvious measurements [21], which is not present in the Bkc\ndata (see Fig. 2 b). Beyond this point, the magnetization\nincreases rapidly and saturates at a field of Bsat\n?c'1.5 T.\nThe anomaly in the magnetization curve at Btsuggests\na metamagnetic transition in which there is a sudden re-\narrangement of the Eu moments within the abplane.\nTo investigate the relationship between charge trans-\nport and Eu magnetism in EuCuAs, we plot the resistiv-\nity (\u001axx) as a function of temperature for three different\nfields (Fig. 2 c). In the absence of a magnetic field, we\nobserve that \u001axxincreases on cooling and reaches a max-\nimum atTN, before dropping sharply as the temperature\napproaches 2 K. We also find that the resistivity peak can\nbe suppressed by an applied magnetic field, as shown\nin the measurements at B= 2:5T and 5 T (Fig. 2 c,d).\nThese observations suggest that the enhanced resistivity\natTNis due to the scattering of charge carriers from co-\noperative Eu spin fluctuations, which start to build up at\ntemperatures of order 50 K and reach a maximum ampli-3\na\n1f\n1Pij\n1Phelix\nij\n1Pobs .\nij\n1\n2K\n1(1,0,1.5)\n1(1,0,4)\n1b\n1c\n1g\n1h\n1e\n1d\n1PND\n1REXS\n1ND\n1ND\n1ND\n1SNP\n1ND\n1(1,0,1.5)\n1τ=0 .5\n1τ=0 .59\n1τ=0 .42\n1REXS\n12K\n18K\n18K\n120 K vs2K\n117 K vs2K\n100L (r.l.u.)\n1\nFigure 3. Unravelling the magnetic order of EuCuAs with neutrons and x-rays. a Powder neutron diffraction (PND). Red\nand blue tick marks indicate structural and magnetic Bragg peaks. bResonant elastic X-ray scattering (REXS). The peaks at\nL= 1:58and2:42are magnetic Bragg peaks. cSingle-crystal neutron diffraction (ND). The peaks at L= 0:5and1:5are magnetic\nBragg peaks. dIntegrated intensity of the Q=(1;0;1:5)magnetic reflection measured with ND. eRefinement of single-crystal ND\ndata, comparing the observed and calculated magnetic Bragg peak intensities for the helical structure. fComparison of observed\n(Pobs:\nij) and calculated ( Phelix\nij) full polarization matrices for four magnetic reflections. For each reflection, the nine elements of\nthe polarization matrix Pijfrom left to right correspond to ij=xx,xy,xz,yx,yy,yz,zx,zyandzz. The data were recorded at\nT= 2K.g,hField dependence of peaks observed with REXS and ND, respectively.\ntude atTN. Spin fluctuations are quenched by a magnetic\nfield comparable to Bsat, and this likely accounts for the\nreduction in the resistivity at TNwith field observed in\nFig. 2 c.\nThe\u001axxmeasurements as a function of field applied\nperpendicular to the c-axis are plotted in Fig. 2 d. At\nT= 2K, the resistivity increases up to Bt= 0:3T, above\nwhich the resistivity decreases sharply with field. As the\ntemperature is increased, the low-field anomaly in the\nresistivity becomes washed out and is completely sup-\npressed at temperatures above TN. Negative magnetore-\nsistance is observed at all temperatures for fields above\nBsat(Fig. 2 d).\nMagnetic diffraction with neutrons and x-rays\nWe investigated the magnetic structure of the Eu\nmoments below TNin EuCuAs using powder neu-\ntron diffraction (PND), single crystal neutron diffraction\n(ND), and resonant X-ray magnetic scattering (REXS),\nas shown in Fig. 3. The PND measurement (Fig. 3 a)\nrevealed magnetic diffraction peaks below TNwhich\ncould be indexed with a magnetic propagation vectorqm= (0;0;\u001c)with\u001c= 0:591(1) reciprocal lattice units\n(r.l.u.). This means that the Eu spins align ferromag-\nnetically within the hexagonal planes and have an in-\ncommensurate period along the caxis. The refinements\ngave good agreement with the data for the proper screw\n(planar helix) structure shown in Fig. 1 d, in which the\nspins lie in the abplane and rotate around the caxis from\nlayer-to-layer. The statistics place an upper bound of 3\u000e\non any out-of-plane tilt.\nWe also found a magnetic propagation vector of the\nform qm= (0;0;\u001c)in the REXS study of a EuCuAs sin-\ngle crystal, but this time we observed \u001c= 0:42(1) r.l.u.,\nFig. 3 b. We note that because there are two Eu layers per\nunit cell, magnetic Bragg peaks are observed at positions\n(H;K; 2L\u0006\u001c), whereH;K;L are integers.\nThe different values of \u001cfound in the PND and REXS\nmeasurements indicate that the period of the helix can\nvary from sample to sample, perhaps due to small dif-\nferences in composition. This finding was reinforced in\nthe single-crystal ND study in which we used a differ-\nent single crystal sample and found yet another value,\n\u001c= 0:50(1) . Figure 3 dshows the temperature depen-\ndence of the integrated intensity of a peak observed at\nq= (1;0;1:5), which demonstrates an order-parameter-4\nlike dependence below TN= 14:5(5)K. This temperature\ncoincides with the peak in the susceptibility (Fig. 2 a),\nand also agrees with TNvalues found in the PND and\nREXS measurements, confirming the magnetic origin of\nthe\u001c= 0:5family of peaks.\nSymmetry analysis for the paramagnetic group\nP63=mmc with propagation vector qm=\u0000\n0;0;1\n2\u0001\nshows\nthat the magnetic group formed by the Eu spin compo-\nnents decomposes into two irreducible representations\n(irreps), \u0000 = \u0000 2+ \u00003. Of these, \u00002describes a pair of\nlongitudinal spin-density wave structures with the spins\npointing along the caxis. This structure can immedi-\nately be ruled out because magnetic peaks of the form\n(0;0;2L\u00061\n2)were observed. The \u00003irrep describes four\ndomains of the planar helix shown in Fig. 1 d. We also\nconsidered the DP-AFM structure shown in Fig. 1 c. This\nstructure is not described by one of the irreps, so based\non Landau’s theory it is not expected to form in a contin-\nuous magnetic phase transition. Nevertheless, we con-\nsider it because after averaging over an equal population\nof domains related by \u0006120\u000ein-plane rotations of the Eu\nmoments about the caxis the magnetic Bragg peak in-\ntensities are identical with those of the helical structure.\nAdditionally, the DP-AFM structure with canted spins is\nfound to be stabilized by an in-plane magnetic field (see\nbelow), so it is important to rule out this structure in zero\nfield.\nThe single-crystal ND data in zero field are described\nwell by both structures assuming equal domain popula-\ntions, with \u001f2\nr= 6:51(\u001f2\nris the usual goodness-of-fit\nstatistic normalized to the number of degrees of free-\ndom). Figure 3 eshows the good agreement between the\nobserved and calculated peak intensities for the helical\nstructure. However, the large shape-dependent correc-\ntions needed to account for the severe neutron absorp-\ntion of Eu in the sample (absorption cross-section 4,530 b\nat 1.8 ˚A wavelength) make a more detailed structural\nanalysis from this data unreliable.\nTo provide a more stringent investigation of the zero-\nfield magnetic order in the EuCuAs sample with \u001c= 0:5\nwe turned to spherical neutron polarimetry (SNP). In\nthe SNP technique [28], the information on the mag-\nnetic structure is obtained from the polarization matrix\nPij. The components of Pijare determined from inten-\nsity ratios (see Methods) and so do not depend on sam-\nple attenuation, which is an advantage for samples with\nstrong neutron absorption like EuCuAs. As with unpo-\nlarized neutron diffraction, SNP cannot distinguish be-\ntween the helical and DP-AFM models when the latter is\naveraged over equal populations of equivalent domains.\nHowever, a sample that is field-cooled through TNinto\nthe DP-AFM phase is expected to preferentially contain\ndomains in which the spins lie perpendicular to the field.\nWith this in mind, we cooled the sample from 25 K to\n2 K in a field of 1 T applied parallel to the baxis, before\nremoving the field, inserting the cryostat into the SNPdevice and measuring the Pijat ten magnetic peak posi-\ntions in zero field. A fit to a single domain of the DP-AFM\nstructure in which all spins are perpendicular to the field\ngave a poor agreement with \u001f2\nr= 649 . Allowing the\ndomain populations to vary, we found that the data con-\nstrain the DP-AFM domain imbalance to be less than 5%.\nIn contrast, a refinement of the helical magnetic struc-\nture against the Pijgave an excellent fit, with \u001f2\nr= 1:85,\nas illustrated in Fig. 3 ffor four of the reflections.\nWe conclude, therefore, that the magnetic structure\nof EuCuAs in zero or very small fields is a planar he-\nlix with a (sample-dependent) period of approximately\nfour Eu layers, as shown in Fig. 1 d. Next, we investigate\nthe metamagnetic transition observed when a magnetic\nfield is applied perpendicular to the caxis (Fig. 2 b). Fig-\nure 3 gshows that with increasing field the incommen-\nsurate (0;0;1:58)peak observed at T= 8K in the REXS\nexperiment changes in position very slightly at first, but\nthen jumps discontinuously at a field of B'0:1T to the\ncommensurate position (0;0;1:5). The ND crystal has a\ncommensurate propagation vector already at zero field,\nbut with increasing field we observe a switch in the mag-\nnetic scattering intensity from L=half-integer to L=\ninteger positions, as illustrated in Fig. 3 h. At fields above\nabout 0.2 T the intensity of the (1;0;1:5)reflection de-\ncreases sharply, while the (1;0;4)reflection increases.\nThe crossover occurs at a field of around 0.3 T at T= 2K,\nwhich corresponds roughly with the metamagnetic tran-\nsition field Bt, see Fig. 2 b. The lower value of Btob-\nserved by REXS is most likely explained by the higher\ntemperature of the sample during the measurement.\nTo clarify the arrangement of the Eu moments above\nthe metamagnetic transition field, a total of 316 reflec-\ntions (both integer and half-integer L) were measured at\nB= 0:4T. The refinement of the model to the observed\nreflections reveals that at this field the Eu moments form\na canted DP-AFM structure, which can be considered as\nthe sum of collinear DP-AFM order and FM order. The\nDP-AFM component is a single domain with the Eu spins\nperpendicular to the field. The integer reflections indi-\ncate a FM component along the direction of the applied\nfield ( Bkb). Therefore, considering our results in zero\nfield and in 0.4 T we conclude that at Bt\u00180:3T the Eu\nmoments reorient within the abplane from the helical to\na canted DP-AFM structure.\nAt higher fields ( B\u00150:8T), the integrated intensity of\n(1;0;1:5)peak remains zero while the (1;0;4)intensity\ncontinues to increase, eventually saturating at B'1:5T,\nwhere the magnetization also plateaus (see Fig. 2 b).\nThus, the behavior of the magnetic intensities above Bt\nis consistent with the gradual canting of the Eu moments\nalong the direction of the applied field into a fully polar-\nized state.5\na\n1b\n1\nEF\n1\n5K\n1\n5K\n1\n5K\n1Binding Energy (eV)\n1\nkx(˚A-1)\n1ky(˚A-1)\n1\nBinding Energy (eV)\n1EF\n1\n0\n10\n10\n10\n10.5\n10.5\n1-0.5\n1-0.5\n10\n10.4\n10.2\n10.5\n1-0.5\n10\n1kx(˚A-1)\n1kx(˚A-1)\n1\n5K\n10\n10.2\n10.2 eV\n10.1\n1\n0.2 eV\n10.4\n10.4\n1-0.4\n1-0.4\n1c\n1d\n1f\n1ky(˚A-1)\n1\n0\n10.4\n1-0.4\n1\nΓ\n1\nK\n1\nM\n1\nΓ\n1\nK\n1\nM\n1kx(˚A-1)\n1\nHelical\n1Binding Energy (eV)\n10\n10.4\n10.2\n10.5\n1-0.5\n10\n1kx(˚A-1)\n1A\n1Γ\n1K\n1kx\n1ky\n1kz\n1M\n1\nA\n1Γ\n10\n10.2\n10.3\n10.1\n1Binding Energy (eV)\n1\nA\n1Γ\n1K\n1M\n1Γ\n1A\n1-0.2\n1-0.3\n1-0.5\n1-1.0\n1-2.0\n1H\n1L\n1-0.1\n1Binding Energy (eV)\n10\n1e\n1g\n1i\n1j\n1k\n1h\n10.5\n1-2.5\n1-1.5\n1\nHelical\n1\nH\n1L\n1\nHelical\n1DP-AFM\n1\n0\n1kx(˚A-1)\n10.4\n1-0.4\n1ky(˚A-1)\n1\n0\n10.4\n1-0.4\n10\n1kx(˚A-1)\n10.4\n1-0.4\n1ky(˚A-1)\n1\n0\n10.4\n1-0.4\n1\nHelical\n1\nHelical\n1Binding Energy (eV)\n1\nkx(˚A-1)\n1ky(˚A-1)\n10\n10\n10\n10.5\n10.5\n1-0.5\n1-0.5\n10.2\n10.1\n1\nΓ\n1\nK\n1\nM\n1\nΓ\n1\nK\n1\nM\n1\nHelical\n1\nFigure 4. Comparison between the calculated and measured electronic band structure of EuCuAs. a -gAb-initio electronic\nstructure calculations. aDispersion along high symmetry lines with helical magnetic order. bComparison between the dispersion\nalong \u0000-A for a helical magnetic configuration (solid line) and DP-AFM structure (dotted line). cHexagonal Brillouin zone of\nEuCuAs. d-gCross-section of the three-dimensional band structure, dispersion along K \u0000\u0000\u0000K, and constant energy maps at EF\nand 0.2 eV, respectively. An upward shift of 0.4 eV has been applied to the calculated bands shown in order to match the ARPES\ndata. The folded bands near \u0000at about\u00000:5eV (unshifted energy) have very little spectral weight and are omitted for clarity (see\nSupplementary Information). h-kmeasured ARPES spectrum at T= 5K.\nElectronic band structure\nTo clarify the nature of the charge carriers in EuCuAs\nbelowTN, we plot in Fig. 4 athe electronic band structure\ncalculated with a commensurate period-4 ( \u001c= 0:5) heli-\ncal arrangement of the Eu magnetic moments (Fig. 1 d).\nOur calculations indicate that the bands that give rise\nto charge transport (i.e. close to the Fermi energy, EF)\nare dominated by states with Cu and As character, while\nthe Eu 4fbands which are responsible for magnetism\nreside\u00181.2 eV below EF(see Supplementary Informa-\ntion). The bands are singly degenerate throughout the\nhexagonal Brillouin zone (Fig. 4 c), with the spin-splitting\ndriven by the chiral spin configuration which breaks all\nof the mirror symmetries of the P63=mmc space group.\nAs such, band crossings can be topologically non-trivial\nand can host Weyl fermions if the nodes are close to the\nFermi energy ( EF). Figure 4 bshows the electron energy\ndispersion along kzfrom the \u0000to A high symmetry points\nof the hexagonal Brillouin zone. We find two Weyl cross-ings that are created by the helical magnetic order, lo-\ncated at the A high-symmetry point about 0.18 eV above\nand belowEF.\nAs a comparison, we also plot in the same energy–\nmomentum window the calculated electronic band struc-\nture with the DP-AFM configuration which we have con-\nsidered but ruled out in the previous section (Fig. 4 b).\nUnlike with the the helical Eu magnetic order, the elec-\ntronic bands in the DP-AFM structure do not manifest\nWeyl points at the A point because the bands that cross\nare doubly degenerate. The Kramers degeneracy is pre-\nserved because the DP-AFM structure possesses mz\u0002T\nsymmetry which leaves kzinvariant, where the mirror\nmzis the CuAs plane between two neighboring Eu atoms\nwith parallel spins.\nIn order to validate our ab initio calculations of the\nelectronic bands, we compare them with angle-resolved\nphotoemission spectroscopy (ARPES) measurements. We\nfind very good agreement providing the calculated bands\nare shifted up in energy by approximately 0.4 eV, indicat-6\ning that this sample is slightly hole-doped. Figure 4 d\nplots the three-dimensional cross-section of the calcu-\nlated electronic dispersion, which shows two concentric\nconical bands centered at the \u0000point, in excellent agree-\nment with the measured spectrum obtained at T= 5 K\n(Fig. 4 h). The calculated dispersion along the K\u0000\u0000\u0000K\nhigh-symmetry direction reveals two pairs of linearly dis-\npersing bands (Fig. 4 e), which we also find in the mea-\nsured spectrum in Fig. 4 i.\nMoreover, the calculated Fermi surface (binding en-\nergyE= 0) displays two concentric circular hole-like\npockets centered at the \u0000point (Fig. 4 f). The constant-\nenergy cut at E= 0:2eV, shown in Fig. 4 g, also has two\nconcentric circular bands but are larger than that at the\nFermi energy. Correspondingly, the measured spectrum\natEFand at 0.2 eV (Fig. 4 j,k) are in good agreement\nwith the dispersion of the calculated bands.\nSpin Hamiltonian\nIn this section we shall develop an approximate spin\nHamiltonian to describe the magnetic behavior of Eu-\nCuAs. Because the spins on Eu are large ( S=7\n2) and\nlocalised we expect that a semiclassical mean-field treat-\nment should account for the static magnetic properties,\nand linear spin-wave theory should give a good descrip-\ntion of the magnetic excitations.\nHelical magnetic structures in which the spins align\nferromagnetically within the layers but rotate from layer-\nto-layer around the axis of the helix are not uncom-\nmon. In materials with inversion symmetry, such that\nthe Dzyaloshinskii–Moriya interaction is absent, they are\nusually considered to arise from competing exchange in-\nteractions. The simplest models are based on the Hamil-\ntonian\nH=X\nhiji\u0000JijSi\u0001Sj+D(Sz\ni)2+g\u0016BSi\u0001B;(1)\nwhich has been analysed in great detail [29–34]. In the\nHeisenberg coupling term it will be sufficient to include\nthree exchange constants, J0, which couples neighbor-\ning spins in the abplane, and J1andJ2, which couple\nnearest and next-nearest neighbor spins along the caxis\n(Fig. 1 a). WithD > 0, the second term ensures easy-\nplane anisotropy, as observed. Later we discuss the pos-\nsibility of small additional terms to H.\nWith zero applied magnetic field ( B= 0), it is well\nknown that the ground state magnetic structure obtained\nfrom Eq. (1) in the mean-field approximation is a planar\nhelix with turn-angle between spins of\n\u0012= cos\u00001\u0012\n\u0000J1\n4J2\u0013\n: (2)\nThe helix is the stable solution for J2<0andjJ1j<\nj4J2j. In EuCuAs, which has two Eu layers per unit cellTable I. Estimated exchange and anisotropy parameters (in\nmeV) for EuCuAs. The two sets of parameters are for samples\nwith different incommensurate propagation vectors \u001c.J0was\ndetermined experimentally for the \u001c= 0:59sample only.\n\u001c J 0 J1 J2 D\n0.42\u0000 0.021\u00000:021 0.009\n0.59 0:060(3)\u00000:0077\u00000:0069 0.011\nalong thecaxis, the propagation vector of the helix is\nqm= (0;0;\u001c)r.l.u. with\u001c=\u0012=\u0019.\nIn general, the behavior of a planar helix in a field ap-\nplied perpendicular to the axis of the helix can be com-\nplex, with several different phases predicted as a func-\ntion of field and helical turn-angle [29–31, 33, 34]. How-\never, for\u0012'\u0019=2, as established here in EuCuAs, it was\nfound that a transition takes place at an intermediate\nfieldBtto a canted double-period AFM structure [30].\nIn other words, at Btthe propagation vector jumps to\nqm= (0;0;1\n2)corresponding to a turn-angle of \u0019=2. This\nis precisely what we have observed in EuCuAs, Fig. 3 g.\nBy applying the mean-field approximation to the Hamil-\ntonian (1), and neglecting any distortion of the ideal he-\nlix forB?c0, which favours a turn-angle of \u0019=2(Ref. 35); (3)\nmetallic (RKKY) exchange.\nNext, we determine the in-plane nearest-neighbour\nexchange parameter J0from the spin-wave spectrum.\nThe inelastic neutron scattering spectrum is reported in\nFig. 5 a, measured on the same sample as used for the\nneutron powder diffraction study, Fig. 3 a. Despite the\npunitive neutron absorption of Eu, the spin-wave scatter-\ning signal is clearly observable in the neutron scattering\ndata, especially for momentum transfers Qbelow about\n1˚A\u00001where a sharp dispersive mode is observed.\nWe simulated the neutron scattering intensity by\npowder-averaging the theoretical expressions obtained\nfrom Eq. (1) by linear spin-wave theory (see Methods).\nWith the Hamiltonian parameters J1,J2andDobtained\nabove (Table I) we find that the inter-layer magnon dis-\npersion has a band width of only 0.1 meV, and so can-\nnot be resolved from the strong elastic scattering that ex-\ntends in energy up to about 0.2 meV (see Fig. 5 a). There-\nfore, the measured spectrum consists almost entirely of\nthe in-plane magnon dispersion.\nThe simulation displayed in Fig. 5 b, performed with\nnearest-neighbour exchange parameter J0= 0:060meV\nand other parameters as given in Table I, is seen to agree\nwell with the observed spectrum. Inclusion of in-plane\nexchange couplings to spins beyond the nearest neigh-bors did not improve the agreement. Our analysis in-\ndicates that the magnitude of the next-nearest-neighbor\nexchange is less than 10% of J0.\nThe modelling presented in this section has found that\nthe nearest-neighbor intra-layer exchange coupling J0is\nbetween 3 and 8 times larger than the nearest- and next-\nnearest-neighbor inter-layer couplings J1andJ2(see Ta-\nble I). Moreover, each Eu spin is coupled to six other\nspins byJ0, and to only two other spins by J1and\nJ2. The magnetism in EuCuAs, therefore, is highly two-\ndimensional. Given this, it is perhaps surprising that\nthe magnetic order is so well correlated in the out-of-\nplane direction, as reflected in the narrow widths of the\npeaks inLscans — see Figs. 3 bandc. We estimate the\ncorrelation length along the caxis to be approximately\n100˚A. This suggests that the material is relatively free\nfrom chemical disorder of the kind that might disrupt\nthe propagation of the magnetic structure.\nDISCUSSION\nOur measurements and analysis provide strong evi-\ndence that the Eu spins in EuCuAs adopt a helical spin\nstructure in zero field below TN= 14:5K. The mag-\nnetic propagation vector qm'(0;0;0:5)is very differ-\nent from the propagation vector qm= (0;0;0)of the A-\ntype AFM structure (Fig. 1 b) which had previously been\nassumed [21] and which is often observed in hexago-\nnal layered Eu compounds, e.g. Refs. [36–41]. At the\nsame time, helical or helical-like magnetic phases have\nbeen observed in a number of other layered Eu com-\npounds, e.g. Refs. [13–16, 42–45]. As it is difficult to\ndistinguish helical and AFM order from bulk magneti-\nzation measurements or ab initio calculations it is im-\nportant to establish the magnetic structure more directly\nfrom a microscopic probe of the spin configuration, such\nas neutron or resonant magnetic x-ray diffraction.\nWe have found that the magnetic order in EuCuAs has\na profound effect on the electronic band topology. The\nP63=mmc space group of EuCuAs has inversion symme-\ntry, so in the paramagnetic phase the bands are doubly\ndegenerate. In the helical phase, however, inversion and\ntime-reversal symmetries are lost, and as a result the\nbands become singly degenerate and the band crossings\nalong \u0000–A can be Weyl nodes.\nIn EuCuAs, therefore, we have an unusual state of af-\nfairs in which a topological phase transition is driven by a\nmagnetic transition from a paramagnet to a helical mag-\nnetic order. This differs from most other known magnetic\nWeyl semimetals, which are induced by ferromagnetism\nor by externally applied magnetic fields [5]. Recently,\nweak helimagnetism was reported in NdAlSi, which is\nalso a magnetic Weyl semimetal [46]. The dominant\ninteractions in NdAlSi favour ferrimagnetic order, but it\nis proposed that Weyl fermions modify the coupling be-8\ntween Nd ions and generate Dzyaloshinkii–Moriya and\nKitaev-type interactions which are responsible for the\nweak chiral tilting of the moments away from the easy di-\nrection. Hence, the interplay between helical magnetism\nand electronic topology is very different in the two mate-\nrials. In EuCuAs, helical magnetism generates the Weyl\nstate, whereas in NdAlSi, the Weyl state generate helical\nmagnetism.\nThe main outcome of this work is the discovery that\nhelical magnetic order induces Weyl nodes in EuCuAs.\nThis finding expands the range of systems in which mag-\nnetic order influences the electronic band topology. Of\ncourse, EuCuAs is far from ideal because the response of\nthe Weyl fermions is complicated by the presence of sev-\neral topologically trivial bands at EF. It would be inter-\nesting, therefore, to find a simpler system in which study\nthe interplay between Weyl fermions and a subsystem of\nhelically ordered local moments.\nMETHODS\nCrystal growth. Single crystalline EuCuAs was grown\nby the self-flux method, as described in Ref. [21]. The\nquality and structure of the single crystals was checked\nwith laboratory x–rays on a 6–circle diffractometer (Ox-\nford Diffraction) and Laue diffractometer (Photonic Sci-\nence). Laboratory single crystal x-ray diffraction con-\nfirmed the hexagonal Ni 2In-type structure (P63=mmc ).\nMagnetization. Magnetization measurements were\nperformed on a Physical Properties Measurements Sys-\ntem (PPMS, Quantum Design) with the vibrating sam-\nple magnetometer (VSM) option. The temperature-\ndependent magnetometry measurements were per-\nformed in the temperature range 2\u0014T\u001450K in an\nexternal field of B= 0:1T in two different field configu-\nrations, with Bparallel and perpendicular to the crystal\ncaxes, respectively. The field-dependent measurements\nwere performed at T=2K in fields up to 5 T applied in\nthe same two field directions, that is BkcandB?c.\nMagnetotransport. Magnetotransport measurements\nwere performed on the PPMS with the resistivity option,\nto shed light on the coupling between charge transport\nand the different spin-configurations. The field was ap-\nplied perpendicular to the crystal caxis, in field strengths\nup to\u00160H= 5T and temperatures down to T= 2K.\nPowder neutron diffraction. The polycrystalline sam-\nple used for neutron powder diffraction and inelastic\nneutron scattering was prepared from single crystals\nwhich were crushed and finely ground. Measurements\nwere made on the WISH diffractometer at the ISIS Neu-\ntron and Muon Facility. A mass of 1.3 g of powder was\ncontained in a thin-walled cylindrical can of diameter\n3 mm, made from vanadium. The can was mounted\ninside a helium cryostat. Data were recorded at tem-\nperatures between 20 K and 2 K. Magnetic Bragg peakswere observed at temperatures below TN= 14:5\u00060:5K\n(see Supplementary Information). Rietveld profile re-\nfinement was performed with the FULLPROF software\nsuite [47]. Data from detector banks 2 to 4 were used\nfor the refinements. The refined lattice parameters at\n20 K in the space group P63=mmc were found to be\na=b= 4:2331(1) ˚A andc= 8:2354(2) ˚A. The Lobanov –\nalte da Veiga function was used to model the absorption\nfor the cylindrical sample, but due to the severe neutron\nabsorption of Eu (absorption cross-section \u001babs= 4;530b\nat wavelength \u0015= 1:8˚A) the absorption correction is not\nsufficiently accurate to give reliable thermal parameters,\nsite occupancies or magnetic moment sizes (see Supple-\nmentary Information).\nSingle-crystal neutron diffraction. Single crystal\nneutron diffraction was performed on the D9 diffrac-\ntometer at the Institut Laue–Langevin (ILL). Hot neu-\ntrons (\u0015= 0:84˚A) were used to reduce the absorp-\ntion due to Eu. To refine the structure in zero field\nwe collected at total of 229 reflections at a tempera-\nture of 2 K with the sample mounted in a four-circle\ncryostat. The full dataset comprised 109 structural re-\nflections consistent with the P63=mmc space group, and\n120 corresponding to the magnetic propagation vector\nqm= (0;0;0:5). After averaging symmetry-equivalent\nreflections, there were a total of 38 unique structural re-\nflections and 40 unique magnetic reflections. Magnetic\nstructure models were refined using the Mag2Pol soft-\nware [48]. Attenuation corrections to the peak intensi-\nties (due to the strong neutron absorption of Eu) were\ncalculated according to the length of the neutron path\nthrough the crystal, whose shape and dimensions were\nmeasured for this purpose. To study how the magnetic\nstructure of EuCuAs evolves with an applied magnetic\nfield, we performed single crystal neutron diffraction at\nvarious field strengths up to B= 2:5T in a vertical field\ncryomagnet. The crystal was mounted with the baxis\nvertical (parallel to the field direction) so that h0lreflec-\ntions were accessible in the horizontal scattering geome-\ntry.\nPolarized neutron diffraction. Polarized neutrons\nof wavelength 0.83 ˚A were employed on the D3 diffrac-\ntometer (ILL) in the spherical neutron polarimetry (SNP)\nset-up with the Cryopad device [49]. The incident beam\nwas polarized by Bragg reflection from the (111) planes\nof a crystal of ferromagnetic Heusler alloy (Cu 2MnAl).\nAn erbium filter was placed in the incident beam to sup-\npress half-wavelength contamination. Nutator and pre-\ncession fields were used to control the direction of the\nbeam polarization. A3He spin filter was used to analyse\nthe polarization of the scattered beam. Standard correc-\ntions for time decay of the filter efficiency were applied\nbased on measurements of the (102) structural Bragg\npeak. The polarization matrix elements Pijare defined9\nas\nPij=Nij\u0000Ni\u0016j\nNij+Ni\u0016j;\nwhereNijandNi\u0016jare the number of counts (at a\nBragg reflection) when the incident neutron polarization\nis alongiand the scattered polarization is measured par-\nallel and antiparallel to j, respectively, with i;jbeing\nthe principal directions x;y;z . The direction xis defined\nas the direction along the scattering vector Q,zis per-\npendicular to the scattering plane, and ycompletes the\nright-handed set.\nResonant elastic x-ray diffraction. The REXS mea-\nsurements were performed at the I16 beamline, Dia-\nmond Light Source. The incident x-ray photon energy\nwas tuned to the Eu L3edge so as to benefit from the\nresonant enhancement of the scattered x-ray intensity\nfrom the Eu2+ions. For the zero-field measurements,\nthe diffractometer was set to the vertical geometry and\nthe\u001b!\u00190scattering channel, to be mainly sensitive to\nmagnetism which can rotate the linear polarization of\nthe incident x-rays. For the magnetic field dependent\nmeasurements, the diffractometer was set to the horizon-\ntal scattering configuration to accommodate the vertical\nfield magnet. The \u0019!\u001b0scattering channel was adopted\nto suppress the charge scattering and enhance the mag-\nnetic scattering.\nDensity functional theory. To clarify the topological\nnature of the electronic band structure in EuCuAs and\nhow it evolves in a magnetic field, we performed den-\nsity functional theory (DFT) calculations of the electronic\nband structure using VASP [50, 51] v.6.2.1. Projector-\naugmented wave pseudopotentials in the generalized\ngradient approximation with Perdew Burke Ernzerhof\nparametrization (PBE) [52] were used. Relativistic\npseudo-potentials were used in the calculations to ac-\ncount for the large spin-orbit coupling (SOC) arising\nfrom the heavy As ions which might lead to band in-\nversion, with a kinetic energy cutoff of 480eV. Further-\nmore, a Hubbard U= 5:0eV was used to model the\nstrong electron-electron correlations and reproduce the\nobserved binding energy of the localized Eu 4fbands\n(see Supplementary Information). A Monkhorst–Pack k-\npoint sampling mesh of 9\u00029\u00027was used [53].\nAngle-resolved photoemission spectroscopy. ARPES\nof EuCuAs was performed on the SIS-ULTRA beamline at\nthe Swiss Light Source (SLS). To determine the \u0000and\nA high symmetry point, we performed a kzdependent\nscan by varying the incident photon energy from 50 eV\nto 150 eV. Samples were cleaved at T\u001815 K under high\nvacuum (\u001810\u00008Torr) and measured at various temper-\natures between 5 K and 22 K at incident photon energy of\n74 eV. Vertical, horizontal, left-handed and right-handed\ncircular incident photon polarization was used. Spectra\nobtained at different temperatures and in each polariza-tion channel are reported in the Supplementary Informa-\ntion.\nInelastic neutron scattering. Measurements were\nperformed at the ISIS Neutron and Muon Facility, U.K.,\non the same 1.3 g polycrystalline sample as used for neu-\ntron powder diffraction. Initial measurements were per-\nformed on the Merlin chopper spectrometer with inci-\ndent neutron energies between 8 and 100 meV. The spec-\ntra established that the magnetic excitation spectrum did\nnot extend beyond 2 meV in energy. Subsequent mea-\nsurements of the same sample were made on the cold-\nneutron chopper spectrometer LET with neutron incident\nenergies of between 2 and 9 meV. The data presented in\nFig. 5 were recorded on LET with 3.7 meV incident neu-\ntrons. During the run, the sample was maintained at a\ntemperature of 2 K in a pumped helium cryostat.\nSpin-wave spectrum . The magnetic excitations of the\nHamiltonian Eq. (1) were calculated by linear spin-wave\ntheory (LSWT). Following Yosida and Miwa [54], we de-\nfine local coordinates which rotate with the spins in the\nhelix. Introduction of Holstein–Primakoff operators in\nthe local coordinates followed by Fourier transformation\nbrings the Hamiltonian into the form\nH=H0+1\n2X\nqXy\nqHqXq; (6)\nwhere\nH0=\u0000NS2Jqm(7)\nXy\nq= (ay\nq;a\u0000q) (8)\nHq=\u0012AqBq\nBqAq\u0013\n: (9)\nSis the spin quantum number, Nis the number of lattice\npoints,ay\nqis a Fourier-transformed boson creation oper-\nator, and\nJq=X\n\u000e6=0J\u000eexp(iq\u0001\u000e) (10)\nAq=S\b\nJqm\u00001\n2Jq\u00001\n4(Jqm+q+Jqm\u0000q) +D\t\n(11)\nBq=S\b1\n2Jq\u00001\n4(Jqm+q+Jqm\u0000q)\u0000D\t\n: (12)\nThe summation in eq. (10) extends over the vectors \u000e\nwhich join a spin to the neighboring spins. Diagonaliza-\ntion of the matrix gHq, where gis the metric tensor\ng=\u00121 0\n0\u00001\u0013\n; (13)\ngives the spin-wave energy\n~!q=q\nA2\nq\u0000B2\nq: (14)10\nEquation (14) was first obtained for a planar helix by\nYosida and Miwa [54]. The eigenvectors of gHqare re-\nlated to the Bose operators via the Bogoliubov transfor-\nmation\n\u0012aq\nay\n\u0000q\u0013\n=\u0012uq\u0000vq\n\u0000vquq\u0013\u0012\u000bq\n\u000by\n\u0000q\u0013\n; (15)\nwhere\nuq=s\nAq+~!q\n2~!q; v q=s\nAq\u0000~!q\n2~!q;(16)\nand\u000by\nqcreates a magnon with wavevector q.\nIn the dipole approximation, the inelastic neutron scat-\ntering intensity is proportional to [28]\nS(q;!) =f2(q)e\u00002WX\n\u000b\f(\u000e\u000b\f\u0000^q\u000b^q\f)S\u000b\f(q;!);\n(17)\n\u000b;\f =x;y;z , wheref(q)is the magnetic form fac-\ntor and e\u00002Wis the Debye–Waller factor. The par-\ntial response functions S\u000b\f(q;!)contain the Fourier-\ntransformed spin operators S\u000b(q),S\f(q). To proceed,\nthese operators are expressed in terms of spin operators\nin local coordinates and hence in terms of magnon cre-\nation and annihilation operators via eq. (15). The final\nexpressions for magnon creation are\nSxx(q;!) =Syy(q;!)\n=g2\u00162\nBNS\n8Aq\u0006qm\u0000Bq\u0006qm\n~!q\u0006qm\n\u0002n(!)\u000e(!\u0000!q\u0006qm) (18)\nSzz(q;!) =g2\u00162\nBNS\n2Aq+Bq\n~!q\n\u0002n(!)\u000e(!\u0000!q): (19)\nThe\u0006terms inSxx=Syyare to be summed. n(!) =\nfexp( ~!=k BT)\u00001)g\u00001is the boson population factor.\nThe partial response functions with \u000b6=\fdo not con-\ntribute to the unpolarized neutron scattering intensity.\nDATA AVAILABILITY\nThe data presented in this paper is available from the\ncorresponding author upon reasonable request.\n[1] N. P. Armitage, Mele E. J., and A. Vishwanath. Weyl and\nDirac semimetals in three-dimensional solids. Rev. Mod.\nPhys. , 90:015001, 2018.[2] B. Yan and C. Felser. Topological materials: Weyl\nsemimetals. Annu. Rev. Condens. 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Soh, J. Sun, and D. Prabhakaran.\nSpin excitations in the candidate Weyl semimetal EuCuAs,\n2019. 10.5286/ISIS.E.RB1920514-1.ACKNOWLEDGMENTS\nThe authors wish to thank Gareth Nisbet, Robert\nPocock, Dan Porter, Sid Parameswaran, Steve Simon and\nRoss Stewart for discussions, Toby Perring for providing\nthe software used to powder-average the spin-wave spec-\ntrum presented in Fig. 5 b, and Pascal Manuel, Dmitry\nKhalyavin and Fabio Orlandi for help with the pow-\nder diffraction experiment on WISH at the ISIS Facil-\nity. The proposal numbers for the data presented in this\nmanuscript are 5-41-1048 (D9, ILL [55]), 5-54-368 (D3,\nILL [56]), 20220501 (SIS-ULTRA, SLS), RB1820237\n(WISH, ISIS [57]), RB2090057 (LET, ISIS [58])\nRB1920514 (Merlin, ISIS [59]), MT20347-1 (I16, DLS).\nD.P. and A.T.B. acknowledge support from the Ox-\nford–ShanghaiTech collaboration project. This work was\nsupported by the U.K. Engineering and Physical Sciences\nResearch Council, grant no. EP/M020517/1. J.-R.S.\nacknowledges support from the Singapore National Sci-\nence Scholarship, Agency for Science Technology and Re-\nsearch and the European Research Council (HERO, Grant\nNo. 810451).\nAUTHOR CONTRIBUTIONS\nJ.-R.S. and A.T.B. conceived the experiments. D.P.\ngrew the single crystals, and J.-R.S., I.V. and D.P. charac-\nterised and performed bulk measurements on the crys-\ntals. Unpolarized neutron powder and single crystal\ndiffraction was carried out by J.-R.S., J.S., A.T.B., J.A.R.-\nV. and O.F. The REXS experiment was conducted by J.-\nR.S., J.S. and A.B., and J.-R.S. and X.P. performed the\nARPES experiment. The INS measurements were made\nby C.B., M.D.L., H.C.W. and A.T.B., and the SNP exper-\niment was performed by J.-R.S., J.A.R.-V. and A.S.. The\nab initio electronic structure calculations and interpreta-\ntion were performed by I.S.-R., F.d.J. and M.G.V., and\nA.T.B. performed the mean-field and spin-wave analysis.\nAll authors reviewed the manuscript.\nCOMPETING INTERESTS\nThe authors declare no competing interests." }, { "title": "1302.5541v2.Gigantic_magnetic_field_polarization_and_magnetoelectric_coupling_in_a_ferrimagnetic_oxide_CaBaCo4O7.pdf", "content": "Gigantic magnetic field induced polari zation and magnetoelectric coupling \nin a ferrimagnetic oxide CaBaCo 4O7 \nV. Caignaert1, A. Maignan1*, K. Singh1,5, Ch. Simon1, V. Pralong1, B. Raveau1, J.F. Mitchell2 \nH. Zheng2, A. Huq3, and L. Chapon4 \n1 Laboratoire CRISMAT, UMR 6508 CNRS/ENSICAEN, 6 bd du Maréchal Juin \nF-14050 CAEN Cedex 4 – France. \n2 Argonne National Laboratory MSD 223 9700 S. 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": "2303.11869v1.Unveiling_the_magnetic_structure_and_phase_transition_of_Cr__2_CoAl_using_neutron_diffraction.pdf", "content": "Unveiling the magnetic structure and phase transition of Cr 2CoAl using neutron di\u000braction\nGuru Dutt Gupt,1Yousef Kareri,2,\u0003James Hester,3Clemens Ulrich,2and R. S. Dhaka1,y\n1Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India\n2School of Physics, The University of New South Wales, Kensington, 2052 NSW, Sydney, Australia\n3Australian Center for Neutron Scattering, Australian Nuclear Science and Technology Organisation (ANSTO),\nNew Illawarra Road, Lucas Heights NSW 2234, Australia\n(Dated: March 22, 2023)\nWe report the detailed analysis of temperature dependent neutron di\u000braction pattern of the\nCr2CoAl inverse Heusler alloy and unveil the magnetic structure up to the phase transition as well\nas its fully compensated ferrimagnetic nature. The Rietveld re\fnement of the di\u000braction pattern\nusing the space group I \u00164m2 con\frm the inverse tetragonal structure over the large temperature\nrange from 100 K to 900 K. The re\fnement of the magnetic phase considering the wave vector k=\n(0, 0, 0) reveals the ferrimagnetic nature of the sample below 730 \u00065 K. This transition temperature\nis obtained from empirical power law \ftting of the variation in the ordered net magnetic moment\nand intensity of (110) peak as a function of temperature. The spin con\fguration of the microscopic\nmagnetic structure suggests the nearly fully compensated ferrimagnetic behavior where the magnetic\nmoments of Cr2 are antiparallel with respect to the Cr1, and Co moments. Moreover, the observed\nanomaly in the thermal expansion and lattice parameters at 730 \u00065 K suggest that the distortion in\ncrystal structure may play an important role in the magnetic phase transition.\nI. INTRODUCTION\nIn the \feld of spin \flters and spintronics, the com-\npensated ferrimagnetic and spin gapless semiconductor\nHeusler alloys emerge as potential candidates because of\ntheir exotic physical properties elegantly controlled by\nthe conduction method of the electrons at the Fermi level\n(EF) [1, 2]. More interestingly, a few Heusler alloys show\nhalf-metallic (HM) nature with 100% spin polarization\nwhere the conduction is only due to one spin channel\nand there are no electrons present with opposite spin\nat E F[3]. Leuken and de Groot [4] theoretically pro-\nposed to realize HM antiferromagnets (AFMs) with full\nspin polarization in Heusler alloys, which are de\fned as\nzero net moments with the fully spin polarization, and\nare also classi\fed as the HM fully compensated ferrimag-\nnets (FCF) [5, 6]. These type of HM-AFM/FCF ma-\nterials have advantage due to their zero stray magnetic\n\feld and therefore no energy losses during device oper-\nation for vaious applications [5]. In recent time, inverse\nHeusler (X 2YZ) alloys, where the atomic number of X is\nsmaller than Y and crystal structure changes from L2 1\nto XA type, are predicted to show such vital magnetic\nproperties as well as spin gapless semiconducting nature,\nand therefore are considered as potential candidates for\npractical applications [7{13]. In this family, Cr 2CoAl was\ntheoretically found to be stabilized in the inverse tetrag-\nonal XA type structure having a negative formation en-\nergy [14, 15], which was later experimentally veri\fed in\nour recent report [16]. Interestingly, the theoretical stud-\nies did predict that the Cr 2{based Heusler alloys possess\na fully spin-polarized band structure, which is highly de-\nsirable in spintronics [1, 5]. On the other hand, Cr-based\n\u0003Present Address: Saudi Electronic University, Saudi Arabia\nyCorresponding author's email: rsdhaka@physics.iitd.ac.incompound such as Cr 3Al exhibits a ferrimagnetic (FIM)\nnature with experimentally observed 84% spin polariza-\ntion [17]. Moreover, Cr 3Al \flms were investigated using\nneutron di\u000braction to explore the magnetic moment of\nthe atoms at di\u000berent sites [18]. In case of Cr 2CoAl, the\nappreciable amount of spin polarization 68% was real-\nized in the compensated ferrimagnetic (CF) state [14].\nTheoretical studies predicted that the e\u000bect of compen-\nsation leads to a decrease in the magnetic moment in the\nCr2CoAl sample where the Cr-Cr neighboring atoms have\nan antiparallel coupling, and the individual magnetic mo-\nments for the nonsymmetric spin structure for the atoms\nat di\u000berent sites were found to be Cr1 (1.36 \u0016B), Cr2\n(-1.49\u0016B), and Co (0.30 \u0016B) in this inverse Heusler al-\nloy [14, 15, 19]. However, the full compensation is not\nexperimentally realized with zero moment in the Heusler\nsamples; for example, Mn 3Ga has a magnetic moment of\n0.65\u0016B/f.u, and MnCoVAl possesses 0.07 \u0016B/f.u [20, 21].\nSince the net moment of the samples is mainly gov-\nerned by the magnetic atoms, in case of inverse Heusler\nalloys the magnetic moment follows the Slater-Pauling\n(SP) relation as M t= (Z t\u000024)\u0016B, where Z tis the to-\ntal number of valance electrons in the unit cell of the\nalloy. Here, for the complete magnetization compensa-\ntion, the value of Z tmust be equal to 24 according to the\nSP relation [22]. Recently, FCF behavior was reported\nin Cr 2CoAl and Cr 2CoGa experimentally as well as by\nab-initio calculations [16, 19, 23, 24]. Our recent report\non Cr 2Co(1\u0000x)CrxAl indicates that the XA structure is\nstable in the single-phase and we observed the signature\nof the FCF state in the x= 0{0.4 samples [16]. At the\nsame time, the magnetization curves show a \fnite hys-\nteresis loop and do not saturate with magnetic \feld at\na temperature of 50 K and 300 K. The magnetization\nbehavior as a function of temperature and magnetic \feld\nhas been classi\fed as antiferromagnetic and/or compen-\nsated ferrimagnetic [16]. However, the crucial fact aboutarXiv:2303.11869v1 [cond-mat.str-el] 21 Mar 20232\nthe Heusler alloys is the presence of antisite disorder,\nwhich can decrease the spin polarization [25{27]. Using\nneutron di\u000braction (ND) [28], L\u0013 azpita et al. determined\nthe atomic distribution in the NiMnGa alloy in the para-\nmagnetic (PM) region [29]. In the same way, an antisite\ndisorder was found in the Mn 2VGa sample between V\nand Ga atoms [30]. Recently, we have determined the\nmagnetic structure of Co 2CrAl sample using powder ND\nacross the phase transition and found the perfect agree-\nment with magnetization behavior [31]. Umetsu et al.\nalso used powder ND to investigate the site occupancies\nin few Co 2based full Heusler alloys both in the FM and\nPM states [32]. Powder ND has also been used to study\nthe antisite disorder and magnetic structure in Co 2based\nHeusler alloys [33, 34]. Interestingly, a structural tran-\nsition, i.e., abrupt change in the lattice parameters was\nobserved by temperature dependent ND, which found to\nbe in agreement with the magnetic phase transition in\nTbCo 2[35]. Also, ND is sensitive to the appearance of\nAFM ordering, and the observed T Nwas found to be con-\nsistent with the magnetization data of CuMnSb [36] as\nwell as in inverse Heusler alloys [37]. The magnetic phase\ntransition in Cr 2CoAl is expected to be around 750 K\n[38]; however, to the best of our knowledge there are no\nreports on magnetization and/or ND measurements at\nhigh temperatures. Therefore, it is of vital importance\nto unveil the magnetic structure, phase transition and\nstructural disorder in the Cr 2CoAl inverse Heusler alloy.\nIn this paper, we present a detailed analysis of the\nND patterns of the Cr 2CoAl sample to determine the\ncrystal structure and the microscopic magnetic behavior\nover the large temperature range from 100 K to 900 K\nacross the magnetic phase transition. The Rietveld re-\n\fnement reveals the inverse tetragonal structure and no\nmeasurable antisite disorder in the sample. The analy-\nsis of the magnetic phase gives the value of net ordered\nmoment around 0.04(4) \u0016B/f.u. at 100 K, which found\nto decrease with temperature and reaches almost zero at\naround 730 K [de\fned as the magnetic ordering temper-\nature T MO]. Also, the intensity plot of the (110) peak\nshows a similar decrease with temperature till 730 \u00065 K\nand then become almost constant. Moreover, the lattice\nparameters increase with an increase in temperature, and\nthe slope of the curve changes near the T MO. A similar\nanamoly is observed in the thermal factor and thermal\nexpansion coe\u000ecient at around T MO. Interestingly, we\n\fnd a nearly FCF structure where the magnetic moment\nof Cr2 shows antiparallel alignment with the Cr1 as well\nas the Co spins. The FIM transition obtained from the\n\ftting of temperature dependence of the magnetic mo-\nment is found to be consistent with the intensity varia-\ntion of the (110) peak with temperature.\nII. EXPERIMENTAL DETAILS\nPolycrystalline Cr 2Co(1\u0000x)CrxAl (x= 0, 0.2) samples\nwere prepared by arc melting (CENTORR, Vacuum In-dustries, USA). The basic characterization of these sam-\nples has been reported in ref. [16]. Powder ND experi-\nments are performed at the high-intensity di\u000bractome-\nter Wombat [39] and the high-resolution di\u000bractome-\nter Echidna [40, 41] at the OPAL research reactor at\nANSTO, Australia, using a cylindrical sample holder.\nA wavelength of \u0015= 1.633 \u0017A and 1.622 \u0017A were se-\nlected with a Ge(113) and a Ge(335) monochromator\nat the instruments Wombat (300-900 K) and Echidna\n(100 K), respectively. The neutron di\u000braction patterns\nwere scanned at various temperatures on heating from\nroom temperature to 900 K in the vacuum furnace. The\nstep size was taken as 0.125oin the 2\u0012range between 25o-\n135oat the Wombat di\u000bractometer for the x= 0 sample.\nThe measured di\u000braction pattern is analyzed with the Ri-\netveld re\fnement method implemented with the FullProf\npackage [42] considering the fundamental aspects of full-\nwidth and half maximum and other reliable parameters\nof the di\u000braction peaks [43]. The magnetic con\fguration\nis generated with neutron powder di\u000braction using the\nbasis irreducible representation (BasIreps) function.\nIII. RESULTS AND DISCUSSION\nIn Fig. 1 we present the high resolution ND pattern\nof the Cr 2Co(1\u0000x)CrxAl (x= 0, 0.2) samples measured\non the Echidna di\u000bractometer in the broad angle range\n20o-150oat 100 K. At \frst glance, we clearly observe the\ntetragonal distortion in the principal re\rections for both\nthe samples, which is consistent with the x-ray di\u000brac-\n(arb. units) (a)high angle x=0\n100 K(312)\n(204)\n(400)\n(224)\n(332)\n(116)\n807060504030\n2θ (deg) x=0.2\n100 K low angle(d) Yobs\n Ycalc\n Yobs-Ycalc\n Bragg Position nuclear\n Bragg Position magnetic(c)low angle\n x=0\n100 K\n(101)\n(110)\n(200)\n(112)\n(220)\n(004)intensity\n14013012011010090\n2θ (deg)(b)\n x=0.2\n100 Khigh angle\nFIG. 1. Rietveld re\fnement (black line) of the powder ND\npattern (red symbols) (a, b) with a nuclear Bragg peaks at\nhigher 2\u0012angles, and (c, d) with both nuclear and magnetic\nBragg re\rections at lower 2 \u0012angles, measured at 100 K for\nboth thex= 0 and 0.2 samples. The di\u000berence pro\fle (blue\nline) and Bragg peak positions (short vertical bars green for\nthe nuclear and magenta for the magnetic) are shown in each\npanel. These high resolution patterns were recorded at the\nEchidna di\u000bractometer ( \u0015= 1.622 \u0017A).3\ntion (XRD) patterns reported in ref. [16]. The mea-\nsurement temperature of 100 K was chosen to be in the\nmagnetic region, as con\frmed in the magnetization data\n[16]. In order to extract the information from the data\nat 100 K, we re\fne the ND pattern following the simi-\nlar procedure as reported in ref. [44]. First, the neutron\npowder-di\u000braction patterns at a higher angle (2 \u0012: 80{\n150o) have been re\fned as the magnetic form factor gen-\nerally is negligible at higher angles above \u001980o[30, 44].\nThe re\fned pattern using the tetragonal structure with\nthe space group I \u00164m2 considering the nuclear contribu-\ntion only [45] are shown in Figs. 1(a, b) for the x= 0 and\n0.2 samples, respectively. We \fnd the lattice parameters\nfor thex= 0 sample; a= 4.051 \u0017A andc= 5.665 \u0017A, and\nfor thex= 0.2 sample; a= 4.075 \u0017A andc= 5.680 \u0017A,\nwhich are consistent with the reports in Refs. [16, 38].\nThe ND data in the AFM state either show new Bragg\npeaks, which appear towards lower 2 \u0012angle in the mag-\nnetically ordered state (below N\u0013 eel temperature) or with\nthe primitive lattice where the magnetic atoms arrange in\nsuch way that their multiplicity is higher than one, hav-\ning a wave vector (k=0) [46{48]. However, in the present\ncase, the magnetic atoms arrange in the multiplicity of\ntwo, which is higher than the multiplicity of the atoms\nin the primitive lattice. Since no additional Bragg peaks\nappear in the magnetically ordered state in our ND pat-\nterns, an AFM structure can be ruled out. This indicates\neither ferromagnetic (FM) or ferrimagnetic (FIM) order-\ning in these samples [25, 31, 46]. The magnetic contri-\nbution is associated with the (110) peak as the intensity\nof this peak increases at lower temperatures due to the\npresence of magnetic ordering [31]. In order to reveal the\nmagnetic structure, the low angle di\u000braction patterns are\nanalyzed incorporating the magnetic contributions and\nusing the lattice structure obtained from the high angle\npatterns, as shown in Figs. 1(c, d) for the x= 0 and 0.2\nsamples, respectively. The extracted net ordered mag-\nnetic moments of the x= 0 andx= 0.2 samples are found\nto be 0.04(4) and 0.05(4) \u0016B/f.u. at 100 K, respectively.\nThese values are reasonably in agreement with the gen-\neralized SP rule considering the total number of valence\nelectrons in the unit cell [49, 50] as well as the value re-\nported in Ref. [51] using the band structure calculations.\nThe antiparallel alignment and the di\u000berent magnitude\nof the magnetic moment vectors of Cr and Co atoms in-\ndicate a nearly FCF structure (discussed later), which\nis consistent with the reported physical nature of FCF\nfor the Cr 2CoAl sample in Ref. [51]. All the extracted\nparameters for the x= 0 sample are listed in Table I of\nthe Supplementary Information [52]. Notably, neutron\ndi\u000braction was also used to study the FCF nature in the\nMn2V1\u0000xCoxGa Heusler alloys [53].\nNow, we mainly focus on the detailed analysis of pow-\nder ND pattern of the x= 0 sample, collected at the\ndi\u000bractometer Wombat in the large temperature range\nfrom 300 K to 900 K, to reveal the magnetic structure and\ntransition temperature. Figs. 2(a-j) show the Rietveld\nre\fned ND patterns considering magnetic plus nuclear\n (i)\n 850K\n120100806040\n2θ (deg) (j)\n 900K\n120100806040\n2θ (deg) (e)\n725Kintensity (arb. units) (c)\n600K (b)\n550K (a)\n300K Yobs\n Ycalc\n Yobs-Ycalc\n Bragg Position nuclear\n Bragg Position magneticx=0\n*\n* (f)\n750K\n*\n*\n (d)\n700K (g)\n775K\n (h)\n 800KFIG. 2. (a-j) The Rietveld re\fned neutron powder di\u000braction\npattern of the x= 0 sample, recorded on the di\u000bractometer\nWombat (\u0015= 1.633 \u0017A). Each pattern is \ftted with magnetic\nplus nuclear phases in the low temperature range 300{725 K,\nand with the nuclear phase only in 750{900 K range. The\nblack asterisk tag indicates the peaks from the Niobium sam-\nple environment. The region 38o{43ohas been removed from\nthe di\u000berence pro\fle (blue line) for clarity in the presentation.\nphases in the temperature range of 300{725 K, and with\nonly the nuclear phase from 750 K to 900 K range. There\nare a few peaks associated with the Niobium sample envi-\nronment, between 2 \u0012= 38o{42o, as well as at \u001981o, which\nare present at all temperatures in Fig. 2. Therefore, for\nthe sake of accuracy of the \ftting parameters, these re-\ngions are excluded from the re\fnement by adjusting the\nrange limit in the Fullprof program [42]. Normally the\nND technique is more sensitive as compared to XRD to\nquantify the antisite disorder due to the distinctly di\u000ber-\nent neutron-bound scattering amplitude of the elements\nCr (3.6 fm), Co (2.5 fm), and Al (3.5 fm) [31, 54]. There-\nfore, we tried to \fnd the antisite disorder between the Co\nand Cr atoms as well as between the Co and Al atoms by4\nre\fning the ND patterns, initially at 900 K (above T C)\nwith the nuclear phase only, as shown in Fig. 2(j). A\nsimilar method was reported to quantify the antisite dis-\norder in the Mn 2VGa and Co 2MnSi Heusler alloys with-\nout a\u000becting the stoichiometry where the atoms also have\ndi\u000berent scattering factors [30, 34]. However, we \fnd no\nsigni\fcant improvement in the re\fnement, which indi-\ncates the absence of measurable antisite disorders. On\nthe other hand, the observed disorder between Cr and Al\nby XRD analysis in Ref. [16] cannot be ruled out from\nthe ND analysis due to their similar neutron scattering\ncross-sections [54]. We also note here that any disorder\nbetween Cr and Al is not expected to a\u000bect the mag-\nnetic moment of these types of samples as predicted in\nRef. [55]. Therefore, the re\fned crystal structure inferred\nfrom the ND pattern measured at 900 K is used for the\nfurther analysis of the successive ND pattern at lower\ntemperatures, as in Ref. [29].\nIn order to analyze the neutron di\u000braction pattern\nin the magnetic region (below \u0019750 K), it should be\nnoted that there are no additional Bragg re\rections in\nthe magnetically ordered state of Cr 2CoAl Heusler alloy,\nsee Fig. 2. However, with decreasing sample tempera-\nture the scattering intensity of the (110) peak increases,\nas plotted in Fig. 3(c), which suggests that the magnetic\nstructure is either FM or FIM at low temperatures and\nexcludes the possibility of a long-range AFM order in\nthisx= 0 sample [30, 46{48]. Thus, to understand the\nmagnetic structure and phase transition, we generate the\nmagnetic moment con\fguration output using BasIREPS\nin the Fullprof program by considering the space group\nI\u00164m2 and the magnetic state of FM or FIM. There are\nthree magnetic atoms Cr1, Cr2, and Co and their corre-\nsponding Wycko\u000b positions are 2 b(0, 0, 0.5), 2 d(0, 0.5,\n0.75), and 2 a(0, 0, 0), respectively [15]. The appropriate\nmagnetic propagation wave vector k= (0, 0, 0) is consid-\nered for the FIM state with the best value by using the k-\nsearch option in the Fullprof program [42]. This method\nprovides the irreducible representation with only one ba-\nsis vector \u0000 4, which is related to the FIM interactions\nwith real and imaginary positions as (0, 0, 1) and (0, 0,\n0), respectively [56]. The basis function helps to reveal\nthe magnetic structure, where the arrangements of the\nmagnetic moments are parallel or anti-parallel [44, 46].\nTo extract the precise values of the magnetic moments\nfrom the ND pattern, it must be noted that the re\fne-\nment is performed with the particular magnetic site of\nthe atoms rather than the individual sites of the disorder\npositions [25, 29]. For example, the moment of the mag-\nnetic atoms with disorder gives the average moment of\nthat atom at di\u000berent Wycko\u000b positions. In re\fning the\nmoment values at Cr1, Cr2, and Co sites, the sizable mo-\nment is found to be related to the (110) re\rection only.\nAlso, within the experimental error bar the magnetic mo-\nments inab\u0000plane were too small to be determined.\nInterestingly, the direction of the magnetic moment of\nCr2 is found to be opposite to the c-axis whereas the\nmoments of Cr1 and Co are parallel. It was theoretically\n800\n400\n0intensity (arb. units)\n353433323130\n2θ (deg) 100 K\n 900 K(110)(d)1100\n1000\n900\n800\n700\n600\n500intensity (arb. units)\n900800700600500400300200\ntemperature (K) intensity (110)\n fit\n(c)-0.75-0.50-0.250.000.25atomic moment ( µB)\n Cr1\n Co\n Cr2\n(a)Tc = 730(5) Kx=0\n(e)\nCr2Cr1\nCo\nabc0.12\n0.08\n0.04\n0.00\n-0.04net moment ( µB/f.u.) net moment\n fit\n(b)FIMPMFIG. 3. (a) The temperature dependence of the ordered mag-\nnetic moments of Cr1, Cr2 and Co sites. (b) The ordered net\nmoment in the temperature range of 300 K to \u0019730 K, and\n(c) temperature dependence of intensity of the (110) re\rection\nfor thex= 0 sample. The blue solid lines are the \ftted curves\nof magnetic moment and intensity with the power law equa-\ntion. (d) The intensity of peak (110) at 100 K and 900 K for\ncomparison, (e) the magnetic spin con\fguration with concor-\ndant ordering wave vector k= (0, 0, 0) along the c-axis in the\nmagnetic unit cell for 100 K. Here, the ND pattern at 100 K\nis recorded at the Echidna di\u000bractometer ( \u0015= 1.622 \u0017A) and\nat high temperatures between 300 K and 900 K are recorded\non the di\u000bractometer Wombat ( \u0015= 1.633 \u0017A).\nreported that the Cr atoms show the opposite polarity\nowing to their mutual antiparallel con\fgurations [14, 15].\nFor the re\fnement of the di\u000braction pattern below 750 K,\nwe have initially taken all the structural parameters ex-\ntracted from high temperature (900 K) data, and then\nre\fned the positions of the magnetic moment sites of the5\natoms to get the accurate microscopic magnetic moment\nvalues at a particular site. However, due to the strong\ndirect interaction of d\u0000states between the neighboring\natoms of nonequivalent Cr atoms, the antiparallel spin\ncon\fguration leads to an almost zero net magnetic mo-\nments [15, 19]. The reliability parameters obtained from\nthe re\fnement of the di\u000braction pattern at 100 K are \u001f2\n= 2.9, Bragg R-factor = 1.4, RF-factor = 1.8, and mag-\nnetic R-factor = 4.1, which proves the good quality of the\nre\fnement [57]. The obtained magnetic moment values\nand lattice parameters from the re\fnement are plotted\nin Figs. 3 and 4, and discussed in detail to understand\nthe magnetic properties and phase transition in Cr 2CoAl\nsample. The ordered magnetic moments of the Cr1, Cr2,\nand Co sites obtained from the re\fnement of the powder\nND patterns are plotted in Fig. 3(a), and the ordered net\nmagnetic moment is shown in Fig. 3(b), which mimic the\nmagnetization behavior and that the magnetic ordering\ndisappear at high temperatures that suggests a transition\nfrom paramagnetic to the commensurate FIM magnetic\nstructure at around T C= 730\u00065 K. We also note here\nthat the observed T C= 730\u00065 K value of Cr 2CoAl us-\ning neutron di\u000braction is found to be consistent with the\nmagnetization study on a similar system, i.e., Cr 2CoGa\nthin \flms, reported in Ref. [58]. However, the authors\nalso observed a signi\fcant change in the T Cvalue de-\npending on the annealing treatment to the thin \flms [58].\nIn Fig. 3(c), the intensity of the (110) peak is plot-\nted with temperature, which clearly increases below the\ntransition temperature and is nearly constant in the PM\nregion. The much higher intensity of the (110) peak ob-\nserved at 100 K manifests the enhancement in the mag-\nnetic ordering at low temperatures. The ordered net\nmagnetic moment and intensity of the (110) Bragg re-\n\rection versus temperature curves are analyzed by \ftting\nan empirical power law: M(T) = M 0(1\u0000T=TC)\fto the\nexperimental data to determine the transition tempera-\nture [48, 59{61]. The FIM transition temperature (T C) is\nfound to be 730 \u00065 K with a critical exponent \f= 0.2\u00060.1,\nwhich is well concordant with the critical exponent of the\nstandard universality classes, assists to get the transition\ntemperature where the intensity approach to zero with\nincreasing temperature. The magnetic scattering is pro-\nportional to the square power of M [31, 62]. In Fig. 3(d),\nthe intensity of (110) peak at 100 K is observed \u001946%\nhigher than at 900 K, which manifests the magnetic or-\ndering at low temperatures. Moreover, Fig. 3(e) shows\nthe orientation of the moment vectors of the individual\natoms in the magnetic unit cell at 100 K where the mag-\nnetic vectors of Cr2 are oppositely aligned with respect\nto thec-axis as well as to the moment vectors of Cr1 and\nCo atoms. This clearly reveals the nearly FCF order [63]\nand is in good agreement with predictions from theoreti-\ncal band structure calculations in Ref. [15] as well as with\nother Cr based alloy [64]. It is interesting to note that\nrecently Xie et al. reported the FCF half-metallic nature\nin the inverse Heusler alloys that shows a spin polarized\nWeyl structure with quadratic nodal lines [51].\n95.5095.0094.5094.00volume (Å3)(b)\n1.251.000.75B (Å2)900800700600500400300200temperature (K)(d)TC1.4051.4041.4031.4021.4011.4001.399c/a ratio(c)5.7255.7005.675lattice parameters (Å)4.0804.0704.060 a c linear fit (a)x=0FIG. 4. (a) The lattice parameters ( aandc) obtained from\nthe Rietveld re\fnement and the solid brown ( \u0014TC) and blue\n(\u0015TC) lines are the linear \ft to the experimental data. (b)\nThe variation in volume of the unit cell, and (c) the tetrag-\nonal ratio ( c=a) as a function of sample temperature. (d)\nThe overall thermal factor (B) variation with the tempera-\nture obtained from the re\fnement. The black vertical dotted\nline shows the boundary of the ferrimagnetic transition. The\nerror bars are standard deviations taken from the re\fnement.\nFurther, in Figs. 4(a{d), we show the obtained lat-\ntice parameters ( aandc), unit cell volume, c=aratio,\nand overall thermal factor (B) inferred from the re\fne-\nment of the ND patterns in the full temperature range\nfor thex= 0 sample. Fig. 4(a) shows a linear increase\nin the lattice parameters with temperature. There is a\nsignature of change in slope at \u0019730 K for both aandc.\nThese \fndings manifest the clear increase in the tetrago-\nnal distortion at this temperature as re\recting from the6\nc=aratio shown in Fig. 4(c). The overall thermal factor\n(B), i.e., the Debye-Waller factor is plotted in Fig. 4(d),\nwhich also shows an increasing trend with temperature.\nThe value of overall thermal factor well concurs with re-\nported for Co 2CrAl and Co 2MnSi at room temperature\n[31, 44]. We \fnd that the thermal expansion in the lat-\ntice parameters has two regions of variation where an\nanomaly is observed at around 730 K. The lattice pa-\nrameters follow the Bose-Einstein statistics for thermal\nexpansion; therefore, the obtained lattice parameters ( a\nandc) are \ftted with a general straight line equation\nin the two di\u000berent regions below and above the phase\ntransition temperature. The linear thermal expansion\ncoe\u000ecient is calculated using the equation \u000b=1\na\u0000@aT\n@T\u0001\n[62, 65, 66], where \u000brepresent the linear thermal expan-\nsion coe\u000ecient and (@aT\n@T) are the values of the slope for\nthe lattice parameters ( aandc). The obtained values\nof\u000b(per K) for aare 0.9 \u000210\u00005and 0.7 \u000210\u00005, and for\ncare 1.2 \u000210\u00005and 1.7 \u000210\u00005below and above \u0019730 K,\nrespectively. The value of \u000bis found to be lower for aside\nthan forcside, which indicates the signi\fcant expansion\non thecaxis. These values show an anomaly in \u000baround\nTC= 730\u00065 K, which is due to the di\u000berent slope of the\nlattice parameters as a result of the distortion in the in-\nverse tetragonal crystal structure around T C. Here, the\n\u000bvalues for Cr 2CoAl are well matched with the similar\nHeusler alloys as reported in refs. [31, 62, 65, 66]. In gen-\neral, the value of \u000bfor the alloys and engineering metals\nis positive and in the order of 4 \u000210{5/K [67].\nIV. CONCLUSIONS\nIn summary, we have investigated the magnetic struc-\nture and phase transition of the inverse Heusler alloysCr2CoAl using powder neutron di\u000braction measurements\nin the large temperature range of 100{900 K. The Ri-\netveld re\fnement of the di\u000braction pattern manifests the\nsingle-phase inverse tetragonal structure of both these\nsamples. We \fnd no signi\fcant antisite disorder between\nCr and Co atoms. More importantly, the ferrimagnetic\n(FIM) ordering is revealed by the re\fnement of the\nmagnetic sites using the space group I \u00164m2 and the\nmagnetic wave vector, k= (0, 0, 0) in the magnetically\nordered state where the direction of the moment vectors\nof Cr2 is opposite to the c-axis as well as the moments\nof Cr1 and Co atoms. Interestingly, the net ordered\nmagnetic moment as a function of temperature reveals\nthe FIM ordering in the sample and the transition\ntemperature is found to be 730 \u00065 K. Moreover, we \fnd\nthe anomaly in the variation in the lattice parameters\nand the thermal expansion factor around the transition\ntemperature, which can be attributed either to the\nmagnetostriction or to the role of structural distortion\nin the magnetic phase transition in inverse Heusler alloys.\nV. ACKNOWLEDGMENTS\nThis work was \fnancially supported by the BRNS\nthrough a DAE Young Scientist Research Award to RSD\nwith Project Sanction No. 34/20/12/2015/BRNS. 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Feng1, Manfred Reehuis2, Peter Adler1, Zhiwei Hu1, Michael Nicklas1, Andreas \nHoser2, Shih-Chang Weng3, Claudia Felser1, Martin Jansen1 \n \n1Max Planck Institute for Chemical Physics of Solids, Dresden, D-01187, Germany \n2Helmholtz -Zentrum Berlin für Materialien und Energie, Berlin, D -14109, Germany \n3National Synchrotron Radiation Research Center (NSRRC), Hsinchu, 30076, Taiwan \n \n \nAbstract: \nThe non -stoichiometric double perovskite oxide La 2Ni1.19Os0.81O6 was synt hesized by solid \nstate reaction and its crystal and magnetic structure s were investigated by powder x-ray \nand neutron diffraction. La 2Ni1.19Os0.81O6 crystallizes in the monoclinic doub le perovskite \nstructure (general formula A2BB’O6) with space group P21/n, where the B site is fully \noccupied by Ni and the B’ site by 19 % Ni and 81 % Os atoms. Using x-ray absorption \nspectroscopy an Os4.5+ oxidation state was established, suggesting presence of about 50 % \nparamagnetic Os5+ (5d3, S = 3/2) and 50 % non -magnetic Os4+ (5d4, Jeff = 0) ions at the B’ \nsites. Magnetization and neutron diffraction measurements on La 2Ni1.19Os0.81O6 provide \nevidence for a ferrimagnetic transition at 125 K . The analysis of the neutron data suggests \na canted ferrimagnetic spin structure with collinear Ni2+ spin chains extending along the c \naxis but a non-collinear spin alignment within th e ab plane. The magnetization curve of \nLa2Ni1.19Os0.81O6 features a hysteresis with a very high coercive field, HC = 41 kOe, at T = \n5 K, which is explained in terms of large magnetocrystalline anisotropy due to the presence \nof Os ions together with atomic disorder. Our results are encouraging to search for rare \nearth free hard magnets in the class of double perovskite oxides. \n \n \n \n 2 I. INTRODUCTION \nDouble perovskite oxides A2BB’O6 containing 3d-5d(4d) transition metal ions at the B \nand B’ sites are attracting great attention, due to the ir interesting physical properties . High-\ntemperature ferrimagnetic half-metal s, ferro - or ferri magnetic insulator s, as well as \nmaterials with large magnetoresistance and exchange bias, are found in this class of \ncompounds [19]. They thus show prospects for spintronic applications. Most of these \ncompounds contain 5 d elements with 5 d2 and 5 d3 electronic configurations . Here, the \ninteresting properties arise from competing interactions within and between the 3 d and 5 d \nsublattices. Studies on 5d4 system s are rare, probably because of the expected nonmagnetic \nground state for 5d4 ions. In the presence of strong spin-orbit coupling (SOC), the orbital \nangular momentum associated with the three t2g orbitals in an octahedral crystal field (l = \n1) entangle s with the spin moments of the electrons which results in an upper jeff = 1/2 \ndoublet and a lower jeff = 3/2 quadruplet. The formation of a SOC -assisted Mott insulating \nstate in Sr2IrO 4 (Ir4+, t2g5) is understood within this picture [10]. According to this scenario, \nthe ground state of a 5d4 system in an octahedral crystal field is a trivial singlet with four \nelectrons filling the lower quadruplet, resulting in a nonmagnetic state with total angular \nmomentum Jeff = 0. Many studies focus on Ir5+ compounds, such as A2BIrO 6 (A = Sr, Ba; \nB = Sc, Y) [11 17]. In contra st to the expectations for Jeff = 0, long-range magnetic order \nat 1.3 K was reported for Sr 2YIrO 6 [11,12] and considered as evidence for intermediate -\nstrength rather than strong spin -orbit coupling which is usu ally anticipated for 5 d ions. \nThese results, however, have been challenged and the observed magnetism in such \nmaterials may arise from defects and /or nonst oichiometry [13,18]. \nBeside Ir5+, also Os4+ ions adopt the 5 d4 electronic configuration . The singlet ground -\nstate magnetism was discussed in AOsO 3 (A = Ca, Sr, Ba) [19] and R2Os2O7 (R = Y, Ho) \n[20]. To the best of our knowledge, the magnetic properties of double perovskite s \ncontaining Os4+ have not been reported yet. In this work, targeting at an Os4+ double \nperovskite La 2NiOsO 6, we synthesize d a nonstoichiometric phase La2Ni1.19Os0.81O6 with \nNi2+ ions at the B site and a mixture of Ni2+, Os4+, and Os5+ ions at the B’ site of the double \nperovskite structure . It was verified by x-ray absorption spectroscopy that the valence state \nof Os is about 4.5+. Magnetization and neutron diffraction studies revealed that \nLa2Ni1.19Os0.81O6 shows a ferrimagnetic transition at 125 K and features an extraordinary \nbroad hysteresis with a giant coercivity of 41 kOe at 5 K. These results are encouraging \nfor the search of hard magnetic materials in the class of 3 d/5d double perovskites. \n 3 \nII. EXPERIMENTAL \nA polycrystalline sample of La2Ni1.19Os0.81O6 was synthesized by solid -state reaction \nfrom La 2O3, NiO, and Os. La 2O3, NiO, and Os in a molar ratio 1 : 1 : 1 were well ground \ntogether and pressed into a pellet. This was loaded into a corundum crucible which was \nplaced into a silica tube along with a second corundum crucible containing MnO 2 (Alfa \n99.9%). The silica tube was then sealed under dynamic vacuum using a H 2/O2 torch, and \nheated at 1250 °C for 48 hours in a tube furnace. MnO 2 decomposes into ½ Mn 2O3 + ¼ O 2 \nat 550 °C and acts as an oxygen source for the reaction. The molar ratio of Os and MnO 2 is \n1 : 4. Small pieces of La2Ni1.19Os0.81O6 were cut from the synthesized pelle t and finely \nground to a fine powder , which was characterized by powder x-ray diffraction using a \nHuberG670 camera [Guinier technique, λ = 1.54056 Å (Cu -Kα1)]. A powder pattern was \ncollected in the 2 θ range between 10.9 and 100.2°. A scanning electron microscope (SEM, \nPhilip s XL30) with an attached energ y dispersive x-ray spectrometer (EDX) was used for \nelemental analysis. \nThe Os -L3 XAS spectra of La2Ni1.19Os0.81O6, an Os5+ Sr2FeOsO 6 reference , and an Os4+ \nreference La 2MgOsO 6 were measured in transmission geometry at the beamline BL07A at \nthe National Synchrotron Radiation Research Center in Taiwan. \n Neutron powder diffraction experiments of La2Ni1.19Os0.81O6 were carried out on the \ninstruments E2, E6, and E9 at the BER II reactor of the Helmholtz -Zentrum Berlin. The \ninstrument E9 uses a Ge monochromator selecting the neutron wavelength λ = 1.309 Å, \nwhile the instrument s E2 and E6 use a pyrolytic graphite (PG) monochromator selecting \nthe neutron wavelength λ = 2.38 and 2.417 Å, respectively . In order to investigate in detail \nthe crystal structure of La2Ni1.19Os0.81O6 at 3.2 and 160 K, well below and above the \nmagnetic ordering temperature , neutron powder patterns were recorded on the instrument \nE9 between the diffraction angles 7.5 and 141 .8°. For a detailed analysis of the magnetic \nstructure and its temperature dependence , we have collected powder patterns between 1.7 \nand 134 K on the instrument E6 between the diffraction angles 5 and 136.4°. In addition , \npowder patterns at 1.7 K and 150 K with higher counting rate and better instrumental \nresolution in the 2 range between 11.8 and 87.3° were obtained on the instrument E2. \nRietveld refinements of the powder diffraction data were carried out with the program \nFullProf [21]. In the case of the data analysis of x-ray diffraction data, we used the atomic \nscattering form factors provided by the program. For the refinements of the neutron \npowder data the nuclear scattering lengths b(O) = 5.803 fm, b(La) = 8.24 fm, b(Ni) = 10.3 4 fm, and b(Os) = 10.7 fm were used [ 22]. The magnetic form factors of the Fe and Os atoms \nwere taken from Refs. [2 3,24]. \nUsing a polycrystalline piece, the electrical resistivity ( ρ) was measured with direct \ncurrent (0.1 mA) in a four-point in -line arrangement (PPMS, Quantum Design). Electrical \ncontacts were made with Au wire s and silver paste. The temperature dependence of the \nmagnetic susceptibility was measured in a SQUID magnetometer (MPMS -5T, Quantum \nDesign). The measurements were conducted in warming after zero -field cooling (ZFC) and \nduring field -cooling (FC) in the temperature range 2 – 300 K under applied magnetic fields \nof 10 kOe. The high-temperature magnetic susceptibility (300 – 565 K) was measured \nusing the same SQUID magnetometer with oven option. Isothermal magnetization curves \nwere initially recorded for fields up to ±50 kOe at temperatures of 5 and 150 K using the \nsame SQUID magnetometer. As the magnetizat ion curves turned out to be unsat urated at 5 \nK and showed a broad hysteresis, M(H) curves were further measured for fields up to ±140 \nkOe at 5 K using a Physical Proper ty Measurement System (PPMS, Qu antum Design) . \n \n \nIII. RESULTS \n \nA Crystal structure \nThe room -temperature crystal structure of La2Ni1.19Os0.81O6 was investigated by x-ray \npowder diffraction as shown in Figure 1 (top) . La2Ni1.19Os0.81O6 was successfully refined \nin the monoclinic space group P21/n (No. 14, standard setting P21/c). In this space group, \nthe La and the three O atoms (O1, O2, and O3) occupy the Wyckoff position 4 e(x,y,z), \nwhile the Ni and Os atoms are at the positions 2 c(½,0,½) and 2 d(½,0,0), respectively. \nDuring the refinement, we used the constraint occ(Ni) + occ(Os) = 1 at the 2 c and 2 d sites. \nThe refinements indicated that the sample is nonstoichiometric compared to the ideal \ncomposition La 2NiOsO 6 and the following occupancies were found: occ(Ni1)/ occ(Os1) = \n0.187(3)/0.813(3) at 2 c, and occ(Ni2)/ occ(Os2) = 1.001(2)/ 0.001(2) at 2 d. This suggests \nthat the 2 d site is fully occupied by Ni, and therefore in the final refinement, the Ni \noccupancy at the 2 d site was fixed to be 1. This gives the composition \nLa2Ni1.187(3) Os0.813(3) O6. The refinements resulted in a residual RF = 0.0180 (defined as RF = \n∑||Fobs| |Fcalc||/∑|Fobs|). The sample contains a smaller amount of La 3OsO 7 (5.6 %) [25,26]. \nFor this compound, only the overall scale factor and the lattice parameters were \nsimultaneously allowed to vary during the Rietveld refinements, whereas the structural 5 parameters, taken from Ref. 2 5, were fixe d. The loss of Os is presumably due to the \nformation of volatile OsO 4 during the synthesis. The EDX analysis revealed that the \nsample contains much more Ni than Os, Ni/Os = 1.17/0.83, which is close to the results \nfrom the Rietveld refinement. \n \n \n \nFigure 1. (color online) Top: Rietveld refinement of the x-ray powder diffraction data of \nLa2Ni1.19Os0.81O6 collected at 295 K . The vertical bars indicate the positions of the nuclear \nBragg reflections for La2Ni1.19Os0.81O6 and impurity phase La 3OsO 7. Bottom: Rietveld \nrefinements of the powder neutron diffraction data of La2Ni1.19Os0.81O6 collected at 160 K \nand 3.2 K (bottom). The strongest peaks of the impurity phase are marked with an asterisk. \n \nDue to the fact that the x-ray scattering power of the O atoms is relatively weak in \ncomparison to those of the much heavier La and Os atom s, the crystal structure was also \ninvestigated by neutron diffraction at 3.2 and 160 K, well below and above the magnetic \n 6 ordering temperature of ab out 125 K . The crystal structure of La2Ni1.19Os0.81O6 could be \nsuccessfully refined in the space group P21/n with residuals RF = 0.0273 (3.2 K) and RF = \n0.0250 (160 K), respectively ( Figure 1 , bottom ). Duri ng the refinement, the occupancies of \nthe Ni and Os atoms at the 2 c and 2 d site were fixed to the values obtained from the x-ray \ndata. Concerning the impurity phase La3OsO 7, only the overall scale factor and the lattice \nparameters were simultaneously allowed to vary during the Rietveld refinements, whereas \nthe structural parameters obtained at lower temperature (taken from Ref. 26 ) were fixed. \nThe obtained crystallographic data are summarized in Table I, and the atomic positions are \nsummarized in Table II. \n \nTABLE I . Crystallographic data obtai ned from refinements of powder x-ray and neutron \ndiffraction data of La 2Ni1.19Os0.81O6 at different temperatures. \n 295 K 160 K 3.2 K \nDiffraction source x-ray Neutron Neutron \nSpace group P21/n P21/n P21/n \nLattice parameters \n \n \n a = 5.58129(8) Å \nb = 5.61271(9) Å \nc = 7.89614(12) Å \n β = 90.075(4) ° a = 5.5742(5) Å \nb = 5.6116(5) Å \nc = 7.8907(7) Å \nβ = 90.01(3) ° a = 5.5700(4) Å \nb = 5.6123(4) Å \nc = 7.8866(6) Å \nβ = 90.04(2) ° \nCell volume 247.356(7) Å3 246.82(4) Å3 246.54(3) \nRF 0.0180 0.0250 0.0273 \n \n \nTABLE II . The atomic positions and isotropic temperature factors ( Biso) of \nLa2Ni1.19Os0.81O6. The refinements were carried out in the monoclinic space group P21/n. \nThe 2c sites ( ½,0,½ ) are occupied by Ni1/ Os1 atoms while the 2d sites(0,0, ½) are occupied \nby Ni2 atoms with the occupanc y fixed to be 1. The Biso were constrained to be equal for \nthe metal atoms and the oxygen atoms, respectively. All parameters marked with an \nasterisk were not allowed to vary during the refinements. \nT (K) 295 160 3.2 \nDiffraction source x-ray Neutron Neutron \nx (La) 0.0060(5) 0.0157(7) 0.0162(6) \ny (La) 0.0379(1) 0.0441(3) 0.0441(3) 7 z (La) 0.2521(3) 0.2508(19) 0.2496(18) \nx (O1) 0.270(3) 0.2843(17) 0.2833(13) \ny (O1) 0.295(3) 0.2809(13) 0.2850(12) \nz (O1) 0.031(5) 0.0460(11) 0.0468(10) \nx (O2) 0.213(3) 0.1970(17) 0.1961(13) \ny (O2) 0.796(2) 0.8001(12) 0.7971(10) \nz (O2) 0.026(5) 0.0347(11) 0.0370(10) \nx (O3) 0.904(3) 0.9218 (11) 0.9218(10) \ny (O3) 0.481(1) 0.4882(9) 0.4846(8) \nz (O3) 0.245* 0.2478(12) 0.2475(10) \nBiso (La/Ni/Os ) 0.16(2) 0.34(2) 0.25(2) \nBiso (O) 1* 0.55(3) 0.46(3) \nOcc(Ni1/Os1) 0.187(3)/0.813(3) 0.187/0.813* 0.187/0.813* \n \n \n \nFigure 2. (color online) Crystal structure of La2Ni1.19Os0.81O6. Blue octahedra are fully \ncentered by Ni2+ ions, grey octahedra contain a majority fraction of Os (Os4+, Os5+) ions \nand a minor fraction of Ni2+ ions. \n \n The inter -octahedral Ni OOs bond angles of La2Ni1.19Os0.81O6 deviate strongly from \n180° (Table III), indicating strong octahedral tilting as shown in Fig . 2. The bond distances \nin the NiO 6 and OsO 6 octahedra are summarized in Table 3. At 295 K, the average Ni O \nbond length, 2.06 Å, is comparable to the literature values for Ni2+ double perovskite \noxides A2NiOsO 6 (A = Ca, Sr, Ba) [6,2 7], which indicates that Ni is divalent in this sample. \n 8 \nTABLE III. Selected bond lengths and bond angles of La2Ni1.19Os0.81O6 derived from x-ray \nand neutron powder patterns at different temperatures. \n \n \n \n \n \n \n \n \n \n \n \n \nB. X-ray absorption spectroscopy \n \n \nFigure 3. (color online) The Os -L3 XAS spectra of La 2Ni1.19Os0.81O6 and of Sr 2FeOsO 6 as \nOs5+ reference and La 2MgOsO 6 as Os4+ reference. \n \nTo explore the valence state of Os, the Os -L3 XAS spectrum of La 2Ni1.19Os0.81O6 was \nrecorded at room temperature. It is well known that XAS spectra are highly sensitive to the \nBond length (Å) 295 K 160 K 3.2 K \nNiO1 2.109(16) × 2 2.016(8) × 2 2.038(7) × 2 \nNiO2 1.981(16) × 2 2.046(9) × 2 2.061(7) × 2 \nNi O3 2.086(5) × 2 2.038(13) × 2 2.040(9) × 2 \nOsO1 1.912(16) × 2 2.038(9) × 2 2.021(7) × 2 \nOsO2 2.052(15) × 2 2.029(8) × 2 2.014(7) × 2 \nOsO3 2.011(5) × 2 2.004(13) × 2 2.002(9) × 2 \nNiO1Os 159.6(6) 154.6(3) 154.0(3) \nNiO2Os 157.8(6) 152.1(3) 151.9(3) \nNiO3Os 149.1(5) 154.8(2) 154.6(2) 9 valence state: an increase of the valence state of the metal ion by one causes a shift of the \nXAS L2,3 spectra by one or more eV toward higher energies [28,29]. Figure 3 shows the \nOs-L3 XAS spectrum of La 2Ni1.19Os0.81O6 together with spectra of Sr2FeOsO 6 and \nLa2MgOsO 6 as Os5+ and Os4+ references, respectively [6]. The energy position of the \nLa2Ni1.19Os0.81O6 spectrum is located exactly in the middle between that of Sr 2FeOsO 6 and \nLa2MgOsO 6 demonstrating the oxidation state of Os4.5+. This is consistent with the \nexpected valence state Os4.46+ for the composition La2Ni1.19Os0.81O6 obtained from the \nRietveld refi nements of the XRD data. \n \nC. Electrical Transport \nThe electrical resistivity of a sintered polycrystalline sample of La2Ni1.19Os0.81O6 is \ndisplayed in Figure 4. The sample shows insulating behavior as the resistivity increases by \nseveral orders of magnitude as the temperature decreases and exceeds the measurement \nlimit for temperatures lower than 100 K. The data was plotted on T1 and T1/4 scales, and \nthe plot is found to be roughly linear on a T1/4 scale (see the inset of Fig . 4), in accordance \nwith a three -dimensional variable range hopping transport model. \n \n \nFigure 4. (color online) Temperature dependence of the electrical resistivity of \nLa2Ni1.19Os0.81O6. The inset shows the corresponding plots on T1 and T1/4 scales. \n \nD. Magnetic properties \n 10 The temperature dependence of the magnetic susceptibility χ of La2Ni1.19Os0.81O6 is \nshown in Figure 5. The sharp increase of χ below about 125 K with cooling indicates a \npossible ferro - or ferrimagnetic transition. Because the minor impurity La 3OsO 7 is \nantiferromagnetic with a Néel temperature of 45 K [2 6], the sharp transition around 125 K \nis intrinsic for La2Ni1.19Os0.81O6. The convex χ1 vs T curve above the magnetic transition \ntemperature indicates a ferrimagnetic transi tion. An attempt to fit the data above 400 K \nwith the Curie -Weiss law resulted in a Curie -Weiss temperature ( θCW) of 73 K and an \neffective magnetic moment ( µeff) per formula unit (f.u.) of 3.63 μB. The negative θCW \nindicates that antiferromagnetic interactions are dominant in this sample, supporting the \nferrimagnetic phase transition. The obtained µeff compares reasonably well with µeff = 3.69 \nμB/f.u., which is calculated from the formula La2Ni1.192+Os0.434+Os0.385+O6 if one assumes a \nspin-only moment for Ni2+ (2.83 μB), and takes the average μeff = 3.28 μB for Os5+ as \nobtained for some Os5+ double perovskites [ 3033]. Any possible Van -Vleck type \nparamagnetic contribution to µeff of the Os4+ ions is neglected in this rough estimate . \n \n \nFigure 5. (color online) Temperature dependence of the magnetic susceptibility of \nLa2Ni1.19Os0.81O6. The inset shows the corresponding χ1 vs T plot. \n \nTo further characterize the magnetic transition, the isothermal magnetization curves of \nLa2Ni1.19Os0.81O6 were measured above and below the transition temperature after zero \nfield cooling. The linear behavior of the M(H) data at 200 K is consistent with the \n 11 paramagnetic state. At 5 K, the M(H) curve collected between 50 kOe and +50 kOe \nshows a large hysteresis , but the hysteresis loop cannot be closed . Thus, the M(H) curve \nwas measured between 140 kOe and +140 kOe , see Figure 6. The magnetization is about \n0.65 μB/f.u. at 5 K and 140 kOe, but still not saturated. A remarkable feature of the M(H) \ncurve of La2Ni1.19Os0.81O6 is the high coercive field HC = 41 kOe at T = 5 K. The M(H) \ncurve was also measured after field cooling (50 kOe) and found to be identical to that \nmeasured after zero field cooling, no exchange bias effect is observed. \n \n \nFigure 6. (color online) The isothermal magnetization of La2Ni1.19Os0.81O6 measured at 200 \nK and 5 K. The small step near H = 0 may be due to a soft magnetic impurity. \n \nE. Magnetic structure \nIn order t o investigate the magnetic structure of La2Ni1.19Os0.81O6, we have collected \nseveral neutron powder patterns on the instrument E6 of BER II ( λ = 2.42 Å) from 1.7 up \nto 134 K . In agreement with our magnetization measurements a magnetic contribution to \nthe powder patterns could be observed below 125 K. The strongest magnetic intensity was \nobserved at the position of the reflections 011 , 101 and 10 1 (Fig. 7). For the Ni and Os \natoms at the positions 2 d(½,0,0; 0,½,½) and 2 c(½,0,½; 0,½,0) magnetic intensity could be \ngenerated with a model, where the spins located at both sites are coupled antiparallel with a \nspin sequence . In the pattern, shown in Fig. 7, addi tional weak intensities were \nobserved at 2 θ = 16.7 and 24 .1 °, which are close to the position of the reflection 001, and \nthat of the reflection pair 100/010 (forbidden in P21/n), respectively. The third reflection \nobserved at 21.5 ° can be indexed as (½½1) M belonging to the impurity La 3OsO 7 [26]. In \n 12 order to characterize these reflections in more detail , we have collected powder patterns at \n1.7 and 150 K on the instrument E2 with a much higher counting rate and better \ninstrumental resolution. From the difference pattern shown in the insert of Fig. 7 magnetic \nintensity was clearly detected for reflection 0 01, whereas for the reflection pair 100 and \n010 the magnetic contribution was found to be negligible. Therefore the intensity observed \nat 24.1 ° should belo ng to another unknown impurity. \nFrom our x-ray diffraction study , it was found that the osmium site 2 c site is partially \noccupied with Ni atoms giving a ratio occ(Ni1)/ occ(Os1) = 0.187(3)/0.813(3), while the 2 d \nsite is fully occupied by Ni2. For the determination of the moment values from the \nrefinements of the magnetic structure we have assumed that Ni1 and Ni2, located at 2 c and \n2d sites, have the same magnetic moment, while initially the Os moments were kept at zero. \nFurther , we used the positional and thermal parameters as well as the monoclinic angle β as \ndetermined more precisely from the analysis of the E9 data . In order to determine the \nmoment direction , we carried out Rietveld refinements using model structures . In the case \nof an ordering along the b and c axes, it could be seen that the magnetic peak significantly \nshifted to a higher 2θ value, while a satisfactory fit was obtained when the moments are \naligned to the a axis. In fact, due to the lattice distortions the reflection pair 101/10 1 is \nshifted to a high er 2θ value in comparison to the 011 (see inset of Fig. 7). From our \ncalculations , we obtained the following intensity ratios I(011/011)/I(101/10 1): ~3/1 (μ // \na), ~1/3 (μ // b), ~1/1 (μ // c). This shows that the best calculated peak position is reached \nwhen the strongest magnetic intensity appears on the reflection 011 generated with a \nmoment direction parallel to the a axis. In the next step, we allowed varying the magnetic \nmoment of the Os atoms at the site 2 c. Independently from the models used above the \nrefined Os moment was found to be 0.00(6) μ B. Therefore in the final refinement , the Os \nmoment was fixed to be zero. \n In order to generate magnetic intensity at the position of the reflection 001 two models \nwere considered, where the Ni and mixed Os/Ni sites (½,0,0 ), (0,½,½) , (½,0,½) and \n(0,½,0) ha ve an additional b spin component with the equences and , \nrespectively. The refinements showed that the spin sequence results in similar ly \ncalculated intensities for the reflections 100 and 001. But the inset of Fig. 7 shows that the \nmagnetic intensity of the 100 is much weaker than that of the 001 reflection . A rea sonable \nfit was obtained by using the spin sequence . Together with the spin component \nalong the a direction with the spin sequence one obtains a spin structure which is \nnoncollinear in the ab plane, but has collinear spin chains along the c direction (Fig. 8). 13 This spin arrangement is compatible with the crystal structure symmetry as shown by a \nrepresentation analysis [ 34]. For both atoms at the sites 2 d(½,0,0; 0,½,½) and 2 c(½,0,½; \n0,½,0) one finds an irreducible representation with the spin sequence along the b \ndirection and a sequence along a and c. Finally for the 1.7 K data set collected on E2 \nwe obtained the moment values μ x(Ni) = 1.77(5) μ B and μ y(Ni) = 0.60(12) μ B resulting in \na total moment μ exp(Ni) = 1.87(5) μ B and a residual RM = 0.073 (defined as RM = ∑|| Iobs| \n|Icalc||/∑|Iobs). In this n oncollinear magnetic structure , the moments form a tilting angle to \nthe a axis of 19(1) °. For the Rietveld refinements of the powder patterns collected on E6, \nwe have fixed the component μ y(Ni). Here we have obtained the component μ x(Ni) = \n1.81(5) μ B and a total moment μ exp(Ni) = 1.90(5) μ B, respectively. This value is only \nslightly smaller than the spin -only moment of 2.0 μ B expected for Ni2+ ions in octahedral \ncoordination environment ( t2g6eg2 electron configuration). The temperature dependence of \nthe obtained Ni moment, shown in Fig. 9, y ields an ordering temperature of 125(3) K. Due \nto the weakness of the magnetic reflection (001) M we only could follow the temperature \ndependence of the moment of the much stronger pronounced x component from E6 data. \n \n \nFigure 7. (color online) Neutron powder pattern of La 2Ni1.19Os0.81O6 taken on E6 at 1.7 K. \nThe calculated patterns [nuclear contribution (blue), sum of the nuclear and the magnetic \n20 30 40 50 60 70400080001200016000\n15 20 25 300500010000\n28 29 30 31 32300040005000\n012 111 001\n100 / 010\n101 / 011\n110 / 002\n200 / 112 / 020\n120 / 210\n122 / 212113 022 / 202211 / 121\n103 / 013La2Ni1.19Os0.81O6 at 1.7 K, E6Intensity (counts)\n2 (deg)N\nM\nIm100 / 010\n101 / 011001Difference\n1.7 - 150 K\nE2\n*\n011\n101 14 contribution (red) ] are compared with the observations (black -filled circles). The positions \n(black bars) of the nuclear ( N) and magnetic ( M) Bragg reflections of La2Ni1.19Os0.81O6, \nand those of the impurity phase La 3OsO 7 (Im) are shown. The region, where the stro ngest \nmagnetic reflection (011) M, and the pair (101) M and (101)M occur is shown in the inset . \nThe second inset (left) shows the presence of the magnetic reflection (001) M as obtained \nfrom the difference pattern (1.7 – 150 K) from E2 data. The weak magnetic intensity \nobserved at 21.5 ° (marked with an asterisk) can be assigned to the magnetic reflection \n(½½1) M of the impurity La 3OsO 7 [26]. All prominent magnetic reflections are labeled in \nred color. \n \nOur data analysis suggest s that the observed ferrimagnetic structure is driven by the \nantiferromagnetic interactions between Ni2+ ions on the 2 d sites and the additional Ni2+ \nions on the 2 c sites, which leads to a partial cancellation of magnetic moments. It is noted, \nhowever, that an unambiguous refinement of Os moments in double perovskites with two \nmagnetic ions at the B and B’ site is difficult [5] and thus it cannot be ruled out that \nordered Os5+ moments contribute to the stabilization of the ferrimagnetic spin structure. \n \n \n \n 15 Figure 8. (color online) Magnetic structure of La2Ni1.19Os0.81O6. For the Ni and mixed \nOs/Ni sites at 2 d(½,0,0; 0,½,½) and 2 c(½,0,½; 0,½,0) only the Ni atoms contribute to the \nmagnetic ordering. The Os site was found to be partially occupied with 19 % Ni. The Ni \nmoments at both sites were constrained to be equal during the Rietveld refinements \nresulting in a moment μ exp(Ni) = 1.90 (5) μ B. Therefore the total moment at the Os /Ni site \nonly reached the value 0.19 × μ exp(Ni) = 0.36 (1) μ B. For clarity , the mag nitude of the Ni \nmoment at the Os /Ni site is exaggerated. The magnetic atoms form ferrimagnetic chains \nalong the c axis, and within the ab plane one finds a noncollinear spin arrangement. \n \n \nFigure 9 . Temperature dependence of the magnetic moments of the Ni2+ ions in \nLa2Ni1.19Os0.81O6. The Wyckoff 2 c site is occupied with 81 % Os1 and 19 % Ni1 , while the \n2d site only contains Ni2. During the refinements of the magnetic moments , we have \napplied the constraint μ(Ni1) = μ(Ni2), while the magnetic moment of Os was set to be \nzero. The bold line is a guide for the eye. \n \n \nIV. DISCUSSION \n0 20 40 60 80 100 120 1400.00.40.81.21.62.0Magnetic moment / Ni2+ ion (B)\nTemperature (K)La2Ni1.19Os0.81O6\nTC = 125(3) K 16 We synthesized a non -stoichiometric double perovskite La 2Ni1.19Os0.81O6 containing \nmerely Ni2+ ions at the 2 d sites but a mixed occupan cy of paramagnetic (Ni2+, Os5+) and \nnonmagnetic (Os4+) ions at the 2 c sites of the monoclinic crystal structure. Formally the \ncompound may be written as La2Ni2+(Ni 0.192+Os0.434+Os0.385+)O6. The most remarkable \nproperty of this insulating compound is that it features a ferrimagnetic transition with an \nextraordinary broad hysteresis at 5 K. The spin structure is noncollinear in the ab plane. \nFor rationalizing the magnetic properties we consider first the so far unknown \nstoichiometric double perovskite La 2Ni2+Os4+O6 with the perfect atomic order , where the \nmagnetism should be entirely determined by antiferromagnetic exchange interactions \nbetween the half -filled eg orbitals of the Ni2+ ions as the Os4+ ions are expected to be \nnonmagnetic ( Jeff = 0). Therefore, for La 2NiOsO 6 antiferromagnetic ordering is anticipated \nas is indeed observed for other A2NiB’O6 compounds with nonmagnetic ions on the B’site \n(B’ = W6+, Ti4+, Ir5+) [3538], c.f. Table IV. The adopted antiferromagnetic spin structure \nwill be determined mainly by the balance between the nearest neighbor Ni2+OONi2+ \ninteractions and the next nearest neighbor Ni2+OOs4+ONi2+ exchange pathways. \nHowever, due to the non -stoichiometry the actual compound La 2Ni1.19Os0.81O6 contains a \nlarge fraction of paramagnetic Ni2+ and Os5+ ions at the 2 c sites, which is the origin for the \nobserved ferrimagnetism with TC = 125 K. Our analysis of the powder neutron diffraction \ndata indicates that the ferrimagnetism may be solely driven by the antiferromagnetic \ninteractions between th e majority of Ni2+ ions at the 2 d sites and the minority of Ni2+ ions \nat the 2 c sites, which implies that the Os5+ moments remain magnetically diso rdered and \npossibly freeze at low temperatures. However, this analysis may not be unambiguous and \nalso antiferromagnetic Ni2+OOs5+ interactions may contribute to the stabilization of the \nnoncollinear ferrimagnetic spin structure. The latter interactions are a consequence of the \ndistorted crystal structure. Since the t2g orbitals of Ni2+ are completely filled, the only \npossible antiferromagnetic nearest neighbor exchange pathway is via virtual hopping \nbetween half -filled Ni -eg and Os -t2g orbitals. The Ni -eg and Os -t2g orbitals are orthogonal in \nthe cubic double perovskite structure, but these interactions become possible in the \nmonoclinic double perovskite where the Ni OOs bond angles are strongly reduced from \n18o°. This me chanism was invoked for explaining the ferrimagnetic state in \nCa2Ni2+Os6+O6 [42]. Antiferromagnetic interactions between Ni2+ (d8) and Os5+ (d3) ions \nappear to be in contradiction to the Goodenough -Kanamori rules which predict a \nferromagnetic interaction due to virtual hopping between the half -filled and empty eg \norbitals at the Ni2+ and Os5+ sites, respectively. However, compared to pure 3 d systems like 17 La2NiMnO 6 [43,44] the c rystal field splitting at the 5 d sites is much enhance d and the \nferromagnetic coupling involving the σ-exchan ge pathway between the eg-orbital s becomes \nweak , in particular in the distorted double perovskite structure [45,46]. \nMost remarkably the magnetization of the present powder sample of La 2Ni1.19Os0.81O6 \nfeatures a broad hysteresis with a coercive field as high as 41 kOe. Similar giant coercive \nfields have been report ed for diver se materials classes . Examples include rare earth \ntransition metal films [ 47], ferrimagnetic transition metal – radical single chain magnets \n[48], intermetallic systems like Mn 1.5Ga films [49] as well as ferrimagnetic Fe 3Se4 \nnanostructures [ 50]. In transition metal oxides extremely high coercivities of 90 and 120 \nkOe were found for single crystals of the ferrimagnet LuFe 2O4 [51] and weak ly \nferromagnetic Sr 5Ru5xO15 [52], respectively. Such mater ials are of interest in the search of \nrare earth free hard magnets. The detailed mechanism of giant coercivity is not generally \nunderstood. However, large magnetocrystalline anisotropy in combination with frustration \nand/or defects and disorder appears to favor huge coercivities [51]. A large \nmagnetocrystalline anisotropy is indeed expected for La 2Ni1.19Os0.81O6 due to the low -\nsymmetry crystal structure and the presence of the heavy Os atoms . The non -collinear spin \narrangement in the ab plane can be attributed to the Dzyaloshinskii -Moriya interaction \nwhich induces spin -canting, whereas the high coercivity is possibly associated with a \npeculiar microstructure and competing exchange interactions introduced by the atomic \ndisorder. \n \nTABLE IV. Comparison of the magnetic properties of La 2Ni1.19Os0.81O6 with those of \nsome Ni2+ based double perovskite oxides. The ions on the B’ site of the double perovskite \nstructure are either non -magnetic (W6+, Ti4+, Ir5+) or magnetic (Os5+, Ir6+). \nMagnetic ions composition Space \ngroup Properties Tm (K) θw (K) µeff (µB) Ref. \nNi2+ Ca2NiWO 6 P21/n AFM 52.5 75 2.85 35 \nNi2+ Sr2NiWO 6 I4/m AFM 54 175 3.05 36 \nNi2+ La2NiTiO 6 P21/n AFM 25 60 3.12 37 \nNi2+ SrLaNiIrO 6 P21/n AFM 74 3.3 38 \nNi2+ + Os4.5+ La2Ni1.19Os0.81O6 P21/n FIM 125 73 3.63 This work \nNi2+ + Os5+ SrLaNiOsO 6 P21/n AFM 60 23 4.13 39 \nNi2+ + Os5+ CaLaNiOsO 6 P21/n AFM 30 83 3.87 40 18 Ni2+ + Ir6+ Sr2NiIrO 6 P21/n AFM 58 - - 41 \n \n \nConclusions \nTargeting the Os4+ (Jeff = 0) compound La2NiOsO 6, we actually obtained \nnonstoichiometric polycrystalline samples of La 2Ni1.19Os0.81O6 by solid st ate reaction . As \nmany other double perovskite oxides , La2Ni1.19Os0.81O6 crystallizes in the monoclinic \nspace group P21/n. Here, the B sites are fully occupied by Ni2+ ions whereas the B’ sites \nfeature a mixed occupancy with paramagnetic Ni2+ and Os5+ as well as non-magnetic Os4+ \nions. Whereas stoichiometric La 2NiOsO 6 is expected to be an antiferromagnet, \nLa2Ni1.19Os0.81O6 reveals a ferrimagnetic transition at 125 K and adopts a ferrimagnetic \nspin arrangement wi th collinear chains along the c axis, but spin canting in the ab plane. \nOur data analysis indicates that only Ni2+ magnetic moments are long -range ordered, but \nthe behavior of the Os5+ moments is not entirely clear . The magnetization curve of \nLa2Ni1.19Os0.81O6 is characterized by a huge coercive field, HC = 41 kOe at T = 5 K which \nis attribute d to the combined influence of large magnetocrystalline anisotropy and atomic \ndisorder . Our results suggest that careful tuning of the chemical composition may lead to \nthe discovery of new rare earth free hard magnets in the class of double perovskite oxides . \n \nAcknowledgement \nThe work in Dresden was partially supported by the Deutsche Forschungsgemeinschaft \nthrough SFB 1143. We thank R. Koban, H. Borrmann, and U. Burkhardt fo r performing \nmagnetization , x-ray, and EDX measurements. \n \nReferences \n1. K. L. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature, 395, \n677 (1998) . \n2. H. Kato, T. Okuda, Y. Okimoto, Y. Tomioka, Y. Takenoya, A. Ohkubo, M. \nKawasaki, and Y. Tokura, Appl. Phys. Lett. 81, 328 (2002) . \n3. Y. Krockenberger, K. Mogare, M. Reehuis, M. Tovar, M. Jansen, G. \nVaitheeswaran, V. Kanchana, F. Bultmark, A. Delin, F. Wilhelm, A. Rogalev, A. \nWinkler, and L. Alff, Phys. Rev. B 75, 020404 (2007) . 19 4. H. L. Feng, M. Arai, Y. Matsush ita, Y. Tsujimoto, Y. F. Guo, C. I. Sathish, X. \nWang, Y. H. Yuan, M. Tanaka, and K. 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Mater. 22, 5712 (2010) . \n " }, { "title": "1609.05855v1.Switching_ferromagnetic_spins_by_an_ultrafast_laser_pulse__Emergence_of_giant_optical_spin_orbit_torque.pdf", "content": "arXiv:1609.05855v1 [cond-mat.mtrl-sci] 19 Sep 2016epl draft\nSwitching ferromagnetic spins by an ultrafast laser pulse:\nEmergence of giant optical spin-orbit torque\nG. P. Zhang1, Y. H. Bai2andThomas F. George3\n1Department of Physics, Indiana State University, Terre Hau te, IN 47809, USA\n2Office of Information Technology, Indiana State University, Terre Haute, IN 47809, USA\n3Office of the Chancellor and Center for Nanoscience\nDepartments of Chemistry & Biochemistry and Physics & Astro nomy\nUniversity of Missouri-St. Louis, St. Louis, MO 63121, USA\nPACS75.78.Jp – Ultrafast magnetization dynamics and switching\nPACS75.40.Gb – Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic\nscaling, etc.)\nPACS78.20.Ls – Magneto-optical effects\nAbstract –Faster magnetic recording technology is indispensable to massive data storage and big\ndata sciences. All-optical spin switching offers a possible solution, but at present it is limited to a\nhandful of expensive and complex rare-earth ferrimagnets. The spin switching in more abundant\nferromagnets may significantly expand the scope of all-opti cal spin switching. Here by studying\n40,000 ferromagnetic spins, we show that it is the optical sp in-orbit torque that determines the\ncourse of spin switching in both ferromagnets and ferrimagn ets. Spin switching occurs only if the\neffectivespinangular momentumofeach constituentinan all oy exceedsacritical value. Because of\nthe strong exchange coupling, the spin switches much faster in ferromagnets than weakly-coupled\nferrimagnets. This establishes a paradigm for all-optical spin switching. The resultant magnetic\nfield (65 T) is so big that it will significantly reduce high cur rent in spintronics, thus representing\nthe beginning of photospintronics.\nIntroduction. – Magnetic switching is the single\nmost important operation for any modern magnetic stor-\nage device, where a magnetic field is employed to switch\nmicroscopicspins from one direction to another. However,\nasthearealdensityincreases,theswitchingspeedbecomes\na major bottleneck for future technological advancement.\nApossible solutionemergedwhenBeaurepaire et al.[1]re-\nported that a 60-fs laser pulse reduced the spin moment of\nferromagnetic nickel films within 1 ps. Their finding her-\nalded the arrival of femtomagnetism [2–4], and research\nefforts intensified immediately [5,6]. However, for over\na decade, the focus has been on demagnetization, not\nmagnetic switching. A major breakthrough came when\nStanciu and coworkers[7] demonstrated that a single laser\npulse could permanently switch the magnetic spin orien-\ntation in amorphous GdFeCo samples. This all-optical\nhelicity-dependent spin switching (AOS) ignited the re-\nsearch community since it may be an alternative to the\ncurrent magnetic storage technology [4]. However, most\nAOSsamplesareamorphous[8,9]andarehardtosimulatewithout significant approximations. To this end, a uni-\nfied understanding is still missing, but several promising\nmechanismshavebeenproposed,whichincludetheinverse\nFaraday effect [7,10], spin-flip stimulated Raman scatter-\ning [11,12], magnetic circular dichroism [13], magnetic\nsublattice competition [14], pure thermal effect [15,16]\nand ultrafast exchange scattering [17]. Recently, Lam-\nbertet al.[18] reported AOS in an ultrathin ferromag-\nnetic [Co(0.4 nm)/Pt(0.7 nm)] 3multilayer. Medapalli et\nal.[19] demonstrated that the helicity-dependent switch-\ning in Co/Pt proceeds in two steps [20]. Such a system is\nmuch more amenable to the simulation without any ma-\njor approximation, and its magnetic properties have been\nwell known for some time [21]. It is likely that a detailed\nstudy of such a system may shed new light on AOS.\nSpin reversal theory. – We employ a thin film of\n101×101×4or 40,804lattice sites in a simple cubic struc-\nture (see the top half of Fig. 1) with an open boundary\ncondition. Each site has a spin Siwhich is exchange-\ncoupled to the nearest neighboring spins through the ex-\np-1G. P. Zhang et al.\nchange interaction Jex. Our Hamiltonian [22–25], which\nis often used in magnetic multilayers [26], is\nH=/summationdisplay\ni/bracketleftbiggp2\ni\n2m+V(ri)+λLi·Si−eE(r,t)·ri/bracketrightbigg\n−/summationdisplay\nijJexSi·Sj, (1)\nwhere the first term is the kinetic energy operator of the\nelectron, the second term is the potential energy operator,\nλisthespin-orbitcouplinginunitsofeV/¯ h2,LiandSiare\nthe orbital and spin angular momenta at site iin the unit\nof ¯h, respectively, and pandrare the momentum and po-\nsition operators of the electron, respectively. To minimize\nthe number ofparameters, we choosea spherical harmonic\npotential V(ri) =1\n2mΩ2r2\niwith system frequency Ω, but\nthis approximation can be lifted when accurate potentials\nare known. Our model represents a small step towards a\ncompletemodel. Weassumethat theelectronmovesalong\nthezaxis with an initial velocity of 1 nm/fs in the har-\nmonic potential, so the initial orbital angular momentum\nis zero. The last term is the exchange interaction, and\nJexis the exchange integral in units of eV/¯ h2. Although\nour main interest is in ferromagnets, the same Hamilto-\nniancandescribeboth antiferromagnetsand ferrimagnets.\nSuch a Hamiltonian contains the necessary ingredients for\nAOS.\nFigure 1 shows that a laser pulse propagates along the\n+zaxis; its amplitude is attenuated according to Beer’s\nlawe−z/d(along + z), where dis the penetration depth.\nThe bottom half of Fig. 1 illustrates our idea of spin\ntorque to switch spins. For convenience, the spatial di-\nmension is measured in the unit of the lattice site number\nalong each direction, so that all the spatial variables are\ndimensionless or in the unit of the site number. The laser\nspot is centered at xc= 51 and yc= 51 with radius rand\nlateral spatial profile [10] e−[(x−xc)2+(y−yc)]2/r2(in thexy\nplane). The laser electric field is described by\nE(r,t) =A(t)exp[−(x−xc)2+(y−yc)2\nr2−z\nd],(2)\nwherexandyare the coordinates in the unit of the site\nnumber. Since in the following our spins are all initial-\nized along the −zaxis, we choose a left-circularly po-\nlarized field A(t) which has a Gaussian shape A(t) =\nA0e−t2/T2[−sin(ωt)ˆx+cos(ωt)ˆy],whereωis the laser car-\nrierfrequency, Tis the laser pulse duration, A0is the laser\nfield amplitude, tis time, ˆxand ˆyare unit vectors, respec-\ntively. We choose T= 100 fs. We only consider a reso-\nnant excitation where the laser photon energy ¯ hω= 1.6\neV matches the system energy ¯ hΩ; for an off-resonant\nexcitation, we refer the reader to a prior study [24]. In\ntransition metals, the penetration depth is about 14 nm,\nwhich corresponds to 30 layers, so we choose d= 30. To\ncompute the spin evolution, we employ the Heisenberg’s\nequation of motion, i¯h˙A= [A,H], where we make thetime-dependent Hartree-Fock approximation, so that all\nthe operators are replaced by their respective expectation\nvalues, and then we solve the equation numerically. Our\ncalculation of the spin change is similar to that of Wien-\nholdtet al.[27] though they used a thermal field.\nDependence of spin switching on spin angular\nmomentum. – We choose eight initial spin momenta\nSz(0) from 0 .2¯hto 1.6¯hin steps of 0.2¯ h, which covers\nmost magnetic materials. For each Sz(0), we vary the\nlaser field amplitude [3,19] A0from 0.01 to 0.08 V /˚A in\nsteps of 0.002 V /˚A. This step is tedious but necessary,\nsince different Sz(0) have different optimal field ampli-\ntudes for spin reversal. We fix the spin-orbit coupling at\nλ= 0.06eV/¯h2, the exchange interaction Jexat 1 eV/¯h2,\nand the spot radius of r= 100. The spins are initialized\nalong the −zaxis, equivalent to applying a magnetic uni-\naxial anisotropy. A spin reversal is considered achieved if\nthezcomponentspin angularmomentum Szchangesfrom\na negative value to a large and positive value at the end\nof the dynamics. Figure 2(a) shows the normalized and\nsystem-averaged spin as a function of time for each Sz(0)\nat its respective optimal laser field amplitude. All the\ncurves, except Sz(0) = 0.2¯h, are vertically shifted for clar-\nity. The dotted horizontal lines denote 0¯ h. We start with\nSz(0) = 0.2¯h, andwesee thatthe spin does notswitch and\nonly oscillates around 0¯ hwith a period determined by the\nproduct of λandSz(0) [24,28]. When we increase Sz(0)\nto 0.4¯h, the oscillation is attenuated and the final spin is\nbarely above 0¯ h. And the situation does not change much\nforSz(0) = 0.6¯h. However, when we continue to increase\nSz(0) above0 .8¯h, the spin ringing is stronglyreduced, and\nthe final spin settles down at a large positive value, an in-\ndication of spin reversal. Above 0 .8¯h, the situation gets\nbetter. For this reason, we define a critical spin angular\nmomentum Sc\nz= 0.8±0.2¯hfor AOS.\nTo quantify AOS, we define the spin switchability as\nη=Sf\nz\nSz(0)×100%,whereSf\nzis the final spin angular mo-\nmentum. This definition is different from that of Vahaplar\net al.[10]. We fix Sz(0) = 1.2¯h, but change the spin-orbit\ncoupling λ. Note that our conclusions are the same for\ndifferent Sz(0) as far as it is above Sc\nz. Figure 2(b) shows\nthat a minimum λof 0.04 eV /¯h2is required to reverse\nspins. Too small a λonly leads to a strong spin oscilla-\ntion, regardlessof the laser field amplitude. This indicates\na unique role of spin-orbit coupling (SOC) in AOS. The\nroles of the exchange interaction and laser field amplitude\nare shown in Fig. 2(c), where we fix Sz(0) = 1.2¯h,r= 100\nandλ= 0.06 eV/˚A2. We notice that as A0increases, Sz\nsharply increases and reaches its maximum. If we increase\nit further, Szis reduced since the spin overshoots, and an\nasymmetric peak is formed. This constitutes our first cri-\nterion that the laser amplitude must fall into a narrow\nregion for AOS to occur. This is consistent with Meda-\npalliet al.’s finding (see Fig. 1(c) of their paper [19]);\nsuch a helicity-dependent switching also agrees with an-\nother study by El Hadri et al.[20]. These agreements do\np-2Spin reversal\nnot necessarily validate all the aspects of our model but\ninstead they simply suggest that our model may offer an\nalternative to the existing models. If we increase A0fur-\nther, a second peak appears since the spin re-switching\nstarts. These double peaks do not appear for a smaller\nSz(0). We find that the exchange does not change this\ndependence a lot.\nPhase diagram of spin reversal. – We construct a\nphase diagram of spin reversal ( η−Sz(0)) in Fig. 3(a) for\nthirteen Sz(0)’s and two radii of the laser spot, r= 100\nand50. For ηtoexceed50-60%, Sz(0)mustbehigherthan\nthe critical value of Sc\nz= 0.8±0.2¯h. The long-dashed line\ndenotesSc\nz. We see that the nickel’s spin momentum is\nwell below Sc\nz, which explains why nickel has never been\nused for AOS. Co is on the threshold. In Co-Pt granular\nsamples [29], the effective spin magnetic moment per 3 d\nholeis 0.77 µB; sincethere are2.49-2.62holes, the spin an-\ngular momentum is 0.96¯ h, satisfying this criterion. In the\nultrathin ferromagnetic [Co(0.4 nm)/Pt(0.7 nm)] 3films\n[18], due to the reduced dimensionality, the enhanced spin\nmoment greatly increases the chance for AOS. The empty\nboxes in Fig. 3(a) represent the case with r= 50 (which\nis close to the switch limit), where only a small portion\nof the sample is exposed to the laser light. We see that\nthe switchability reduces sharply since the laser fluence\non lattice sites away from the center of the laser beam\nbecomes very weak and is not strong enough to reverse\nspins on those sites. Since the essence of AOS is rooted in\nspin-orbit coupling and all the switchabilities are obtained\nat the optimal field amplitude, we do not expect that a\nmore accurate potential would change the phase diagram\nstrongly. Our criterion not only applies to ferromagnets,\nbut also to ferrimagnets. Figure 3(b) illustrates that each\nof the major elements in all the 11 GdFeCo and TbFe al-\nloys [30] has the effective spin above Sc\nz. This constitutes\nstrong evidence that our finding has a broader impact on\nthe ongoing research in all-optical spin switching.\nEmergence of optical spin-orbit torque. – While\nthe effect of the laser field amplitude on AOS is obvious\n[31], how the initial spin Sz(0) affects the spin switching\nis not obvious. We examine how the spin evolves with\ntime. For a spin at site i, the spin angular momentum S\nprecedes according to\ndSi\ndt=/summationdisplay\nj(i)JexSi×Sj+λ(Li×Si), (3)\nwhere the two driving terms on the right-hand side rep-\nresent two torques. The first is the Heisenberg exchange\ntorqueτex=/summationtext\nj(i)JexSi×Sj. Since all the spins are fer-\nromagneticallyordered, this torque is verysmall. The sec-\nond one is the spin-orbit torque (SOT), τsoc=λ(Li×Si),\nwhich may serve as a source term for the inverse Faraday\neffect [32,33]. Before the laser excitation, τsocis small,\nsince in solids the orbital angular momentum Lis largely\nquenched. With the arrival of the laser pulse, Lis boostedsharply [32] (see Fig. 4) and helicity-dependent, where\nJex= 1eV/¯h2, andSz(0) = 1.2¯h, but three components\nof the orbital angular momentum behave differently. Lx\nandLyare mostly negative, but Lzis positive. Around 50\nfs,Lxreaches−0.24¯h, whileLyswings to −0.16¯hand the\nchange in Lzis smaller, around 0 .04¯h. All three compo-\nnents settle down to zero around 200 fs. This is very im-\nportant, since if the orbital momentum were big after the\nlaser field is gone, the spin would oscillate very strongly\nand could not be reversed faithfully. Thus, through the\nspin-orbit coupling, the laser field increases the orbital an-\ngular momentum, and subsequently τsocis boosted. For\nthis reason, Tesavova et al.[34]called τsocthe optical spin-\norbit torque, or femtosecond spin-orbit torque by Lingos\net al.[35].\nWe choose two initial spin momenta, Sz(0) = 0.3¯hand\n1.2¯h, with all the spins initialized along the −zaxis (see\nthe light blue arrows in Figs. 5(a) and (b)). Figure 5(a)\nshows that at 0 .3¯hthe spin undergoes strong oscillations\nand shows many spirals, but does not settle down to the\n+zaxis after the laser pulse is gone (see the red arrow).\nBy contrast, at 1 .2¯hthe spin flips over from the −zto +z\naxis within 110 fs, without strong oscillation (see the solid\nred arrow). To understand why the initial spin angular\nmomentum has such a strong effect on AOS, Figure 5(c)\nshows that τsocat 0.3¯his very weak, around 0.01 ¯ h/fs,\nand more importantly, it rapidly swings between positive\nand negative values, both of which are detrimental to the\nspin reversal. At Sz(0) = 1.2¯h,τsocis positive and large,\nwhich allows the spin to switch over successfully. This\nsuggests that SOT offers an alternative path to AOS (see\nthe bottom figure of Fig. 1), and it acts like an effec-\ntive magnetic field, which has been sought after in the\nliterature [10,15] for nearly a decade. At 1.2¯ h, we time-\nintegratethe torquefrom -200to +200fs and find that the\ntime-averaged torque corresponds to 65 T of a magnetic\nfield. In spintronics, the spin transfer torque heavily relies\non the high electric current [26,36]. Such a large SOT,\nif implemented in real experiments, should significantly\nreduce the requirement of huge electric current for spin-\ntronics [37], and thus opens a door for rapid applications\nin storage technology [38].\nConclusion. – We have investigated all-optical spin\nswitching in 40,000 ferromagnetic spins. We identify that\nit is the laser-induced optical spin-orbit torque that de-\ntermines the fate of spin switching. The spin-orbit torque\nsensitively depends on the value of the initial spin mo-\nmentum of each active element in a sample, regardless\nof the types of magnets. To switch, each active element\nmusthaveitseffectivespinangularmomentumlargerthan\n(0.8±0.2)¯h. This means that the switchability in Fe,\nGd and Tb is likely to be higher than Co and Ni. PMA\nobserved in various AOS materials [9] seems to be an indi-\ncation of enhanced spin moment, which is in line with our\ntheory. The ps all-optical spin switching observed in fer-\nrimagnets is associated with the weak exchange coupling;\np-3G. P. Zhang et al.\nin ferromagnets, with a stronger coupling, the switching\nis much faster. SOT is so large that it will significantly\nreduce the electric current used in spintronics. After our\npresentstudy wasfinished, we noticedarecentpublication\nby Bokor’s group [37] to use a laser to assist magnetiza-\ntion reversal. A combination of photonics and spintronics\nrepresents the arrival of photospintronics [39].\n∗∗∗\nWe would like to thank Dr. Hassdenteufel for sending\nus the experimental results [30]. This workwas solely sup-\nported by the U.S. Department of Energy under Contract\nNo. DE-FG02-06ER46304. Part of the work was done\non Indiana State University’s quantum cluster and high-\nperformance computers. The research used resources of\nthe National Energy Research Scientific Computing Cen-\nter, which is supported by the Office of Science of the U.S.\nDepartment of Energy under Contract No. DE-AC02-\n05CH11231. This work was performed, in part, at the\nCenter for Integrated Nanotechnologies, an Office of Sci-\nence User Facility operated for the U.S. Department of\nEnergy (DOE) Office of Science by Los Alamos National\nLaboratory (Contract DE-AC52-06NA25396) and Sandia\nNational Laboratories (Contract DE-AC04-94AL85000).\nREFERENCES\n[1] E. Beaurepaire, J. C. Merle, A. Daunois, and J.-Y. 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Novak, P.\nMalyet al., Nat. Photonics 7, 492 (2013).\n[35] P. C. Lingos, J. Wang, and I. E. Perakis, Phys. Rev. B\n91, 195203 (2015).\n[36] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater.\n320, 1190 (2008).\n[37] J. Bokor (private communication). Here the laser-indu ced\nspin-orbit torque was used to reverse magnetization.\n[38] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n[39] P. C. Mondal, P. Roy, D. Kim, E. E. Fullerton, H. Cohen,\nand R. Naaman, Nano Lett. 16, 2806 (2016).\np-4Spin reversal\nNyNzNxd\nJrx\ny\nz\nλ\nSLJex\nSpin switchingτ\nFig. 1: All-optical spin switching in ferromagnets. (Top) T he\nsimulated sample has the dimensions ( Nx= 101)×(Ny=\n101)×(Nz= 4), more than 40,000 spins. The light propagates\nalong the + zdirection with penetration depth dand radius\nof the spot r. (Bottom) The laser-induced optical spin-orbit\ntorque provides the necessary torque to reverse the spin.\n[40] B. Szpunar and B. Kozarzewski, Phys. Stat. Sol. (b) 82,\n205 (1977).−200 0 200 400\nTime (fs)−20246810121416Normalized spin0 0.020.040.060.08 0.1λ(eV/h−2)\n020406080100\nSwitchability η (%)\n0 0.02 0.04 0.06 0.08\nA0(V/Å)−2−101\nSz(h− )\nJex=0.5 eV/h−2\nJex=1.0 eV/h−2\nJex=1.5 eV/h−21.4h−1.6h−\n1.2h−\n1.0h−\n0.8h−\n0.6h−\n0.4h−\n0.2h−(a) (b)\n(c)Sz(0)=1.2h−\nSz(0)=1.2h−\n1\nFig. 2: (a) Time evolution of the zcomponent of the normal-\nized and system-averaged spin angular momentum for eight\ninitial spin values Sz(0) from 0.2¯ hto 1.6¯h. The spin rever-\nsal realized starts once Sz(0) is around and above 0.8¯ h. (b)\nSwitchability as a function of spin-orbit coupling. The cri tical\nvalue is around 0.04 eV/¯ h2. (c) Dependence of the final spin\non the laser field amplitude for three values of the exchange\nintegralJex.\n0.2 0.6 1 1.4 1.8 2.2\n Sz(0)[h− ]−20020406080100Switchability η (%)Theory\nr=100\nr=50\n−2 −1 0 1 2 3\nSzeff[h− ]Gd28Fe63Co9Gd26Fe64.7Co9.3Gd25Fe65.6Co9.4Gd24Fe66.5Co9.5Gd22Fe68.2Co9.8Gd22Fe74.6Co3.4Tb30Fe70Tb29Fe71Tb27Fe73Tb24Fe76Tb22Fe78Experimentthreshold(a) (b)\nSc\nz(0)=0.8h−\nnon−reversal region−0.8h−\n0.8h−NiCoFeGd,Tb\nGd\nFeFeTb\nFig. 3: (a) Phase diagram of the spin switchability versus th e\ninitial spin angular momentum Sz(0) at the respective optimal\nlaser field amplitudes. The empty circles and boxes refer to t he\nresults with r= 100 and r= 50, respectively. The long-dashed\nline denotes the critical spin Sc\nz. Two thin vertical lines repre-\nsent the spins for Ni and Fe. Co is on the border line, while Gd\nand Tb are way above Sc\nz. (b) Computed experimental effec-\ntive spin angular momentum for each element in 11 GdFeCo\nand TbFe alloys [30]. Without exception, all elements have\nspin larger than Sc\nz.\np-5G. P. Zhang et al.\n−200 0 200 400\nTime (fs)−0.4−0.3−0.2−0.100.1Orbital angular momentum (h− )\nLx\nLy\nLz\nFig. 4: Orbital angular momentum change as a function of\ntime. Here Jex= 1eV/¯h2, andSz(0) = 1.2¯h. The laser am-\nplitude is at its optimal value of 0.018V /˚A. The solid, dotted\nand dashed lines denote Lx,LyandLzcomponents [25], re-\nspectively.\n−101−101−101−\nSySz(0)=0.3h\nSxSz(a)\n−101−101−101−\nSySz(0)=1.2h\nSxSz(b)\n−101−101−101\nτyτxx10−2τz(c)\n−202−2020123\nτyτxx10−2τz(d)\nFig. 5: (a) Precession of the normalized spin angular mo-\nmentum in 3-dimensional space at Sz(0) = 0.3¯h. The blue\narrow denotes the initial spin, and the red one the final spin.\nThe trace of the final spin forms a spiral and the spin does\nnot switch. (b) The normalized spin angular momentum at\nSz(0) = 1.2¯his directly switched from the −zto +zaxis with-\nout precession [25]. (c) Time evolution of the spin-orbit to rque\natSz(0) = 0.3¯h. The torque is zero in the beginning. All\nthe torques are in the units of ¯ h/fs. (d) Same as (c) but for\nSz(0) = 1.2¯h[25].\np-6Spin reversal\nSupplementary Materials\nA main difference between ferrimagnets and ferromag-\nnets is that ferrimagnets have magnetic sublattices while\nferromagnetsdo not. To directly apply ourresults to ferri-\nmagnets, we need to understand whether these magnetic\nsublattice spins behave similarly to those spins in ferro-\nmagnets. Fortunately, we find that according to our sim-\nulation, at least in the weak laser field limit, left(right)-\ncircularlypolarizedlightonlyswitchesspinfromdown(up)\nto up(down), not the other way around. Therefore, flip-\nping a sublattice spin in ferrimagnets is equivalent to flip-\nping a spin in ferromagnets. This is the theoretical basis\nto apply our theory to ferrimagnets.\nIn the following, we explain how the spin angular mo-\nmentum is computed from the experimental data. Ex-\nperimentally, the measured magnetic property is often the\nremanent magnetization mRin units of 103A/m, or equiv-\nalently emu/cc [30]. To convert magnetization to spin an-\ngular momentum, we need the volume of the sample, but\nexperimentallythevolumeofthesampleisnotgiven. This\nmakes a quantitative comparison nearly impossible.\nWe find a simple and powerful method which does not\nrely on the experimental volume. Take R xT1−xalloy as\nan example, where R stands for Tb or Gd and T stands\nfor Fe. We ignore Co since its concentration is too low.\nWe first compute the effective volume\nVeff=xVR+(1−x)VT, (4)\nwhereVRandVTare the supercell volume of pure element\nRandT.Tbhasahcpstructure, withthelatticeconstants\na= 3.601˚A andc= 5.6936˚A; Fe has a bcc structure\nwitha= 2.8665˚A. Then we multiply mRbyVeffto\nget the effective spin moment for the alloy, i.e., Meff=\nmRVeff. SinceMeffis in the units of [Am2], we convert\nit to the Bohr magneton µB, with the conversion factor of\n0.10783×10−3.\nSzpunar and Kozarzewski [40] carried out extensive cal-\nculations on transition-metal and rare-earth intermetallic\ncompounds by comparing their results with the experi-\nmentalones, and concludedthat it is reasonabletoassume\nthat the average magnetic moments of the transition met-\nals and of the rare earth metals are roughly independent\nof structures. Then, the effective spin moment Meffcan\nbe approximately written as\nMeff=xMR+(1−x)MT≡Meff\nR+Meff\nT,(5)\nwhereMRandMTare the spin moments of pure R and T,\nrespectively. Here the last equation defines the effective\nspin moment for R and T.\nHowever, this single equation is not enough to compute\nMRandMTsince there are two unknowns for a single\nequation. The trick is that we use two sets of composi-\ntions,x1andx2, so we have two equations,\nM(1)\neff=x1MR+(1−x1)MT (6)\nM(2)\neff=x2MR+(1−x2)MT, (7)whereM(1)\neff=m(1)\nRV(1)\neffandM(2)\neff=M(2)\nRV(2)\neff. Here\nagain we rely on the assumptions that MRandMTdo\nnot change much with composition change from x1tox2.\nWhen we choose x1andx2, we are always careful whether\nMRorMTchangessign, since experimentallythe reported\nvalues are the absolute value. In addition, it is always\nbetter to choose those x1andx2which have the same\nsign ofMRandMT. Choosing several different pairs of\n(x1,x2) is crucial to a reliable result. Solving the above\ntwo equations, we can find MRandMT.\nBefore we compute the spin angular momentum, we\ncheck whether the computed spin moments MRandMT\n(in the units of µB) are within the respective value of\neach pure element, i.e., M◦\nGd= 7.63µB,M◦\nTb= 9.34µB,\nandM◦\nFe= 2.2µB. If the computed spin moment ( MR\nandMT) is far off from those spin moments, this indicates\nthat either our method or the experimental result is not\nreliable; as a result, their spin angular momentum is not\nincluded in our figure, but is included here. Once the spin\nmoment passes this test, we proceed to convert the spin\nmoment to spin angular momentum.\nOur method works better for Gd alloys than Tb alloys,\nsince the former has zero orbital angular momentum but\nthe latter has a nonzero orbital angular momentum. For\nGd and Fe, the orbital momentum is largely quenched.\nAssuming that the Lande g-factor is 2, we divide the spin\nmoments MRandMTby 2 to get the spin angular mo-\nmentum SRandSTin the unit of ¯ h. To get the effective\nspin angular momentum, we multiply SRandSTwithx\nand 1−x, respectively, i.e.,\nSeff\nR=xSR (8)\nSeff\nT= (1−x)ST. (9)\nIt is these two effective spin angular momenta that we ap-\nply our above criterion to. For Tb, our results have an\nuncertainty since its orbital angular momentum in its al-\nloys is unknown, although its orbital angular momentum\nin pure Tb metal is 3.03¯ h. Table 1 shows the orbital-free\nspin angular momentum for 11 alloys, where we adopt a\nsimple cubic structure for Fe since it matches the exper-\nimental values better. These data are used to plot Fig.\n3(b) of the main paper.\nBefore we show all the details of our results, we wish\nto present an example how the spin moment changes with\nthe concentration under our assumption. Since R and T\nare ferrimagnetically coupled and MRandMTdiffer by\na sign,Meffchanges from a positive value to a negative\nas the composition xchanges. Figure 6 shows such an\nexample, where we use the experimental value MGd=\n7.63µBof pure Gd and MFe= 2.2µBof pure Fe. A V\nshape curve is formed, the same as the experiment [30].\nMeffis close to zero around x= 0.22. The effective spin\nmomenta Seff\nGdfor Gd and Seff\nFefor Fe are also shown (use\nthe right axis). As xincreases, Seff\nGdincreases but |Seff\nFe|is\nreduced. Two long-dashed lines denote our critical values\n±Sc\nz. We use the dotted line box to bracket the narrow\np-7G. P. Zhang et al.\nwindow for spin switching. Since this window is very close\nto the compensation point (in term of the concentration\nx), this explains why Hassdenteufel et al.[8] found that\nthelowremnantmagnetizationforAOSmustbebelow125\nemu/cc. This is the direct consequence of the requirement\nof the critical value of Sc\nz.\nInthefollowing,wetabulateallthecomputedresultsfor\nboth GdFeCo and TbFe alloys, respectively. All the tables\nstart with the spin moment for each element, followed by\nthe effective spin angular momentum for each element in\nthe alloys. To reduce possible errorsin those experimental\ndata, we always choose multiple pairs of data for the same\nalloy.\nGd alloys. We start with a pair of Gd 24Fe66.5Co9.5\nand Gd 22Fe68.2Co9.8. Table 2 (the first two rows) shows\nthat Gd has a magnetic moment of -10.2187 µBand Fe\n3.9149µB, respectively,wherewepurposelykeepmoresig-\nnificant figures to show the accuracy of our results. Note\nthat Gd and Fe are ferrimagnetically coupled, so they dif-\nfer by a negative sign. By comparing them with their re-\nspective element values, we conclude that these moments\nare reasonable. We then compute the effective spin angu-\nlar momentum for Gd and Fe (see the third and fourth\ncolumns). Clearly, both numbers are larger than our criti-\ncal spin angular momentum. Then we compute four addi-\ntional combinations of alloys. If the experimental results\nwere exact and free of any error, the obtained effective\nspin angular momentum should not change. However, in\nreality, they do change, but we find that the changefor Gd\nalloys is very small. For instance, Gd 22Fe68.2Co9.8is used\ntwice, but each case has a similar Seff\nGdandSeff\nFe(compare\nthe first pair and third pair). The same is also true for\nGd22Fe74.6Co3.4, but when we pair Gd 22Fe68.2Co9.8with\nGd22Fe74.6Co3.4, wefind a slightlylargerchange. The rea-\nson is easy to understand since these two compounds have\na very similar composition, and the relative error becomes\nlarger.\nTable 3 assumes a bcc structure for Fe. Here the values\nare all reduced somewhat, but the main conclusion re-\nmains the same. We also find that the biggest error comes\nfrom the alloy pairs with a similar composition (see the\nlast pair). We notice that Seff\nFeis slightly below our criti-\ncalvalue. Itislikelythatthelargerrelativeerrorwhentwo\ncompositions are close is responsible for the discrepancy.\nTb alloys. In comparison with Gd alloys, Tb alloys\nare more prone to errors, since their orbital momentum is\nnot completely quenched. Table 4 shows two different sto-\nries. For the first seven pairs, we see that all the moments\nfor Tb and Fe are reasonably close to their respective ele-\nment moments. But for the last two pairs, their values are\nway too low. We know why this occurs. Tb 36Fe64is not\nthe AOS compound, and only shows the pure thermal de-\nmagnetization. From the first seven pairs, we see that the\nexperimental result for Tb 30Fe70is reliable, since different\npairs give a similar spin moment. But when it is paired\nwith Tb 36Fe64, it leads to an unreasonable result. Thismeansthat the structure-propertyofTb 36Fe64isquite dif-\nferent from the AOS compounds such as Tb 30Fe70, and it\nmay not have the linear relation between the spin moment\nand the composition xas we assume above. Our finding\nis backed by the last pair, where Tb 34Fe66is not an AOS\ncompound initially, and only after the heating does it be-\ncome AOS. If we look at Hassdenteufel’s Fig. 7 [30], we\nfind that Tb 34Fe66does not follow the trend of the rest\nof the TbFe alloys. For this reason, they are not included\nin Table 1. Table 4 shows all the spin angular moments\nthat are computed, with zero orbital angular momentum.\nTable 5 shows the same data but with bcc structure for\nFe. The main conclusion is the same as Table 4.\nHelicity-dependent and helicity-independent all-\noptical spin switchings. – There is enormous interest\nin both the all-optical helicity-dependent spin switching\n(AO-HDS) and the all-optical helicity-independent spin\nswitching (AO-HIDS). Experimentally, Stanicu et al.[7]\nfirst demonstrated a clear helicity-dependent switching in\nGdFeCo, but when they [15] later increased the laser in-\ntensity, the switching became helicity-independent. In\nother words, there is a clear transition from a helicity-\ndependent switching to a helicity-independent switching\nin GdFeCo when one increases the laser fluence. The un-\nderlying reasonof this transition has been unclear, though\nthere are several mechanisms proposed [15]. On the other\nhand, Lambert et al.[18] demonstrated a clear helicity-\ndependent switching in their Co(0 .4 nm)/Pt(0.7 nm)] 3\nfilm. At low laser power (362 nW), they showed that a re-\nversed domain is written for σ+, but not for σ−or linearly\npolarized light. When they increased the laser power, re-\ngionsofdemagnetizedrandom domainsdeveloped. There-\nfore, the helicity-independent switching does not occur in\nCoPt films. This may suggest that the difference between\nAO-HDSand AO-HIDS canboth be laser-intensitydepen-\ndent and material-dependent. Our model, which is solely\nbased on a ferromagnet, does show a helicity-dependent\nswitching. The light helicity is important for our model\nto work, as far as the laser fluence is small. For instance, if\nthe spin points down (the −zaxis), only the left-circularly\npolarized light can efficiently switch the spin up, if the\nlaserfieldamplitudeisweak. Thisfindingappearstoagree\nwith the experimental results reasonably well [18]. If the\nlaser field becomes too stronger, we are not completely\nconfident whether our model can describe the physics cor-\nrectly, though we did test the model in a single site case\n[24], where we found that the switching becomes highly\nnonlinear, and even the linearly polarized light can switch\nthe spin. For this reason, our present paper exclusively fo-\ncuses on the lower laser field limit and ferromagnets. We\nare currently exploring whether our model can describe\nGdFeCo. Our present model, without further change, is\nunsuitable for GdFeCo since GdFeCo is amorphous, ferri-\nmagnetic and much more complicated. At minimum, we\nhave to include the magnetic sublattices.\np-8Spin reversal\nTable 1: Computed effective spin angular momentum for each\nelement in GdFeCo and TbFe alloys. Multiple pairs of alloys\nare used to compute the effective spin angular momentum for\nseveral compounds to demonstrate the range of the change in\nthe spin angular momentum. The sign convention of the spin\nangularmomentumis thateither Gdor Tbhasapositivevalue,\nwhile Fe has a negative value. The original signs of those spi n\nangular momentum are shown in Tables 2 through 5. A simple\ncubic structure is adopted for Fe.\nAlloy Seff\nGd(¯h)Seff\nFe(¯h)Seff\nTb(¯h) (orbital free) Seff\nFe(¯h) (orbital free)\nGd28Fe63Co91.3414 -1.1691 – –\nGd26Fe64.7Co9.31.2456 -1.2006 – –\nGd25Fe65.6Co9.41.1517 -1.1777 – –\nGd24Fe66.5Co9.51.2262 -1.3017 – –\nGd24Fe66.5Co9.51.0867 -1.0113 – –\nGd22Fe68.2Co9.81.1241 -1.3350 – –\nGd22Fe68.2Co9.81.0135 -1.2244 – –\nGd22Fe68.2Co9.80.9846 -0.7737 – –\nGd22Fe74.6Co3.40.9846 -0.8463 – –\nTb30Fe70 – – 2.1506 -1.8385\nTb30Fe70 – – 1.7594 -1.4473\nTb30Fe70 – – 1.4952 -1.1831\nTb30Fe70 – – 1.4698 -1.1577\nTb29Fe71 – – 2.0789 -1.8648\nTb27Fe73 – – 1.5835 -1.5093\nTb27Fe73 – – 1.1867 -1.1125\nTb24Fe76 – – 1.2641 -1.3452\nTb24Fe76 – – 1.1758 -1.2569\nTb22Fe78 – – 1.2789 -1.5007\nTb22Fe78 – – 1.1587 -1.3806\nTb22Fe78 – – 1.0965 -1.3183\nTb22Fe78 – – 0.9669 -1.1887\np-9G. P. Zhang et al.\nTable 2: Computed effective spin angular momentum for each\nelement in GdFeCo alloys for simple cubic Fe structure.\nSeff\nGdSeff\nFe\nGd -10.2187 µB –\nFe 3.9149 µB –\nGd24Fe66.5Co9.5 -1.2262¯ h1.3017¯h\nGd22Fe68.2Co9.8 -1.1241¯ h1.3350¯h\nGd 9.5818 µB –\nFe -3.7114 µB –\nGd28Fe63Co9 1.3414¯ h-1.1691¯h\nGd26Fe64.7Co9.3 1.2456¯ h-1.2006¯h\nGd -9.2139 µB –\nFe 3.5907 µB –\nGd25Fe65.6Co9.4 -1.1517¯ h1.1777¯h\nGd22Fe68.2Co9.8 -1.0135¯ h1.2244¯h\nGd -9.0562 µB –\nFe 3.0415 µB –\nGd24Fe66.5Co9.5 -1.0867¯ h1.0113¯h\nGd22Fe74.6Co3.4 -0.9962¯ h1.1345¯h\nGd 8.9509 µB –\nFe -2.2689 µB –\nGd22Fe68.2Co9.8 0.9846¯ h-0.7737¯h\nGd22Fe74.6Co3.4 0.9846¯ h-0.8463¯h\nTable 3: Computed effective spin angular momentum for each\nelement in GdFeCo alloys for bcc Fe structure.\nSeff\nGdSeff\nFe\nGd -7.4560 µB –\nFe 2.8615 µB –\nGd24Fe66.5Co9.5 -0.8947¯ h0.9514¯h\nGd22Fe68.2Co9.8 -0.8202¯ h0.9758¯h\nGd 7.4926 µB –\nFe -2.9045 µB –\nGd28Fe63Co9 1.0490¯ h-0.9149¯h\nGd26Fe64.7Co9.3 0.9740¯ h-0.9396¯h\nGd -6.7695 µB –\nFe 2.6400 µB –\nGd25Fe65.6Co9.4 -0.8462¯ h0.8659¯h\nGd22Fe68.2Co9.8 -0.7446¯ h0.9002¯h\nGd -6.6669 µB –\nFe 2.2355 µB –\nGd24Fe66.5Co9.5 -0.8000¯ h0.7433¯h\nGd22Fe74.6Co3.4 -0.7334¯ h0.8339¯h\nGd 6.7532 µB –\nFe -1.7221 µB –\nGd22Fe68.2Co9.8 0.7428¯ h-0.5873¯h\nGd22Fe74.6Co3.4 0.7428¯ h-0.6424¯hTable 4: Computed effective spin angular momentum for each\nelement in TbFe alloys for simple cubic Fe structure.\nSeff\nTbSeff\nFe\nTb 14.3375 µB –\nFe -5.2529 µB –\nTb30Fe70 2.1506¯ h-1.8385¯h\nTb29Fe71 2.0789¯ h-1.8648¯h\nTb 11.7296 µB –\nFe -4.1352 µB –\nTb30Fe70 1.7594¯ h-1.4473¯h\nTb27Fe73 1.5835¯ h-1.5093¯h\nTb -11.6262 µB –\nFe 3.8479 µB –\nTb22Fe78 -1.2789¯ h1.5007¯h\nTb19Fe81 -1.1045¯ h1.5584¯h\nTb -10.5340 µB –\nFe 3.5399 µB –\nTb22Fe78 -1.1587¯ h1.3806¯h\nTb24Fe76 -1.2641¯ h1.3452¯h\nTb 9.9679 µB –\nFe -3.3802 µB –\nTb22Fe78 1.0965¯ h-1.3183¯h\nTb30Fe70 1.4952¯ h-1.1831¯h\nTb 9.7986 µB –\nFe -3.3076 µB –\nTb30Fe70 1.4698¯ h-1.1577¯h\nTb24Fe76 1.1758¯ h-1.2569¯h\nTb -8.7901 µB –\nFe 3.0480 µB –\nTb22Fe78 -0.9669¯ h1.1887¯h\nTb27Fe73 -1.1867¯ h1.1125¯h\nTb 5.3268 µB –\nFe -1.3912 µB –\nTb36Fe64 0.9588¯ h-0.4452¯h\nTb30Fe70 0.7990¯ h-0.4869¯h\nTb 2.1818 µB –\nFe -0.4756 µB –\nTb34Fe66 0.3709�� h-0.1570¯h\nTb24Fe76 0.2618¯ h-0.1807¯h\np-10Spin reversal\nTable 5: Computed effective spin angular momentum for each\nelement in TbFe alloys for bcc Fe structure.\nSeff\nTbSeff\nFe\nTb 11.2034 µB –\nFe -4.1158 µB –\nTb30Fe70 1.6805¯ h-1.4405¯h\nTb29Fe71 1.6245¯ h-1.4611¯h\nTb 9.0822 µB –\nFe -3.2067 µB –\nTb30Fe70 1.3623¯ h-1.1224¯h\nTb27Fe73 1.2261¯ h-1.1705¯h\nTb -7.8075 µB –\nFe 2.6098 µB –\nTb22Fe78 -0.8588¯ h1.0178¯h\nTb19Fe81 -0.7417¯ h1.0570¯h\nTb -7.4627 µB –\nFe 2.5126 µB –\nTb22Fe78 -0.8209¯ h0.9799¯h\nTb24Fe76 -0.8955¯ h0.9548¯h\nTb 7.4621 µB –\nFe -2.5124 µB –\nTb22Fe78 0.8208¯ h-0.9798¯h\nTb30Fe70 1.1193¯ h-0.8793¯h\nTb 7.4619 µB –\nFe -2.5123 µB –\nTb30Fe70 1.1193¯ h-0.8793¯h\nTb24Fe76 0.8954¯ h-0.9547¯h\nTb -6.3789 µB –\nFe 2.2069 µB –\nTb22Fe78 -0.7017¯ h0.8607¯h\nTb27Fe73 -0.8611¯ h0.8055¯h\nTb 4.4942 µB –\nFe -1.2404 µB –\nTb36Fe64 0.8090¯ h-0.3969¯h\nTb30Fe70 0.6741¯ h-0.4342¯h\nTb 1.7919 µB –\nFe -0.4100 µB –\nTb34Fe66 0.3046¯ h-0.1353¯h\nTb24Fe76 0.2150¯ h-0.1558¯h\np-11G. P. Zhang et al.\n0 0.2 0.4 0.6 0.8 1\nGd−Concentration x−202468|Meff(µΒ)|\n0 0.2 0.4 0.6 0.8 1−202468\nSGdeff(h− ), SFeeff(h− )\nGd\nFe0.8h−\n−0.8h−\nFig. 6: Effective spin moment change as a function of Gd\nconcentration x(circles). The two solid lines represent the\neffectivespinangular momentumfor GdandFe(usingtheright\naxis). The two horizontal dashed lines denote the predicted\ncritical spin angular momentum ( ±0.8¯h). The dotted line box\nhighlights the narrow region of the Gd concentration where\nspin angular momentum satisfies our criterion and the spin\nreversal occurs.\np-12" }, { "title": "1604.07534v1.Weak_localization_effect_in_topological_insulator_micro_flakes_grown_on_insulating_ferrimagnet_BaFe12O19.pdf", "content": "1\nScientific RepoRts | 6:21334 | DOI: 10.1038/srep21334www.nature.com/scientificreportsWeak localization effect in \ntopological insulator micro flakes \ngrown on insulating ferrimagnet \nBaFe12O19\nGuolin Zheng1,*, Ning Wang1,*, Jiyong Yang1, Weike Wang1, Haifeng Du1, Wei Ning1, \nZhaorong Yang1,2, Hai-Zhou Lu3, Yuheng Zhang1,2 & Mingliang Tian1,2,4\nMany exotic physics anticipated in topological insulators require a gap to be opened for their topological \nsurface states by breaking time reversal symmetry. The gap opening has been achieved by doping magnetic impurities, which however inevitably create extra carriers and disorder that undermine the \nelectronic transport. 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 /< 10 \ntesla [15]) in antiferromagnets (AFMs) offers room -temperature intrinsic THz frequency [16-\n19]. It has been theoretically predicted that the domain -wall (DW) velocity in AFMs is limited \nby the maximum spin -wave group velocity ( vg,max) [19]. As the DW velocity ( vDW) approaches \nvg,max, its increment becomes smaller, and the excessive energy provided b y the driving force \nis dissipated in the form of THz spin-wave emission . Althou gh the abundant AFMs are good \nplatform s for theoretical studies, the complete magnetization cancellation prevents \nexperimental det ermination of magnetic states. By contrast, the ferrimagnets (FiMs) [20-23] \nwith antiparallel exchange coupled rare earth (RE) and transition metal (TM) alloys possess \nfinite magnetization s due to the unequal sublattices, and many interesting phenomenon s have \nbeen predicted in FiMs, such as the self -focusing skyrmion racetracks [24], and the fast DW \nmotion without Walker breakdown [21,25 -27]. Therefore, the FiMs are suitable materials for \ndeveloping ultrafast spintronic devices by taking advantages of both ferromagnets (FMs) and \nAFMs properties. 3 \n In this article , we theoretically study the current -driven DW motion in the \nFiM[Gd x(FeCo) 100‒ x]/heavy metal tungsten (W) bilayer (see Fig. 1(a)) , where vDW firstly \nincreases and then saturates at large current density ( Jc), in contrast to the linear trend predicted \nby the analytical model based on the collective coordinate approach [28], which describes the \nDW motion in one dimension using two variables (i.e., the location and magnetization angle \nof the DW center) and assumes a rigid DW profile . We then show that this deviation can be \ncorrected by applying the Lorentz contraction on the DW width ( λ) to incorporate the upper \nbound of vg,max due to the relativistic effect . In addition, the DW motion in the velocity \nsatura tion region is accompanied by the emission of THz spin waves. Although many \nexperimental studies on the current -driven DW mot ion in FiMs have been conducted [25-\n27,29,30] , evidence of spin -wave emission s has not been identified. To understand the \ncondition s of spin -wave emissions, the parametric effects are studied, and we find that the \ncritical current density ( Jsw) required to excite the spin wave increases dramatically with the \nexchange constant ( Aex) and exceeds 1012 A m–2, which can lead to device breakdown and thus \nprevents experimental verifications. To excite THz spin waves at small Jsw, the suitable \nmaterial requires small Aex, large crystal anisotropy ( K), and large Dzyaloshinskii -Moriya \ninteractions constant ( D). Moreover, we propose a three -terminal device structure to \nelectrically detect the THz signal, where the collective magnetization oscillation is translated \ninto voltage signals through the tunnel -magnetoresistance (TMR) effect. Finally, we show that \na unifo rm continuous signal with improved output power can be obtained by red ucing the \ndamping constant ( α), and a high speed DW motion with vDW = 1.5 km s–1 is predicted . These \nresults could provide insights for the experimental investigation of THz spin wave, fast \nracetrack memory, and applications using the THz spintronic oscillator. \n \nII. Methods 4 \n The spin -orbit torque driven domain -wall motion in FiM is modelled using the one-\ndimensional atomistic model [31-33], which includes antiferromagnetic coupled TM and RE \nelements. The Hamiltonian is given by\n22\nex 1 1ˆ ˆ ˆ ( ) ( ) ( )i i i j i j i i i\ni i i iE A K D S S S z S x y S S\n, where κj is the d omain -\nwall hard -axis anisotropy . The spin dynamics of each sublattice is described by the atomistic \nLandau -Lifshitz -Gilbert equation 𝜕𝐒𝑖/𝜕𝑡=−𝛾𝑖𝐒𝑖×𝐁eff,𝑖+𝛼𝑖𝐒𝑖×𝜕𝐒𝑖/𝜕𝑡−𝛾𝑖ℏ𝐽c𝜃SH/\n(2𝑒𝑀S,𝑖𝑡FiM)[𝐒𝑖×(𝐒𝑖×𝐲̂)−𝛽𝐒𝑖×𝐲̂], where ћ is the reduced P lanck constant , θSH is the spin -\nHall angle, e is the electron charge, Ms,i is the saturation magnetization, tFiM is the thickness of \nthe ferrimagne tic layer, γi =giμB/ћ is the gyromagnetic ratio where gi is the g-factor and μB is \nthe Bohr magneton, Beff,i = –(1/μi)∂Ei/∂Si is the effect ive field where μi = Ms,id 3, and β denotes \nthe ratio between the field-like torque (FLT) and damping -like torque ( DLT) . The four terms \non the right -hand s ide are precession, damping, DLT , and FLT, respectively. The parameters \nused in our simulation are summarized as follows. Aex = 3 meV, KTM = K RE = 0.04 meV, κTM \n= κRE = 0.2 μeV, D = 0.128 meV, αTM = αRE = 0.015 , tFiM = 0.4 nm, gTM = 2.2, gRE = 2, θSH = \n0.2, ρTa = 200 × 10–8 Ω m, ρFiM = 248× 10–8 Ω m [34], TMR = 100% [35], RP = 500 Ω [36,37] . \nThe thickness of the W and FiM layer are 5 nm and 0.4 nm, respectively. \nThe temperature , which can be tuned to change the material from Gd to FeCo \ndominated regions, is an important parameter in FiMs. The effect of temperature is generally \ntaken into account by describing the change of magnetization using a power -law [27,38,39] . \nRecent studies [27,40] show that the temperature change induced by Joule heating might \nchange the net magnetization from the Gd to FeCo dominant . However, in this study, we \nassume a negligible effect from Joule heating by limiting Jc below 1012 A m–2 and choosing the \nsubstrate with high cooling efficiency [41,42] . Therefore, we focus on the magnetization 5 \n dynamics in the FiM at a stable temperature of 300 K with Ms,TM = 1149 ×103 A m–1 and Ms,RE \n=1012 ×103 A m–1. \n \nIII. Device structure and THz spin-wave generation \nAs shown in Fig. 1 (a), a bilayer structure with the FiM deposited on top of the tungsten \nlayer is first studied to understand the spin -wave generation . With sufficient Dzyaloshinskii -\nMoriya interactions (DMI ), the initial magnetic state in the FiM is Né el wall with right -handed \nchirality due to the positive sign of DMI in W (Note 1 in [43]), in agreement with ref. [44]. Jc \nin the x direction generates vertical spin current polarized along the y axis, and t he resulting \nspin-orbit torque (SOT) efficiently moves the DW in one direction depending on the current \npolarity . According to the theoretical study based on the collective coordinate approach [28], \nvDW of a stable Né el wall is a linear function of Jc as \nvDW=(sFeCo+sGd)πλBD/(4α), (1) \nwhere sFeCo(Gd) = Ms,FeCo(Gd) /γFeCo(Gd) is the angular momentum for FeCo (Gd), and BD = \nћθSHJc/(2etFiMsFeCosGd) is the effective SOT field. In contrast to th e straight line predicted by \nequation (1), numerical simulations using the atomistic model (see Fig. 1 (b)) show an increase \nand saturation trend . For Jc > Jsw, the spin wave emerges at the DW and vanishes in the dire ction \nopposite to the DW motion (see Note 2 in [43]). The frequency of the spin wave is identified \nin the THz range by performing fast Fourier transform (FFT) on the magnet ization . The spatial \nprofile of the spin wave is shown in Fig. 1 (c), where the DW is located at 148 nm and moves \nagainst the electron flow [44]. On the left side of the DW, mx and my show strong spatial \nvariations , corresponding to the atoms precessing at different phase and amplitude as \nschematically depicted in Fig. 1 (a). The spatial profile of spin waves is related to the generation \nmetho d. For example, the spin wave with uniform amplitudes can be generated by applying 6 \n microwave field to the whole sample [45], whereas the one generated by spin -torque nano -\noscillators [46-48] has the largest amplitude at the current injection point and propagates to the \nsurroundings with reduced amplitude due to the inevitable damping. Similar to the second case, \nthe source of spin waves in this study is located at the DW, and the oscillation amplitude \nreduces in atoms which are far away from the source. Therefore, the generation of spin waves \nreduces the DW energy, inhibiting the linear increase of vDW as a function of Jc. Theoretical \nstudies have identified that vDW is limited by vg,max due t o the rel ativistic effect [19,21] , which \nis supported by the result that the velocity trend shown in Fig. 1 (b) can be well explained by \nadding the Lorentz contraction into the collective coordinate approach (see more discussions \nin Fig. 3 ). Furthermore, we have numerically verified that the spin wave generated in our device \nis able to move another chiral DW in FiM. Similar phenomenon has been predicted in FM [49] \nand AFM [50]. The interplay between spin wave and DW [51-55] can then be explore d to \noptimize the racetrack memory. \nDespite these theoretical predictions and numerical results, no direct evidence of spin -\nwave emission has yet been experimentally observed [25-27]. Therefore, we investigate the \ncondition s for spin-wave emission by studying the dependence of Jsw on material parameters . \nAs shown in Fig. 1 (d), Jsw increases dramatically and rapidly surpasses 1012 A m–2 for Aex > 8 \nmeV, and this trend remains the same when K or D is cha nged ( shown in Note 3 in [43]). The \nlarge Jsw, as a primary obstacle in experiments , can easily lead to sample breakdown . In \naddition, since the spin current generated by the spin -orbit coupling is polarized in the y \ndirection, large Jc would distort the Né el wall, resulting in inefficient DW motion [19]. \nTherefore, the sample with small Aex, large K, and large D is required for a stable Né el wall \nwith small Jsw, which can facilitate the e xperimental exploration of spin -wave emission. \n \nIV. Electrical detection using a three -terminal structure 7 \n Spin wave s are mainly detected using microstrip antennas [56] or optical approaches \n[57], which are unsuitable for the integrated device s. In contrast, the giant magnetoresistance \n(GMR) and TMR are widely used in spintronics for detecting magnetization states by measuring \nvoltage signals [58-64]. Limited by the lateral size of magnetic stacks (> 30 nm), the collective \nspins with nonuniform amplitude have to be utilized (as shown in Fig. 1 (c)). Using the three -\nterminal structure schematically illustrated in Fig. 2 (a), we show that the THz spin wave can \nbe electrically detected, and the frequency of the output signal is identical to that of the single \nspin. The generation of spin wave has been discussed in the previous section, and the detection \nof magnetization states can be realized by passing a small current ( IMTJ) throug h the magnetic \ntunne l junction ( MTJ ) consist ing of the FM, MgO, and F iM layers . When the DW passes \nthrough the MTJ , the dynamics of collective spins are manifested as the change in resistance \ndue to the TMR effect , resulting in alternating electrical signal s (Vo) across the MTJ . To extract \nthe alternating component of VMTJ, a bias-tee is used to measure the THz signal at the radio \nfrequency (RF) port. Two independent current sources are used in two separate channels, \nenabling the independent control of vDW and Vo. The effect of IMTJ on DW motion is negligible, \nand Ic does not direc tly affect Vo because the vertical current is vanished (Note 5 in [43]). As a \nresult, the output power can be enhanced by simply increasing IMTJ. In order t o correctly capture \nthe current distributions , a dist ributed resistance model , as shown in Fig. 2 (b), is developed and \nsolved analytically (Note 4 in [43]). When the current distribution is obtained , the ave raged \ncurrent passing through the W layer is used to calculate the SOT , followed by the simulation \nof magnetization dynamics using the atomist ic spin model ( Methods ). The MTJ resistance , as \na function of the averaged FiM magnetization, is recomputed at each time step, and then the \ncurrent distribution is calculated again . This process is repeated to get the time evolution of \nmagnetizations . In addition , Vo can also be improved using the MTJ with large TMR , which \nhas been reported in several studies , such as 15% in the TbCoFe/CoFeB/MgO/CoFeB/TbCoFe 8 \n MTJ [36], and 55% in the GdFeCo/CoFe/Al 2O3/CoFe/TbFeCo MTJ [35]. It is worth noting \nthat the detailed MTJ stack is less important as long as it can provide sufficient TMR. \nTherefore, the se FiM -based MTJ s with large TMR can be used readily in this proposal to \nenhance the output signal. \nThe proof -of-concept oscillation for the averaged my and ΔVo, defined as Vo – 252.465 \nmV, are shown in Fig s. 2(c) and 2 (d), respectively. A clear oscillation pattern appears when \nthe DW passes through the MTJ at 95 ps ( referr ed to Note 6 in [43]), and then it gradually \nvanishes as the DW moves further away. As discussed in Fig. 1 (b), the spin wave is originated \nfrom the energy dissipation of DW. With the DW moving away, the atom s lose energy and \nslowly return to the stable state due to the damping . The resulting Vo oscillates with a peak -to-\npeak amplitude of 14.5 μV. As shown in the inset of Fig. 2 (d), the main frequency ( f) of Vo is \n437 GHz , which is identical to tha t of single spin precession (cf. Note 6 in [43]), thus supporting \nthat Vo has the same origin with the spin wave . Since these atoms also experience the SOT, \nthey precess with negligible amplitude at fSOT. This is similar to the current -induced jiggling in \nFMs, which is determined by the combined field of SOT, exchange, and anisotropy (cf. the \ninset of Fig. 2 (d) and Note 6 in [43]). \n \nV. Appearance of relativistic effect and performance improvement \nTo elucidate the origin of the spin wave emission , we first study the relation between \nvDW and Jc. According to equation (1), vDW, plotted in F ig. 3(a) using the solid line, is a linear \nfunction of Jc, contradict ing to the saturation trend as shown in Fig. 1 (b). In addition, vDW \nwithout Lorentz contraction at Aex = 8 meV and Jc = 1012 A m–2 is 5.9 km s–1, which is four \ntimes larger than that from the atomistic simulation. To resolve these discrepancies , it has been \npointed out that the DW motion in AFMs or FiMs with large DMI requires additional \nconsideration of the relativistic effect [19,21] , which imposes the Loren tz contraction on λ, 9 \n resulting in the deviation from its equilibrium value ( λeq) by a factor of √1−(𝑣DW/𝑣g,max)2, \nwhere vg,max is obtained from the dispersion relation of the spin wave. Such correction is similar \nto the length measurement of a fast moving object, which, according to the special relativity, \nshould be modified by the Lorentz factor √1−(𝑣/𝑐)2 with v and c denoting the speed of \nobject and the speed of light, respectively. According, the Lorentz contraction is applied to vDW \nof the current -driven DW motion in FiM, and then equation (1) is modified as \n𝑣DW=𝑏(𝑠FeCo+𝑠Gd)π𝜆eq√1−(𝑣DW/𝑣g,max)2𝐵D/(4𝛼), (2) \nwhere vg,max=8Aex/[ d 2(sFeCo+sGd)] with the lattice constant d = 0.4 nm , and b = 0.27 is a fitting \ncoefficient. Assuming that 𝜆eq=𝑑√2𝐴ex/𝐾, the corrected vDW, plotted as the dash line in Fig. \n3(a), shows a clear saturation trend with the maximum vDW below 1 km s–1. \nIn addition, the atomistic simulation predicts that vDW increases linearly with Aex (shown \nas the circles in Fig. 3 (b)), whereas it is a square root relation according to equation (1), as \nshown in Fig. 3 (b) using the solid line. By correcting vDW using the Lorentz contraction, vDW \npredicted by equation (2) (see the dash line of Fig. 3 (b)) shows a good agreement with the \nnumerical result. Due to the large Jc = 1012 A m–2 used in Fig. 3 (b), the Né el wall is distorted \nand a sizeable Bloch component (i.e., my) appears in all the studied sam ples. It has been \ndiscussed in r ef. [19] that the damping -like SOT cannot move the Bloch wall, and hence it is \nreasonable to use b < 1 to get quantitative agreements. Therefore, equation (1) from the \ncollective coordinate approach is insufficient to explain the numerical results, and the \nintroduction of the relativistic correction leads to excellent agreements. \nNext, w e study vDW as a function of Jc at different Aex. As shown in Fig. 3 (c), high speed \nDW motion s with a maximum vDW = 1.5 km s–1 appears at Aex = 8 meV , which can be useful \nin applications such as the race track memory [65]. Similar to Fig. 1 (d), where the MTJ structure 10 \n and distributed resistance model are excluded , Jsw increases dramatically with Aex. Although \nthere are debates on the relative amplitude of the DLT and FLT in the magnetic layer attached \nto the heavy metal [66-69], it is well accepted that both torques are sizeable. Using the FLT to \nDLT ratio β = 1.1 , which is in accordance with the Rashba -Edelstein effect [67], we find that \nthe FLT , albeit wi th larger magnitude, has negligible effect on vDW (cf. the third panel of Fig. \n3(c)). The conclusion holds under different FLT strength or even the FLT with an opposite sign \n[70]. This is consistent with the study in AFM that the FLT does not a ffect the dynamics of \nNé el wall [19]. Despite the substantial Bloch component at large Jc, the FLT in this study is \ninsufficient to change vDW [19]. \nIn addition to the fast DW motion, the frequency of Vo can be tuned in a wide range, \nwhich is advantage ous over existing devices. For example, the wireless communication system \nat 200 to 300 GHz relies on the SiGe or Si -CMOS techno logy [4,71] , whereas GaN, InP, or \nphotonics devices oscillated at higher frequency are required for applications above 500 GHz \n[4]. The necessity of integrating different technologies complicates the transceiver design. As \nshown in Fig . 4(a), f, as a function of Aex, can be changed in a wide range from 264 GHz to 1.1 \nTHz, which can be intuitively understood as consequences of the increased effective field . The \nreduced frequency below 300 GHz may also enable experimental veri fication using real -time \noscilloscope. Therefore, the large frequency window in the spin -wave based spintronic \noscillator offers a unified platform for the THz applications. \nBesides the frequency tunability, a uniform signal with large output power is also \npreferred to simplify the peripheral circuits [71]. Since the vanish ing of spin wave is induced \nby the loss of energy input when the DW moves away, the uniformity of oscillation signal can \nbe improved by either reducing vDW or α. Since α is directly associated with material properties , \nwe investigate the effect of α in this study . As shown in Figs. 4(b) and 4 (c), Vo, in which the \nmain frequency remains the same, becomes more uniform and its magnitude is four times larger 11 \n when α is changed from 0.015 to 0.01 . Moreover , fSOT becomes more manifested at smaller α. \nTo understand this, the atoms precessing at different frequencies are investigated . The \nreduction of α increases the relaxation time of all atoms, most of which are precessing at fSOT \nwithout emitting spin waves . As a result , the relative amplitude change is larger in fSOT \ncompared to that in f. However , Vo is still dominated by f because of its larger averaged \noscillation amplitude . Therefore , the engineering of material parameters can significantly \nimprove the device performance, opening up possibilities of using spintronic oscillators in \nwideband applications. \n \nVI. Discussion and Conclusion \nIn conclusion, we have studied the condi tions of THz spin-wave emission in the FiM/W \nbilayer under the SOT. The DW velocity is limited by the maximum group velocity due to the \nrelativistic effect, and the excessive energy is dissipated in the form of THz spin wave s. The \ncritical current requi red for the spi n-wave emission increases dramatically in materials with \nstrong exchange coupling, and this trend remains the same when K or D is changed. Therefore, \nthe FiM with small Aex, large K, and large D is suitable for the experimental demonstration. We \nthen propose the electrical detection of the THz spin wave in a three -terminal structure by \ntranslating the collective magnetization oscillation into voltage signals. We show that a wide \nfrequency range from 264 GHz to 1.1 THz , a uniform continuous signal with improved output \npower , and a fast DW motion at 1.5 km s–1 can be obtained by optimizing material parameters . \nThis work promotes understanding of magnetization dynamics in FiM , provides candidates for \nthe fast racetrack memory, and should stimulate the desi gn of THz oscillator for wideband \napplications using a unified platform . \n \n 12 \n References \n[1]. P.H. Siegel. Terahertz technology in biology and medicine. 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Terahertz Sci. \nTechnol. 1, 477 (2011). \n \nCorresponding Authors : *a0132576@u.nus.edu , †elelg@nus.edu.sg \n \n Acknowledgements \nThis work at the National University of Singapore was supported by CRP award no. NRF -\nCRP12 -2013 -01, and MOE -2017 -T2-2-114. \n \n \n 16 \n Figures \n \nFigure 1. Illustration and condition for the spin wave emission. (a), Sche matic view of the THz \nspin wave generated in the FiM/W bilayer , where the red and blue color in the FiM layer \ndistinguish the up and down domain respectively . The length and width of W and FiM are 400 \nnm and 40 nm, respectively. The current moves the Né el wall along the +x direction. The spin \nwave is initiated at the DW and vanishes in –x direction. The precession frequency of each spin \nis identified in THz range. (b), The DW velocity as a function of Jc. The dash line separates \nregions without (region 1) and with (region 2) spin wave. (c), The spatial profile of \nmagnetization at Jc = 4× 1011 A m–2 and A = 5 meV , where the DW is located at 148 nm . The \nDW moves to the right as marked by the brown arrow . (d), Critical current density for the spin -\nwave emission as a function of exchange constant. \n \n \n17 \n \nFigure 2 . Electrical detection of the THz signal. (a), Schematics of the three -terminal structure . \nThe length and width of both MgO and FM layer are 40 nm. The FM layer has in -plane easy \naxis along in the +y direction. The bias tee is a three -port diplexer consisting of a capacitor and \nan inductor. Ic = 160 μA and IMTJ = 16 μA are used in this study unless otherwise stated . (b), \nSimulation workflow. The current distribution is calculated using the distributed resistance \nmode l, and then t he averag e current passing through the W layer is input to the atomistic spin \nmodel to get the magnetization dynamics. At each time step, the MTJ resistance (RM) is updated \nbased on the magnetization of the FiM layer (mFiM), followed by the recalculation of current \ndistribution. This process forms a closed loop and free running to get the time e volution of \nmagnetizations. The number of segmentation nA = 10, nB = 10, n = 13 which gives 0.03% \ndifference in IHA compared to n = 193 ( cf. Note 5 in [43]). Refer to Note 4 in [43] for the \nnotation of parameters. (c), Time evolution of my and ΔVo, defined as Vo – 252.465 mV, at Ic = \n160 μA and IMTJ = 16 μA. my is obtained by averaging spins under the MTJ. The reduction of \noscillation amplitude is induced by the loss of energy i nput when the DW moves away. (d), \nThe frequency spectrum for the time r egion starting from 97.26 ps, marked by the black dot, to \n200 ps. The main frequency is originated from spin wave at 437 GHz and a smaller SOT \ninduced precession with negligible amplitude appears at 213 GHz , showing as the inset . \n18 \n \nFigure 3. Relativistic effect on the domain -wall motion. (a), vDW as a function of Jc at Aex = 5 \nmeV for systems with (dash line) and without (solid line) the Lorentz contraction . (b), vDW as \na function of Aex with Ic = 160 μA and IMTJ = 16 μA. Results from the atomistic model are \nshowing as the blue dots. The analytical results based on equations (1) and (2) are plotted as \nthe solid and dash line, respectively. Aex above 8 meV is not considered since the Jsw is above \n1012 A m–2, which can easi ly leads to device breakdown. (c), vDW as a function of Jc at different \nAex. The dash lines define the critical current density, above which the spin waves are emitted . \nThe effect of FLT is studied in the sample with Aex = 5 meV, where the black squares denote \nvDW with β = 1.1 . \n19 \n \nFigure 4 . Parametric effect on the oscillator performance. (a), Numerical results of f as a \nfunction of Aex with Ic = 160 μA and IMTJ = 16 μA. The maximum frequency is 1.1 THz at Aex \n= 8 meV. (b, c), Time evolution (b) and frequency spectrum (c) of Vo in samples with different \ndamping constant at Ic = 160 μA and IMTJ = 16 μA. The curves are vertically offset for clarity, \ni.e., 36 μV in (b) and 3.4 μV in (c). Vo becomes more uniform and maintains for a longer time \nwhen the damping constant is reduced. f remains nearly unchanged for different α, i.e., f = 437 \nGHz , 442 GHz, and 428 GHz for α = 0.015, 0.012, and 0.01, respectively. A substantial fSOT \nappears at 209 GHz for α = 0.01. \n" }, { "title": "1706.07118v2.Spin_pumping_and_shot_noise_in_ferrimagnets__bridging_ferro__and_antiferromagnets.pdf", "content": "Spin pumping and shot noise in ferrimagnets: bridging ferro- and antiferromagnets\nAkashdeep Kamra\u0003and Wolfgang Belzigy\nDepartment of Physics, University of Konstanz, D-78457 Konstanz, Germany\nA combination of novel technological and fundamental physics prospects has sparked a huge in-\nterest in pure spin transport in magnets, starting with ferromagnets and spreading to antiferro- and\nferrimagnets. We present a theoretical study of spin transport across a ferrimagnet jnon-magnetic\nconductor interface, when a magnetic eigenmode is driven into a coherent state. The obtained spin\ncurrent expression includes intra- as well as cross-sublattice terms, both of which are essential for a\nquantitative understanding of spin-pumping. The dc current is found to be sensitive to the asym-\nmetry in interfacial coupling between the two sublattice magnetizations and the mobile electrons,\nespecially for antiferromagnets. We further \fnd that the concomitant shot noise provides a useful\ntool for probing the quasiparticle spin and interfacial coupling.\nIntroduction. The quest for energy e\u000ecient informa-\ntion technology has driven scientists to examine uncon-\nventional means of data transmission and processing.\nPure spin current transport in magnetic insulators has\nemerged as one of the most promising candidates[1{4].\nHeterostructures composed of an insulating magnet and\na non-magnetic conductor (N) enable conversion of the\nmagnonic spin current in the former to the electronic\nin the latter, thereby allowing for their integration with\nconventional electronics. In conjunction with the techno-\nlogical pull, these low dissipation systems have provided\na fertile playground for fundamental physics [5{7].\nCommencing the exploration with ferromagnets (Fs),\nthe focus in recent years has been shifting towards an-\ntiferromagnets (AFs) [8{10] due to their technological\nadvantages [11]. While a qualitative understanding of\nsome aspects of AFs, such as spin pumping [12, 13],\nhas been borrowed without much change from Fs, the\nleading order e\u000bects in several other phenomena, such\nas spin transfer torque [13] and magnetization dynam-\nics [8], bear major qualitative di\u000berences. Thus, several\nphenomena, already known for Fs, are now being gener-\nalized for AFs [14].\nAlthough ferrimagnets ( Fs) have been the subject of\ncomparatively fewer works [7, 15, 16], their high potential\nis undoubted. The additional complexity of their mag-\nnetic structure comes hand in hand with broader pos-\nsibilities and still newer phenomena. The spin Seebeck\ne\u000bect [17{19] in an Fwith magnetic compensation tem-\nperature has unveiled rich physics due to the interplay\nbetween the opposite spin excitations in the magnet [16].\nFurther studies have asserted an important role of the\ninterfacial coupling between the magnet and the conduc-\ntor [20]. While yttrium iron garnet is a ferrimagnet and\nhas been the subject of several studies [1, 3, 4, 21{23],\nit is often treated as a ferromagnet on the grounds that\nonly the low energy magnons are important [24].\nIn this Letter, we evaluate the spin pumping current\n(Isz) and the concomitant spin current shot noise [ S(\n)]\nin aF-N bilayer [Fig. 1(a)], when one of the Feigen-\nmodes is driven into a coherent sate. A two-sublattice\nmodel with easy-axis anisotropy and collinear ground\n(a)\n (b)\nFIG. 1. (a) Schematic of the magnet (M) jnon-magnetic con-\nductor (N) heterostructure under investigation. Equilibrium\nmagnetization for sublattcies A (blue) and B (red) point\nalong ^zzzand\u0000^zzz, respectively. An eigenmode in M is driven\ncoherently and injects z-polarized spin current into N. (b)\nSchematics of possible interface microstructures. Shaded re-\ngions around each spin represent the wavefunction cloud of\nthe localized electrons composing the spin. Our model en-\ncompasses compensated as well as uncompensated interfaces\nincluding lattice disorder.\nstate is employed. Our model continuously encompasses\nsystems from ferromagnets to antiferromagnets, thereby\nallowing analytical results for the full range of materials\nwithin a uni\fed description. It further allows arbitrary\n(disordered) interfaces. In addition to the bulk asym-\nmetry, stemming from inequivalent sublattices, we \fnd a\ncrucial role for the interfacial coupling asymmetry (Fig.\n2), consistent with the existing experiments [16, 20] and\ntheoretical proposals [25]. Such an asymmetry may oc-\ncur even in a perfect crystalline interface [Fig. 1(b)] due\nto the nature of the termination or the di\u000berent wave-\nfunction clouds of the electrons constituting the local-\nized spins in the two sublattices. Spin transport in AF-N\nbilayers is found to be particularly sensitive to the inter-\nfacial asymmetry, with spin current nearly vanishing for\nsymmetrical coupling of the two sublattices with N cor-arXiv:1706.07118v2 [cond-mat.mes-hall] 7 Nov 20172\nFIG. 2. Normalized spin current vs. bulk ( tB=MA0=MB0)\nand interfacial ( tI= \u0000AA=\u0000BB) asymmetries for lower fre-\nquency uniform mode in coherent state. All other bulk pa-\nrameters are kept constant, no external magnetic \feld is ap-\nplied, andIN= 2~j\u001fj2!qqq\u000bAB. The spin current for tB= 1\n(also depicted in the inset for clarity) is small due to the\nspin-zero quasiparticles in symmetric AFs, and it abruptly\nincreases with a small bulk symmetry breaking due to quasi-\nparticle transformation into spin ~magnons [7]. The di\u000berent\nparameter values employed are given in the supplemental ma-\nterial [26].\nresponding to the case of a compensated interface (Fig.\n2).\nA key result of our work is the following semi-classical\nexpression for the spin current injected into N [27]:\ne\n~Isz=X\ni;j=fA;BgGij(^mmmi\u0002_^mmmj)z=X\ni;j=fm;ngGij(iii\u0002_jjj)z;\n(1)\nwhere ^mmmA(B)is the unit vector along sublattice A (B)\nmagnetization, mmm= [^mmmA+^mmmB]=2,nnn= [^mmmA\u0000^mmmB]=2,\nGmm=GAA+GBB+2GAB,Gnn=GAA+GBB\u00002GAB,\nandGmn=Gnm=GAA\u0000GBB. Employing GAB=\nGBA=pGAAGBB, which is derived, along with the\nexpressions for GAAandGBB, in subsequent discussion\nbelow, we further obtain Gmm= (pGAA+pGBB)2and\nGnn= (pGAA\u0000pGBB)2. Our result [Eq. (1)] for the\ninjected spin current adds upon the existing understand-\ning of spin pumping via AFs [13] by (i) providing analytic\nand intuitive expressions for the conductances, (ii) incor-\nporating the cross terms characterized by GABandGmn,\n(iii) deriving the relation GAB=pGAAGBBbased upon\na microscopic interfacial exchange coupling model, (iv)\naccommodating compensated ( GAA=GBB) as well as\nuncompensated interfaces, and (v) allowing for interfacial\ndisorder. As detailed in the supplemental material [26],\nthe spin pumping expression given in Ref. [13] is recov-\nered from Eq. (1) by substituting GAB=GBA= 0 and\nGAA=GBB, and yields results qualitatively di\u000berentfrom what is reported herein [26]. This di\u000berence in re-\nsults stems from the assumption made in Ref. [13] that\n^mmmAand ^mmmBare independent variables, which is equiva-\nlent to setting GAB=GBA= 0 implicitly. ^mmmAand ^mmmB\nare coupled via inter-sublattice exchange and hence can-\nnot be treated as independent when considering system\ndynamics.\nWe de\fne the dynamical spin correction factor SDvia\nthe relation SD\u0011limT!0S(0)=2~Isz, whereTis the\ntemperature and S(0) is the low frequency spin current\nshot noise. When the e\u000bect of either the dipolar inter-\naction [28] or the sublattice coupling on the eigenmode\nunder consideration can be disregarded, SD~coincides\nwith the spin of the eigenmode. In other words, when\na full 4-dimensional (4-D) Bogoliubov transform [7] is\nrequired to obtain the relevant eigenmode, SDis a prop-\nerty of the entire heterostructure and depends upon the\nbulk as well as the interface. Thus, shot noise o\u000bers a\nuseful experimental probe of the interfacial properties as\ndiscussed below.\nModel. The model we study consists of a two-sublattice\nmagnet coupled via interfacial exchange interaction to a\nnon-magnetic conductor [Fig. 1(a)]. We assume MA0\u0015\nMB0with the respective sublattice saturation magnetiza-\ntionsMA0;MB0. The bulk of the magnet is characterized\nby a classical free energy density which is then quantized,\nusing the Holstein-Primako\u000b transformations [29{31], to\nyield the magnetic contribution to the quantum Hamil-\ntonian ~HMin terms of the magnon ladder operators.\nWe consider Zeeman ( HZ), easy-axis anisotropy ( Han),\nexchange (Hex) and dipolar interaction ( Hdip) (see foot-\nnote [28]) in the magnetic free energy density written in\nterms of the A and B sublattice magnetizations MA(rrr)\nandMB(rrr). With an applied magnetic \feld H0^zzzand\n\u00160the permeability of free space, the Zeeman energy\ndensity reads HZ=\u0000\u00160H0(MAz+MBz). The easy-\naxis anisotropy is parametrized in terms of the con-\nstantsKuA; KuBasHan=\u0000KuAM2\nAz\u0000KuBM2\nBz[31].\nThe exchange energy density is expressed in terms of\nthe constantsJA;JB;JABandJ[31]:Hex=P\nxi=x;y;z[JA(@MMMA=@xi)\u0001(@MMMA=@xi) +JB(@MMMB=@xi)\u0001\n(@MMMB=@xi) +JAB(@MMMA=@xi)\u0001(@MMMB=@xi)] +JMMMA\u0001\nMMMB. The dipolar interaction energy density is obtained\nin terms of the demagnetization \feld HHHmthat obeys\nMaxwell's equations in the magnetostatic approximation:\nHdip=\u0000(1=2)\u00160HHHm\u0001(MMMA+MMMB) [7, 30, 31]. Quantiz-\ning the magnetization \felds and employing the Holstein-\nPrimako\u000b transformation, we obtain the quantum Hamil-\ntonian for the magnet:\n~HM=X\nqqq\u0014Aqqq\n2~ay\nqqq~aqqq+Bqqq\n2~by\nqqq~bqqq+Cqqq~aqqq~b\u0000qqq+Dqqq~aqqq~a\u0000qqq\n+Eqqq~bqqq~b\u0000qqq+Fqqq~aqqq~by\nqqqi\n+ h:c: ; (2)\nwhere ~aqqqand ~bqqqare, respectively, sublattice A and\nB magnon annihilation operators corresponding to3\nwavevector qqq. Relegating the detailed expressions for\nthe coe\u000ecients Aqqq; Bqqq\u0001\u0001\u0001to the supplemental mate-\nrial [26], we note that Cqqqis dominated by the intersub-\nlattice exchange while Dqqq,Eqqq,Fqqqresult entirely from\ndipolar interaction. The magnetic Hamiltonian is di-\nagonalized via a 4-D Bogoliubov transform to new op-\nerators [7] ~ \u000bqqq=ulqqq~aqqq+vlqqq~by\n\u0000qqq+wlqqq~ay\n\u0000qqq+xlqqq~bqqqand\nsimilar for ~\fqqq:~HM=P\nqqq~!lq~\u000by\nqqq~\u000bqqq+~!uq~\fy\nqqq~\fqqq. The\nsubscriptslandurefer to lower and upper modes thus\nassigning the lower energy to ~ \u000bmodes. The diago-\nnal eigenmodes are dressed magnons with spin given by\n~(juqqqj2\u0000jvqqqj2+jwqqqj2\u0000jxqqqj2) [7]. Disregarding dipolar\ninteraction, the eigenmode spin is plus or minus ~. Incor-\nporating dipolar contribution, the spin magnitude varies\nbetween 0 and greater than ~[7].\nThe non-magnetic conductor is modeled as a bath\nof non-interacting electrons: ~HN=P\nkkk;s=\u0006~!kkk~cy\nkkk;s~ckkk;s,\nwhere ~ckkk;sis the annihilation operator corresponding to\nan electron state with spin s~=2 along z-direction and\norbital wavefunction kkk(rrr). The conductor is coupled\nto the two sublattices in the magnet via an interfacial\nexchange interaction parameterized by JiA,JiB:\n~Hint=\u00001\n~2Z\nAd2%X\nG=A;B\u0010\nJiG~SSSG(%%%)\u0001~SSSN(%%%)\u0011\n;(3)\nwhere%%%is interfacial position vector, Ais the interfacial\narea, ~SSSA,~SSSBand ~SSSNrepresent spin density operators\ncorresponding to the magnetic sublattices A, B and the\nconductor, respectively. In terms of the eigenmode ladder\noperators, the interfacial exchange Hamiltonian reduces\nto [32]:\n~Hint=~X\nkkk1;kkk2;qqq1\u0010\n~Pkkk1kkk2qqq1+~Py\nkkk1kkk2qqq1\u0011\n; (4)\nwhere ~Pkkk1kkk2qqq1\u0011~cy\nkkk1;+~ckkk2;\u0000\u0010\nWA\nkkk1kkk2qqq1~aqqq1+WB\nkkk1kkk2qqq1~by\n\u0000qqq1\u0011\n,\n~WG\nkkk1kkk2qqq1=JiGp\nMG0=2j\rGj~R\nAd2%%%[ \u0003\nkkk1(%%%) kkk2(%%%)\u001eqqq1(%%%)]\nwith\rGthe typically negative gyromagnetic ratio cor-\nresponding to sublattice G(= A,B), and \u001eqqq1(rrr) is\nwavefunction of the magnon eigenmode with wavevector\nqqq1. Our goal is to examine the spin [33] current and\nits noise when one of the magnetic eigenmodes is in a\ncoherent state. We may, for example, achieve the \u000bqqq\nmode in a coherent state by including a driving term in\nthe Hamiltonian: ~Hdrive\u0018cos(!qqqt)(~\u000bqqq+ ~\u000by\nqqq) [34].\nThe operator corresponding to the z-polarized spin\ncurrent injected by M into N is obtained from the in-\nterfacial contribution to the time derivative of the total\nelectronic spin ( ~SSS):\n~Isz=1\ni~h\n~Sz;~Hinti\n=~X\nkkk1;kkk2;qqq1\u0010\n\u0000i~Pkkk1kkk2qqq1+i~Py\nkkk1kkk2qqq1\u0011\n:\n(5)\nThe above de\fnition captures the spin pumping\ncontribution to the current injected into N and\nFIG. 3. (a) Dispersion, (b) quasiparticle spin, (c) spin cur-\nrent injected into N and (d) dynamical spin correction factor\nvs. wavenumber (along x-direction) around the anti-crossing\npoint in a ferrimagnet. 2 \u0019fN=j\rAj\u00160MA0andfl(qN) =\n2fl(0) de\fne the normalizations fN; qNwithfl(q) the lower\ndispersion band. IN= 2~j\u001fj2!qqq\u000bABandtI\u0011\u0000AA=\u0000BB= 1,\nunless stated otherwise. The inset in (a) depicts the full dis-\npersion diagram. Dashed lines in (c) depict the spin current\nI0\nszdisregarding the cross-sublattice terms. Dashed lines in\n(d) depict the quasparticle spin, once again, to help compar-\nison. The parameters employed in the plot are given in the\nsupplemental material [26].\ndisregards the e\u000bect of interfacial spin-orbit cou-\npling [35]. The power spectral density of spin\ncurrent noise S(\n) is given by [36]: S(\n) =R1\n\u00001lim\u001c0!1(1=2\u001c0)R\u001c0\n\u0000\u001c0h~\u000eIsz(\u001c)~\u000eIsz(\u001c\u0000t) +~\u000eIsz(\u001c\u0000\nt)~\u000eIsz(\u001c)id\u001c ei\ntdt, wherehidenotes the expectation\nvalue and ~\u000eIsz=~Isz\u0000h~Isziis the spin current \ructua-\ntion operator.\nResults and Discussion. The spin current Iszin steady\nstate is obtained by evaluating the expectation value of\nthe spin current operator ~Isz[Eq. (5)] assuming a mag-\nnetic mode, e.g. \u000bqqq, in coherent state so that ~ \u000bqqqmay be\nsubstituted by a c-number \u001f[37]:\nIsz= 2~j\u001fj2\u0002\n\u0000AA\u0000\njuj2\u0000jwj2\u0001\n+ \u0000BB\u0000\njvj2\u0000jxj2\u0001\n\u00002\u0000AB<(u\u0003v\u0000wx\u0003)]; (6)\nwhereu;v;w;x correspond to the excited eigen-\nmode, \u0000 ij =\u0019P\nkkk1;kkk2Wi\nkkk1kkk2qqq\u0010\nWj\nkkk1kkk2qqq\u0011\u0003\n(nkkk2\u0000\nnkkk1)\u000e(!kkk1\u0000!kkk2\u0000!qqq) [38], with i;j=fA;Bg, and\nnkkkrepresenting the occupancy of the corresponding\nelectron state given by the Fermi-Dirac distribution.\nAssuming (i) WG\nkkk1kkk2qqqdepends only on the electron chem-\nical potential \u0016in N such that it may be substituted\nbyWG\n\u0016, and (ii) the electron density of states around\nthe chemical potential g(\u0016) is essentially constant, we\nobtain the simpli\fed relations: \u0000 ij=\u000bij!qqq. Here,\n\u000bij=\u0019~2Wi\n\u0016(Wj\n\u0016)\u0003V2\nNg2(\u0016), withVNthe volume of N.4\nFIG. 4. Dynamical spin correction factor SDvs. wavenumber\n(along x-direction) for a symmetrical AF. Dashed line depicts\nthe zero spin of the magnetic quasiparticles. fl(qN) = 2fl(0)\nde\fnes the normalization qNwithfl(q) the lower dispersion\nband. The parameters employed in the plot are given in the\nsupplemental material [26].\nThis also entails \u000bAB=\u000bBA=p\u000bAA\u000bBB. Since the\nclassical dynamics of a harmonic mode is captured by the\nsystem being in a coherent state [39], the spin current\nevaluated within our quantum model [Eq. (6)] must\nbe identical to the semi-classical expression expected\nfrom the spin pumping theory [12] generalized to a\ntwo-sublattice system. As detailed in the supplemental\nmaterial [26], we evaluate the semiclassical expression\ngiven by Eq. (1) for such a coherent state. The result\nthus obtained is identical to Eq. (6), provided we\nidentifyGij= (\u000bije=~)p\nMi0Mj0=j\rijj\rjj. Since\u000bAB=p\u000bAA\u000bBB, we obtain GAB=GBA=pGAAGBB[40].\nThese relations along with Eq. (1) constitute one of the\nmain results of this Letter.\nIn order to gain an understanding of the qualitative\nphysics at play, we examine the injected spin current nor-\nmalized by IN= 2~j\u001fj2!qqq\u000bABaround the anti-crossing\npoint in the dispersion of a ferrimagnet (Fig. 3) for sym-\nmetric interfacial coupling (\u0000 AA= \u0000BB). Due to dipo-\nlar interaction [28], the dressed magnon spin smoothly\nchanges between plus and minus ~resulting in a similar\nsmooth transition in the spin current [7]. Figure 2 de-\npicts the normalized spin current injected by the lower\nfrequency uniform mode ( qqq= 000) with respect to asym-\nmetries in the bulk tB(=MA0=MB0) and the interface tI\n(= \u0000AA=\u0000BB). For simplicity, we keep all other bulk pa-\nrameters constant and assume the applied \feld to vanish.\nFor the case of a perfect AF ( tB= 1) [41], we \fnd a small\ncurrent with varying tIthat vanishes at tI= 1 (inset in\nFig. 2). The small magnitude of the current is attributed\nto the dipolar interaction mediated spin-zero magnons in\nperfect AFs. The spin current has much larger values\nwhentB6= 0 since the dressed magnons acquire spin ~with a small bulk symmetry breaking [7]. The spin cur-\nrent in this case is highly sensitive to tI. This sensitivity\nis particularly pronounced for AFs, for which the bulk\nsymmetry can also be broken by an applied magnetic\n\feld.\nThe shot noise accompanying the dc spin current in-\njected into N is evaluated for a temperature T:\nS(\n) = 2 ~j\u001fj2\u0002\n\u000bAA\u0000\njuj2+jwj2\u0001\n+\u000bBB\u0000\njvj2+jxj2\u0001\n\u00002\u000bAB<(u\u0003v+wx\u0003)] [F(\n) +F(\u0000\n)]; (7)\nwhereF(\n)\u0011~(\n +!qqq) coth( ~[\n +!qqq]=[2kBT]) with\nkBthe Boltzmann constant. F(\n)!~j\n +!qqqjwhen\nT!0. When the dipolar interaction e\u000bect is neglected,\ni.e.w;x!0, limT!0S(0)!2~Isz[Eqs. (6) and (7)]\nsuch that the dynamical spin correction factor SD!1.\nAnd when the mode under consideration is not a\u000bected\nby sublattice B, we have v;x!0 andSD~approaches\nthe spin of the squeezed-magnon [37]. In the general\ncase,SD(\u00151) depends upon the magnetic mode, in-\nterfacial interaction as well as the eigenmodes in N, and\nis thus a property of the entire heterostructure. Fig-\nure 3(d) depicts SDfor a ferrimagnet around the anti-\ncrossing point in its dispersion. SD\u00191 away from the\nanti-crossing, and diverges at some wavenumber which\ndepends upon the interfacial asymmetry tI. This diver-\ngence results from a vanishing Isz.SDvs. wavenumber\nfor a symmetric AF with varying interfacial asymmetry\nis depicted in Fig. 4. Thus a combined knowledge of\nIszandSDmay allow to probe interfacial asymmetries\nexperimentally [42]. Since deviations of SDfrom 1 are\nnecessarily accompanied by quasiparticles with spin dif-\nferent from ~, it also o\u000bers an indirect signature of their\nformation.\nIn order to simplify expressions, we have employed the\napproximation WG\nkkk1kkk2qqq\u0019WG\n\u0016, which is commonly used in\nthe tunneling Hamiltonian description of spin [36, 37, 43,\n44] and charge [45] transport. This approximation pro-\nvides a reasonable description in the limit of strong scat-\ntering in N and a disordered interface. The opposite limit\nof quasi-ballistic transport in N and an ideal AF jN inter-\nface has been described numerically [13, 25, 46] as well as\nanalytically [47]. Our approximation further disregards\nthe dependence of the spin conductances on qqq[48, 49].\nSummary. We have presented a theoretical discussion\nof spin transport across a magnet jnon-magnetic conduc-\ntor interface when a magnetic eigenmode is driven to a\ncoherent state. Analytical expressions for the dc spin\ncurrent, including cross terms which were disregarded in\nRef. [13], and spin conductances have been obtained.\nOur theory takes into account the important role of bulk\nand interfacial sublattice-asymmetries as well as lattice\ndisorder at the interface. The spin current, especially\nin antiferromagnets, is found to be sensitive to interfa-\ncial asymmetry. We have evaluated the spin current shot\nnoise at \fnite temperatures and shown that it can be em-5\nployed to gain essential insights into quasi-particle spin\nand interfacial asymmetry.\nAcknowledgments. We thank Utkarsh Agrawal, So\nTakei, Scott Bender, Arne Brataas, Ran Cheng, Niklas\nRohling, Eirik L\u001chaugen Fj\u001arbu, Hannes Maier-Flaig,\nHans Huebl, Rudolf Gross, and Sebastian Goennenwein\nfor valuable discussions. We acknowledge \fnancial sup-\nport from the Alexander von Humboldt Foundation and\nthe DFG through SFB 767 and SPP 1538 SpinCaT.\nNote added in proof. Recently, Liu and co-workers re-\nported [50] a \frst principles calculation of damping in\nmetallic antiferromagnets. Their conclusions are fully\nconsistent with our work and show the important role\nof cross-sublattice terms.\n\u0003akashdeep.kamra@uni-konstanz.de\nywolfgang.belzig@uni-konstanz.de\n[1] Gerrit E. W. 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(2)], values\nof the various parameters used in the plots, a plot analo-\ngous to Fig. 2 employing the spin current expression ( I0\nsz)\nfrom Ref. [13], and detailed calculations demonstrating\nequivalence between Eqs. (1) and (6).\n[27] We emphasize that this expression is restricted to the\nz-component of spin, and may not be employed for the\nfull spin vector. It has been shown that the microscopic\nmatrix elements corresponding to x and y polarized spin\ntransport are in general di\u000berent [51]. This distinction is\noften not made in literature.\n[28] Here we use the term \\dipolar interaction\" to represent\nany contribution to the magnetic Hamiltonian that re-\nsults in spin non-conserving terms up to the second or-\nder in the ladder operators. Depending upon the ma-\nterial, these terms may predominantly have di\u000berent\nphysical origin such as magnetocrystalline anisotropy,\nDzyaloshinksii-Moriya interaction and so on.\n[29] T. Holstein and H. Primako\u000b, \\Field dependence of the\nintrinsic domain magnetization of a ferromagnet,\" Phys.\nRev.58, 1098{1113 (1940).\n[30] C. Kittel, Quantum theory of solids (Wiley, New York,\n1963).\n[31] A.I. Akhiezer, V.G. Bar'iakhtar, and S.V. Peletminski,\nSpin waves (North-Holland Publishing Company, Ams-\nterdam, 1968).\n[32] We have retained only the terms which contribute to z-\npolarized spin transport. The disregarded terms lead to\nminor shifts in magnon and electron energies, and are\nimportant for x and y polarized spin transport [51].\n[33] In the following discussion, the term `spin' is intended to\nmean z-component of the spin unless stated otherwise.\n[34] A typical method for driving the uniform mode is fer-\nromagnetic resonance. Exciting a non-uniform mode is\nrelatively di\u000ecult. Our goal, however, is to understand\nthe nature of individual modes, for which a `theoretical'\ndrive su\u000eces.\n[35] Tianxiang Nan, Satoru Emori, Carl T. Boone, Xinjun\nWang, Trevor M. Oxholm, John G. Jones, Brandon M.\nHowe, Gail J. Brown, and Nian X. Sun, \\Comparison of\nspin-orbit torques and spin pumping across nife/pt and\nnife/cu/pt interfaces,\" Phys. Rev. B 91, 214416 (2015).\n[36] Akashdeep Kamra and Wolfgang Belzig, \\Magnon-\nmediated spin current noise in ferromagnet jnonmagnetic\nconductor hybrids,\" Phys. Rev. B 94, 014419 (2016).\n[37] Akashdeep Kamra and Wolfgang Belzig, \\Super-\npoissonian shot noise of squeezed-magnon mediated spintransport,\" Phys. Rev. Lett. 116, 146601 (2016).\n[38] Note that Wi\nkkk1kkk2qqq\u0010\nWj\nkkk1kkk2qqq\u0011\u0003\nis real.\n[39] C. Gerry and P. Knight, Introductory Quantum Optics\n(Cambridge University Press, 2004).\n[40] The relation GAB=GBA=pGAAGBBholds generally\nand without making the approximation Wp\nkkk1kkk2qqq\u0019Wp\n\u0016.\n[41] The case of an antiferromagnet corresponds to identical\nparameters for both the sublattices. A compensated fer-\nrimagnet, on the other hand, is represented by identical\nsaturation magnetizations, while the remaining parame-\nters are in general di\u000berent, for the two sublattices.\n[42] Akashdeep Kamra, Friedrich P. Witek, Sibylle Meyer,\nHans Huebl, Stephan Gepr ags, Rudolf Gross, Gerrit\nE. W. Bauer, and Sebastian T. B. Goennenwein, \\Spin\nhall noise,\" Phys. Rev. B 90, 214419 (2014).\n[43] S Takahashi, E Saitoh, and S Maekawa, \\Spin current\nthrough a normal-metal/insulating-ferromagnet junc-\ntion,\" Journal of Physics: Conference Series 200, 062030\n(2010).\n[44] Steven S.-L. Zhang and Shufeng Zhang, \\Spin conver-\ntance at magnetic interfaces,\" Phys. Rev. B 86, 214424\n(2012).\n[45] G.D. Mahan, Many-Particle Physics , Physics of Solids\nand Liquids (Springer, 2000).\n[46] So Takei, Bertrand I. Halperin, Amir Yacoby,\nand Yaroslav Tserkovnyak, \\Super\ruid spin transport\nthrough antiferromagnetic insulators,\" Phys. Rev. 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B 91, 140402 (2015).1\nSupplementary material with the manuscript Spin pumping and shot noise in\nferrimagnets: bridging ferro- and antiferromagnets by\nAkashdeep Kamra and Wolfgang Belzig\nROLE OF CROSS TERMS IN SPIN PUMPING\nThe semi-classical expression for spin current injected into a conductor ( N) by an adjacent ferrimagnet ( F), when\nan eigenmode of the latter is driven into a coherent state, is reproduced below (Eq. (1) in the main text).\ne\n~Isz=GAA(^mmmA\u0002_^mmmA)z+GBB(^mmmB\u0002_^mmmB)z+GAB(^mmmA\u0002_^mmmB+^mmmB\u0002_^mmmA)z; (S1)\n=e\n~I0\nsz+GAB(^mmmA\u0002_^mmmB+^mmmB\u0002_^mmmA)z; (S2)\nwhere ^mmmA(B)is the unit vector along sublattice A (B) magnetization, and we have de\fned the spin current expression\ndisregarding the cross terms as I0\nsz. Employing mmm= [^mmmA+^mmmB]=2, andnnn= [^mmmA\u0000^mmmB]=2, Eq. (S1) can be recast in\nthe following form:\ne\n~Isz=Gmm(mmm\u0002_mmm)z+Gnn(nnn\u0002_nnn)z+Gmn(mmm\u0002_nnn+nnn\u0002_mmm)z; (S3)\nwhereGmm=GAA+GBB+ 2GAB,Gnn=GAA+GBB\u00002GAB, andGmn=GAA\u0000GBB. We note that substituting\nGAB=pGAAGBB, as derived in the main text, yields the expressions for Gmm,GnnandGmnas speci\fed in the\nmain text. On the other hand, substituting GAB= 0 andGAA=GBBleads to an expression ( I0\nsz) identical to the one\nobtained in Ref. [13]. To compare the two cases, we plot I0\nszvs. bulk and interfacial asymmetries (Fig. 1) analogous\nto the Fig. 2 in the main text. Clear qualitative di\u000berences can be seen with I0\nszoverestimating the injected spin\ncurrent and underestimating the sensitivity to interfacial asymmetry.\nFIG. 1. Normalized spin current (disregarding the cross-sublattice terms) vs. bulk ( tB=MA0=MB0) and interfacial ( tI=\n\u0000AA=\u0000BB) asymmetries for lower frequency uniform mode in coherent state. All other bulk parameters are kept constant, no\nexternal magnetic \feld is applied, and IN= 2~j\u001fj2!qqq\u000bAB. The spin current for tB= 1 is small due to the spin-zero quasiparticles\nin symmetric AFs, and it abruptly increases with a small bulk symmetry breaking due to quasiparticle transformation into spin\n~magnons [7].2\nDERIVATION OF THE MAGNETIC HAMILTONIAN\nThe classical Hamiltonian for the system is given by the integral of energy density over the entire volume V:\nHM=Z\nVd3r(HZ+Han+Hex+Hdip); (S4)\n=HZ+Han+Hex+Hdip; (S5)\nwith contributions from Zeeman, anisotropy, exchange and dipolar interaction energies, as discussed in the main text.\nQuantization of Hamiltonian is achieved by replacing the classical variables MMMA;MMMBwith the corresponding quantum\noperators ~MMMA;~MMMB. The Holstein-Primako\u000b (HP) transformation [29, 30] given by:\n~MA+(rrr) =p\n2j\rAj~MA0~a(rrr); (S6)\n~MB+(rrr) =p\n2j\rBj~MB0~by(rrr); (S7)\n~MAz(rrr) =MA0\u0000~j\rAj~ay(rrr)~a(rrr); (S8)\n~MBz(rrr) =\u0000MB0+~j\rBj~by(rrr)~b(rrr); (S9)\nexpresses the magnetization in terms of the magnonic ladder operators ~ a(rrr);~b(rrr) corresponding, respectively, to the\ntwo sublattices A; B . In the above transformation, ~MP+=~My\nP\u0000=~MPx+ (\rP=j\rPj)i~MPy, and\rP,MP0are the\ngyromagnetic ratio and the saturation magnetization corresponding to sublattice P. Carrying out the quantization\nprocedure, the magnetic Hamiltonian is obtained:\n~HM=X\nqqq\u0014Aqqq\n2~ay\nqqq~aqqq+Bqqq\n2~by\nqqq~bqqq+Cqqq~aqqq~b\u0000qqq+Dqqq~aqqq~a\u0000qqq+Eqqq~bqqq~b\u0000qqq+Fqqq~aqqq~by\nqqq\u0015\n+ h:c: ; (S10)\nwhere\nAqqq\n~=\u00160H0j\rAj+ 2KuAj\rAjMA0+ 2JAq2j\rAjMA0+Jj\rBjMB0\n+\u00160j\rAj\u0014\nNz(MB0\u0000MA0) +\u000eqqq;000Nx+Ny\n2MA0+ (1\u0000\u000eqqq;000)sin2(\u0012qqq)\n2MA0\u0015\n; (S11)\nBqqq\n~=\u0000\u00160H0j\rBj+ 2KuBj\rBjMB0+ 2JBq2j\rBjMB0+Jj\rAjMA0\n+\u00160j\rBj\u0014\nNz(MA0\u0000MB0) +\u000eqqq;000Nx+Ny\n2MB0+ (1\u0000\u000eqqq;000)sin2(\u0012qqq)\n2MB0\u0015\n; (S12)\nCqqq\n~=p\nj\rAjMA0j\rBjMB0\u0014\nJ+JABq2+\u00160\u000eqqq;000Nx+Ny\n2+\u00160(1\u0000\u000eqqq;000)sin2(\u0012qqq)\n2\u0015\n; (S13)\nDqqq\n~=\u00160j\rAjMA0\u0014\n\u000eqqq;000Nx\u0000Ny\n4+ (1\u0000\u000eqqq;000)sin2(\u0012qqq)\n4ei2\u001eqqq\u0015\n; (S14)\nEqqq\n~=\u00160j\rBjMB0\u0014\n\u000eqqq;000Nx\u0000Ny\n4+ (1\u0000\u000eqqq;000)sin2(\u0012qqq)\n4e\u0000i2\u001eqqq\u0015\n; (S15)\nFqqq=2q\nDqqqE\u0003qqq: (S16)\nNx;y;z in the expressions above are the components of the demagnetization tensor in its diagonal form, \u0012qqq; \u001eqqqare\nrespectively the polar and azimuthal angles of qqq, and all remaining symbols have been de\fned in the main text.3\nVALUES OF MODEL PARAMETERS\nParameter Fig. 2 Fig. 3 Fig. 4 Units\n\u00160H0 0 0.05 0 T\nNx,Ny,Nz1,0,0 1,0,0 1,0,0 Dimensionless\n\rA 1.8 1.8 1.8\u00021011s\u00001T\u00001\n\rB 1.8 1.8 1.8\u00021011s\u00001T\u00001\nMA 5 5 5\u0002105A/m\nMBMA=tB2.5 5\u0002105A/m\nJA 1 5 1\u000210\u000023J\u0001mA\u00002\nJB 1 1 1\u000210\u000023J\u0001mA\u00002\nJAB 0.1 0.1 0.1\u000210\u000023J\u0001mA\u00002\nJ 5 1 5\u000210\u00004Jm\u00001A\u00002\nKuA 2 2 2\u000210\u00007Jm\u00001A\u00002\nKuB 2 2 2\u000210\u00007Jm\u00001A\u00002\nSEMI-CLASSICAL AND QUANTUM EXPRESSIONS FOR SPIN CURRENT\nA key result of our work is the semi-classical expression [Eq. (S1)] for the spin current injected by the ferrimagnet\ninto the conductor in terms of the sublattice magnetizations. This has been derived under the assumption that one\nmagnetic mode is driven into a coherent state. Since a coherent state emulates the classical dynamics of a harmonic\noscillator, this semi-classical result should be identical to an analogous expression for spin current obtained within a\nquasi-classical theory. Here, we demonstrate this equivalence rigorously and identify the spin conductances in terms\nof the parameters within our microscopic model.\nThe magnetic Hamiltonian [Eq. (S10)] can be diagonalized by a four-dimensional Bogoliubov transform [7]:\n0\nBBB@~\u000b\u0014\u0014\u0014\n~\fy\n\u0000\u0014\u0014\u0014\n~\u000by\n\u0000\u0014\u0014\u0014\n~\f\u0014\u0014\u00141\nCCCA=0\nBBB@u1v1w1x1\nu2v2w2x2\nu3v3w3x3\nu4v4w4x41\nCCCA0\nBBB@~a\u0014\u0014\u0014\n~by\n\u0000\u0014\u0014\u0014\n~ay\n\u0000\u0014\u0014\u0014\n~b\u0014\u0014\u00141\nCCCA=S0\nBBB@~a\u0014\u0014\u0014\n~by\n\u0000\u0014\u0014\u0014\n~ay\n\u0000\u0014\u0014\u0014\n~b\u0014\u0014\u00141\nCCCA; (S17)\nwhere\u0014\u0014\u0014denotes the wavevector qqqrunning over half space [7, 29]. The transformation matrix Sis obtained by imposing\nthe requirement that the Hamiltonian should reduce to:\n~HM=X\n\u0014\u0014\u0014[~!l\u0014\u0014\u0014(~\u000by\n\u0014\u0014\u0014~\u000b\u0014\u0014\u0014+ ~\u000by\n\u0000\u0014\u0000\u0014\u0000\u0014~\u000b\u0000\u0014\u0000\u0014\u0000\u0014) +~!u\u0014\u0014\u0014(~\fy\n\u0014\u0014\u0014~\f\u0014\u0014\u0014+~\fy\n\u0000\u0014\u0000\u0014\u0000\u0014~\f\u0000\u0014\u0000\u0014\u0000\u0014)]: (S18)\nHere, we have employed the invariance of the coe\u000ecients A\u0014\u0014\u0014;B\u0014\u0014\u0014;\u0001\u0001\u0001, appearing in the magnetic Hamiltonian [Eq.\n(S10)], under the replacement \u0014\u0014\u0014!\u0000\u0014\u0014\u0014. This invariance also leads to the following properties of the transformation\nmatrixS:\nS22=S\u0003\n11; S 21=S\u0003\n12; (S19)\nwhereSijare the 2\u00022 block matrices constituting the 4 \u00024 matrixS. SinceStransforms a set of bosonic operators\ninto a di\u000berent set of bosonic operators, the corresponding commutation rules impose yet another constraint on the\ntransformation matrix:\nSYSy=Y=)S\u00001=YSyY\u00001; (S20)\nwhereY=\u001bz\n\u001bz, with\u001bzthe third Pauli matrix.\nWe consider that the mode ~ \u000bqqqis in a coherent state so that the operator ~ \u000bqqqcan be replaced by a c-number \u001f.\nAll other modes are assumed to be in equilibrium. The dynamics of this coherent mode is captured by replacing all\nquantum operators by their expectation values. Employing Eqs. (S17), (S19) and (S20), we obtain:\nh~aqqqi=u\u0003\n1\u001f\u0000w1\u001f\u0003; (S21)D\n~bqqqE\n=x\u0003\n1\u001f\u0000v1\u001f\u0003: (S22)4\nThe above two equations in conjunction with Eqs. (S6) and (S7) express the expectation values of the magnetization\noperators. Employing \u001f=j\u001fje\u0000i!qqqt, we thus evaluate:\n\u0012\nh^mmmAi\u0002d\ndth^mmmAi\u0013\nz=2~!qj\rAj\nMA0j\u001fj2(ju1j2\u0000jw1j2); (S23)\n\u0012\nh^mmmBi\u0002d\ndth^mmmBi\u0013\nz=2~!qj\rBj\nMB0j\u001fj2(jv1j2\u0000jx1j2); (S24)\n\u0012\nh^mmmAi\u0002d\ndth^mmmBi\u0013\nz+\u0012\nh^mmmBi\u0002d\ndth^mmmAi\u0013\nz=2~!qs\nj\rAjj\rBj\nMA0MB0j\u001fj2[\u00002<(u\u0003\n1v1\u0000w1x\u0003\n1)]: (S25)\nThe equations (S23) - (S25) obtained above demonstrate the equivalence between the semi-classical (Eq. (1) in the\nmain text) and the quantum (Eq. (6) in the main text) expressions for the spin pumping current, and allow us to\nidentify the spin conductances in terms of the parameters in the quantum model." }, { "title": "1206.0527v1.Effects_of_Zeroline_and_Ferrimagnetic_Fluctuation_on_Nuclear_Magnetic_Resonance_for_Dirac_Electrons_in_Molecular_Conductor_alpha__BEDT_TTF_2I3.pdf", "content": "arXiv:1206.0527v1 [cond-mat.mes-hall] 4 Jun 2012Effects of Zeroline and Ferrimagnetic Fluctuation\non Nuclear Magnetic Resonance for Dirac Electrons\nin Molecular Conductor α-(BEDT-TTF) 2I3\nA. Kobayashi, Y. Suzumura\nDepartment of Physics, Nagoya University, Furo-cho, Chiku sa-ku, Nagoya, 464-8602 Japan\n(Dated: November 26, 2018)\nWere-examinethewavefunctionoftwo-dimensional massles s Diracelectron in α-(BEDT-TTF) 2I3\nconsisting of four molecules A, A’, B and C in a unit cell, usin g a tight-binding model. We find\nzerolines in the Brillouin zone, on which the component of the wave func tion becomes zero for Bor\nCsites. The zerolines, which are bounded by two Dirac points a t±k0and pass through the M- or\nY-points, result in a fact that the density of states of the B site exhibits no the Van Hove sin gularity\nnear the energy of the Dirac points. By taking account of the on-site Coulomb interaction within\nthe random phase approximation, we examine the spin fluctuation in order to investigate prope rties\nof the nuclear magnetic resonance for temperatures T >50K. In the region for 100 < T <300K, it\nis shown that the Knight sift for B-site monotonously decreases with decreasing temperature , owing\nto lack of the Van Hove singularity, while it shows a maximum f or the other sites ( A,A′andC\nsites). In the region for 50 < T <100K, it is shown that the Knight sift is convex downward and t he\nKorringa ratio increases with decreasing temperature for B-site. Such a behavior originates from\nthe ferrimagnetic spin fluctuation related to the zerolines . These results are consistent with those\nof the nuclear magnetic resonance experiments.\nI. INTRODUCTION\nMolecular conductor α-(BEDT-TTF) 2I31has highly\ntwo-dimensionalelectronicsystem in the plane ofBEDT-\nTTF1/2molecule owing to layered structure with the\nplane of I−\n3anion, and has brought much interest by\nthe variety of electronic states, such as a charge ordered\nstate2, a superconducting state in the presence of charge\nordering, and a zero gap state (ZGS) with a massless\nDirac electron3.\nThe charge ordered state was suggested theoretically\nusing an extended Hubbard model4–6, and was confirmed\nby NMR experiment7. The superconducting state with\nthe charge ordering under uniaxial pressure along the\nstacking axis ( a-axis)8was also investigated theoretically\nusing the extended Hubbard model9. A narrowgap state\n(NGS) was suggestedto explain both anomalousincrease\nof Hall coefficient and almost-constant resistivity with\ndecreasing temperature at high pressures10.From the\ncalculation of the NGS, it was obtained that the charge\ngap disappears at high pressure, leading to the density\nof state (DOS) vanishing linearly at the Fermi energy11.\nFurher, the ZGS with a Dirac cone in energy dispersion\nwas found theoretically12using the transfer energies of\nref. 13, and was also confirmed by the first principle\ncalculations14,15. The energy spectrum near the Fermi\nenergyexhibits two tilted Dirac cones, which are de-\nscribed by the tilted Weyl equation16,17. The tilt of the\nDirac cone has been confirmed by a comparison between\nthe theoretical and experimental results for the temper-\nature (T) dependence of the Hall coefficient18,19and the\nangular dependence of the magnetoresistance19,20.From\nthe calculation of the variation of the Dirac point, the\nemergence of a pair of massive Dirac points is predicted\nin the charge ordering state at low pressures21, the merg-\ning of two massless Dirac electrons is shown at extremelyhighpressures16, suggesting robustnessofDiracelectrons\nagainst uniaxial pressures. Effects of electron correla-\ntion for long range Coulomb repulsion in the tilted Dirac\nelectron system have been investigated theoretically in\nthe absence of magnetic field22,23and in the presence of\nmagnetic field24.As for the short rangepart of Coulomb\nrepulsion,ontheotherhand, effectoffluctuationsinZGS\nhas not yet investigated theoretically in the context of\ninequivalence of BEDT-TTF sites (A = A′∝negationslash= B∝negationslash= C as\nshown in Fig. 1).\nThe property of the wave function plays important\nroles in the DOS and the electron-hole excitation in\nα-(BEDT-TTF) 2I3. The anomalous behavior of the\nBloch wave functions exists in the vicinity of the Dirac\npoint. The momentum dependences of the velocity ma-\ntrix and the charge density exhibit a singularity at the\npoint.16The angular dependence of the wave functions\nfor each site reveals a fact that the absolute value be-\ncomeszerofor B site or C site at a special direction ofthe\npoint28. The theoretical calculation of Knight shift Kα\nand (1/T1T)α28. has been performed based on a tight-\nbinding model for α-(BEDT-TTF) 2I3whereKα∝Tand\n(1/T1T)α∝T2owing to the Dirac cone spectrum. The\nKnight shift for the respective site reveals a relation,\nKC> KA> KB, which is consistent with experimen-\ntal results,25originates from both the tilt of Dirac cone\nand property of wave function in the vicinity of the Dirac\npoint. Thus, the inequivalence of BEDT-TTF sites ob-\nserved in NMR experiments reveals inner degree of free-\ndom of Dirac electron in molecular conductors.25–27\nHowever, the tight binding model is not enough to\nexplain the details, e.g, the non-linear temperature de-\npendences of Kαas shown in the following experiment.\nIn high temperature region ( T >100K), the Knight\nshiftKαwith decreasing temperature monotonously de-\ncreases for α= B site, while it exhibits a maximum2\nforα= A and C sites. In medium temperature region\n(50< T <100K), the Knight shift is convex downward\nwith decreasing temperature for B site, while the compo-\nnents forAand Csites exhibit linear T-dependences.25,27\nFor a local NMR relaxation rate 1 /T1, the difference in\n(1/T1T)αinα= A, B, and C sites, is small in low tem-\nperature region ( T <50K), i.e., all components are con-\nvex downward with T2-dependences for T <100K. As\na result, the Korringa ratio (1 /T1TK2)αforα= B site\nclearly increases with decreasing Tin medium temper-\nature region, and exhibits an inequality (1 /T1TK2)B>\n(1/T1TK2)A>(1/T1TK2)C27.\nIn the present paper, we re-examine the wave func-\ntion inα-(BEDT-TTF) 2I3using a tight-binding model\nwhere the transfer integrals are given by the first prin-\nciple calculation14, and calculate the spin fluctuation\nwithin the random phase approximation on the on-site\nCoulomb interaction to investigate Kαand (1/T1T)αfor\nT >50K.Based on the formulation in §II, we demon-\nstrate following new results in §III.We find zerolines in\nthe two-dimensionalBrillouin zone, where the wavefunc-\ntion is zero for the components of BorCsite. They are\nbounded by the two Dirac points at ±k0. The zerolines\npassing through the M- orY-points give rise to the ab-\nsence of two Van Hove singularities for only the B-site.\nIn the high temperature region for 100 < T < 300K,\nthe present numerical results on T-dependences of Kα,\nare consistent with the experimental results25–27, where\nexistence (absence) of Van Hove singularities for α= A\nand C sites ( α= B site) is essential. In the region for\n50< T <100K,it isshownthat KBis convexdownward,\n(1/T1TK2)Bincreases with decreasing T, and then the\ninequality (1 /T1TK2)B>(1/T1TK2)A>(1/T1TK2)C\nis reproduced, owing to the ferrimagnetic spin fluctua-\ntion related to the zerolines and enhanced by the on-site\nCoulomb interaction. These results are also consistent\nwith those experimental results25–27. In§IV, summary\nand discussion are given.\nII. FORMULATION\nThe model describing the two-dimensional electronic\nsystem in α-(BEDT-TTF) 2I3is shown in Fig. 114,29,30.\nThe unit cell consists of four BEDT-TTF molecules on\nsites A, A′, B and C, where A is equivalent to A′so\nthat inversion symmetry is preserved, while the sites A,\nB and C are inequivalent. There are six electrons for the\nfourmoleculesinaunit cell, thusthe bandsare3 /4-filled.\nOn the basisofthe HOMO orbitalsofthese sites4,5, these\nelectrons are described by a Hubbard model with the on-\nsite Coulomb interaction U,\nH=/summationdisplay\n(iα:jβ),σ(tiα;jβa†\niασajβσ+h.c.)\n+/summationdisplay\niαU a†\niα↑a†\niα↓aiα↓aiα↑, (2.1)y\nx\na1\na1b3\nb3b2\nb2A\nA’B\nCa3\na2b1b4\nb4b1a1’\na1’a3’\na4’\nFIG. 1: The model describing the electronic system of α-\n(BEDT-TTF) 2I314,29,30. The unit cell consists of four BEDT-\nTTF molecules A, A′, BandCwithtentransfer energies. The\na- andb-axis in the conventional notation correspond to the\ny- andx-axis in the present paper. The molecules A and A′\nare equivalent in the presence of the inversion symmetry in\nthe ZGS. (Color Online)\nwherei,jdenote indices of a given unit cell, and α,β(=\nA, A′, B and C) are indices of BEDT-TTF sites in the\nunit cell. In the first term, a†\niασdenotes a creation oper-\nator with spin σ(=↑,↓) andtiα;jβis the transfer energy\nbetween the ( i,α) site and the ( j,β) site. Throughout\nthe paper, /planckover2pi1and the lattice constant aaretaken as unity.\nHereafter, the energies are given in eV, and the tempera-\nture is also given by kBTin eV,i. e.1eV =104K, where\nkBis the Boltzmann factor.\nThe transfer energies at finite temperature Tis esti-\nmated by the interpolation formula28\ntX(T) =tX(LT)+(tX(RT)−tX(LT))(T−LT)/(RT−LT).\n(2.2)\nThe transfer energies tX(RT) and tX(LT)\nare given by the 1st principle calculation14\n, where ta1(RT) = −0.0101,ta2(RT) =\n−0.0476,ta3(RT) = 0 .0093,tb1(RT) = 0 .1081,tb2(RT) =\n0.1109,tb3(RT) = 0 .0551,tb4(RT) = 0 .0151,ta1′(RT) =\n0.0088,ta3′(RT) = 0 .0019,ta4′(RT) = 0 .0009,and\nta1(LT) = −0.0267,ta2(LT) = −0.0511,ta3(LT) =\n0.0323,tb1(LT) = 0 .1241,tb2(LT) = 0 .1296,tb3(LT) =\n0.0513,tb4(LT) = 0 .0512,ta1′(LT) = 0 .0119,ta3′(LT) =\n0.0046,ta4′(LT) = 0 .0060,, where low temperature\nLT = 0.0008 and we put room temperature RT = 0 .03.\nThe Hamiltonian is diagonalized numerically for a\ngivenkin each spin subspace, according to\n4/summationdisplay\nβ=1˜ǫαβσ(k)dβγσ(k) =ξγσ(k)dαγσ(k),(2.3)\nwhereξγσare the eigenenergies ordered such that,\nξ1σ(k)> ξ2σ(k)> ξ3σ(k)> ξ4σ(k) (γ= 1,2,3,4 is the3\nbandindex), and dαγσ(k)arethecorrespondingeigenvec-\ntors. The T-dependence of the chemical potential owing\nto the electron-hole asymmetry18is taken into account\nin the present calculation, although it has not been con-\nsidered in 28.\nThe bare susceptibility on the site representation is\ngiven by11\n[ˆχ0]αβ=χ0\nαβ(q,ωl)\n=−T\nNL/summationdisplay\nknGαβ(k+q,ǫn+ωl)Gβα(k,ǫn) (2.4)\n=−T\nNL/summationdisplay\nknFαβ(k,q)\n(i(ǫn+ωl)−ξγ(k+q))(iǫn−ξγ(k)),(2.5)\nwhere the bare Green function on the site representation\nis defined by\nGαβ(k,ǫn)≡/summationdisplay\nγdαγ(k)d∗\nβγ(k)/(iǫn−ξγ(k)) (2.6)\nwithǫn= (2n+1)πTandωl= 2nπTare the Matsubara\nfrequencies, and the form factor is defined by\nFαβ(k,q) =F(1)\nαβ(k,q)+F(2)\nαβ(k,q), (2.7)\nF(1)\nαβ(k,q) =/summationdisplay\nγ=1,2Fγγ\nαβ(k,q), (2.8)\nF(2)\nαβ(k,q) =/summationdisplay\nγ=1,2Fγ¯γ\nαβ(k,q), (2.9)\nFγγ′\nαβ(k,q) =dαγ(k+q)dβγ(k+q)∗dβγ′(k)dαγ′(k)∗,\n(2.10)\nwith ¯γ∝negationslash=γindicating another band within the conduc-\ntion (γ= 1) and valence ( γ= 2) bands. The form fac-\ntor reflects the character of each site for the intraband\n(F(1)\nαβ(k,q)) andinterband (F(2)\nαβ(k,q)) fluctuations.\nIn the presenceof spin symmetry, the longitudinal spin\nsusceptibility, ˆ χs, and the transverse spin susceptibility,\nˆχ±, are given by11\nˆχs= ˆχ±= (1−ˆχ0ˆU)−1ˆχ0(2.11)\nwithˆU=UˆIandˆIis the unit matrix.\nThe knight shift Kα(the local spin susceptibility) and\nthe local NMR relaxation rate (1 /T1)αwith A, A′, B and\nC sites are given by28\nKα=/summationdisplay\nβχs\nαβ(0,0), (2.12)\n(1/T1)α=Tlim\nω→0/summationdisplay\nqIm[χ±\nαα(q,ω)]/ω.(2.13)\nIII. RESULTS\nA. Zeroline and density of states\nFigure 2(a), (b), (c) and (d) show absolute values of\nthe wave functions of the conduction band ( γ= 1) for\n;\n/:\nC\u000b\n\u0012\u0010\u0012\u0013\u0010\u0012\n; /:\nD\u000b\n\u000fM\u0012\nM\u0012Γ\n;\n/:\nE\u000b\n\u0012\u0010\u0012\u0013\u0010\u0012\n\nF\u000b\n; /:\u000fM\u0012\nM\u0012Γ\nFIG. 2: Absolute values of the wave functions of the con-\nduction band ( ξ= 1) for the components of Bsite(\n|dB1(k)d∗\nB1(k)|(a) and and their coutour plot(b)) and those\nofCsite ((c) and (d)) in the Brillouin zone. The zerolines of\nthe conduction band are bounded by two Dirac points at ±k0\nas shown in the dark region of (c) and (d), and in schematic\nfigure (e) for Bsite (green solid line) and Csite (red dashed\nline).(Color Online)\nthe components of Bsite (Figs 2(a) and (b)) and Csite\n(Figs 2(c) and (d)). We find ”zerolines”which are curved\nlines where absolute value of the wave function is zero,\nonly for the components of BandCsites. The zerolines\nfor both BandCsites are bounded by two Dirac points\nat±k0in the Brillouin zone, where the wave functions\nare discontinuous . In the conduction band, the zeroline\nforB(C) site passes through the M- (Y-)point as shown\nin Fig 2(e). In the valence band, on the contrary, the\nzeroline for B(C) site passes through the Y- (M-)point.\nThe zeroline for B(C) site in conduction band coincides\nwith the zeroline for C(B) site in valence band in the\nvicinity of the Dirac points. , while these are slightly\ndifferent far from the Dirac points, since the conduction\nand valence bands are not equivalent in α-type organic\nconductors.\nIn Fig 3(a), the conduction band and the valence band\nare shown The chemical potential with ω= 0 is situated\non the Dirac points at T= 0, while it decreases with\nincreasing temperature owing to the asymmetry of the4\nω(a)\nMZM[\n−0.05 0 0.050510\nωC\nA\nBρα(ω)(b)\nFIG. 3: (a) Energy dispersions in the Brillouin zone for the\nconduction and valence bands ( γ= 1 and 2), which has a\ncontact point with the tilted Dirac cones at ±k0. The chem-\nical potential is given by with ω= 0. There are the saddle\npoints at the M-point in the conduction band and at Y-point\nin the valence band near ω= 0. (b) The density of states\nραnear the energy of the Dirac point ( ω= 0) at T= 0 for\nthe components of α=Asite (blue dotted line), α=Bsite\n(green solid line), and α=Csite (red dashed line). (Color\nOnline)\nconduction and valence bands.18The saddle points close\nto the Fermi energy are seen at the M-point and at Y-\npoint, respectively. The saddle points give rise to the\nVan Hove singularities ω= 0.012eV in the conduction\nband, and at ω=−0.026eV in the valence band. Such a\nsingularity appears in the components of density of state\natAandCsites as shown in Fig 3(b), where the energy\nof the Dirac points is ω= 0. For the component of B\nsite, however,thesetwoVanHovesingularitiesdisappear,\nsince the zeroline for Bsite passes through the location\nof the saddle point.B. Knight shift and NMR relaxation rate\nFigure 4(a) shows temperature dependences of the\nKnight shift Kαforα=A,B, andCsites with U= 0.12\nandK0\nαwithU= 0. The value of U= 0.12 is cho-\nsen so that the temperature dependences of Kαin the\npresent calculation reproduce those of experimental re-\nsults for T >50K. The Knight shift K0\nα(withU= 0)\nforT <0.015 has been calculated in ref. 28, which\ngives slightly different results due to ignoring the T-\ndependence of the chemical potential. In high tempera-\ntureregion(100 < T <300K),With decreasingtempera-\nture, the Knight shift monotonously decreases for B-site\nwhilethere isamaximumin the temperaturedependence\nof the Knight shift for AandCsites. Such behavior is\nindependent of Uoriginates from the combined effect of\nthe VanHovesingularityand the zerolinesasdiscussedin\nFig 3. In medium temperature region (50 < T <100K),\non the other hand, it is shown that the Knight shift is\nconvexdownwardwith decreasingtemperature for B-site\nwithU= 0.12, while the components for AandCsites\nexhibit linear T-dependences in this region. Such a effect\nofelectron correlation originates from the ferrimagnetic\nspin fluctuation described later. Figure 4(b) shows tem-\nperature dependences of (1 /T1T)αforα=A,B, and\nCsites with U= 0.12 and (1 /T1T)0\nαwithU= 0. It is\nshown that (1 /T1T)αexhibit linear T2-dependences with\nT <100Kapproximately, and allcomponents areweakly\nenhanced by U. Figure 4(c) shows temperature depen-\ndences of the Korringa ratio (1 /T1TK2)αforα=A,B,\nandCsites with U= 0.12 and (1 /T1TK2)0\nαwithU= 0.\nIt is clearly shown that the Korringa ratio for Bsite is\nstrongly enhanced by Uand increases with decreasing T,\nowing to the anomalous T-dependence of KB. The elec-\ntron correlation effect results in the enhancement of the\nfollowing inequality,\n(1/T1TK2)B>(1/T1TK2)A>(1/T1TK2)C.(3.1)\nC. Ferrimagnetic fluctuation in spin susceptibility\nWe examine static spin susceptibility at low energy by\nchoosing T= 0.02 andω= 0. In Fig 5(a), the mo-\nmentum dependences of the diagonal components of the\nspin susceptibilities χs\nααand the baresusceptibilities χ0\nαα\n(U= 0) for α=A,B, andCare shown forα=A,B,\nandCwithU= 0.12. The diagonal components are\npositive in the Brillouin zone and are enhanced by U.\nThe momentum dependences of the off-diagonal compo-\nnents of the spin susceptibilities χs\nαβfor (α,β) = (A,B),\n(B,C), and (A,C) withU= 0.12, and the bare suscepti-\nbilitiesχ0\nαβfor (α,β) = (A,B), (B,C), and (A,C) with\nU= 0 are shown in Fig 5(b). It is foundthatχs\nBCand\nχs\nABare negative in the Brillouin zone, while χs\nAChas\nboth positive and negative value. Those absolute val-\nues are enhanced by U. Especially, χs\nBCplays the most5\n0 0.01 0.02 0.03024KαKC\nKBKA\nTKC0\nKA0\nKB0(a)\n10−310−2 10−1100101102(1/T1T)α(1/T1T)C\nT(1/T1T)C0(1/T1T)A\n(1/T1T)A0\n(1/T1T)B\n(1/T1T)B0(b)\n0 0.01 0.02 0.0301020(1/T1TK2)α\nT(c)\n(1/T1TK2)B\n(1/T1TK2)B0\n(1/T1TK2)C(1/T1TK2)C0(1/T1TK2)A\n(1/T1TK2)A0\nFIG. 4: (a) Temperature dependences of the Knight shift Kα\nwithU= 0.12 (thick line) and K0\nα(thin line) for α=A(red\ndashed line), B(green solid line), and Csites (blue dotted\nline), respectively. (b) Temperature dependences of (1 /T1T)α\nwithU= 0.12 (thick line) and (1 /T1T)0\nα(thin line) for α=\nA(red dashed line), B(green solid line), and Csites (blue\ndotted line), respectively. (c) Temperature dependences o f\nKorringa ratio (1 /T1TK2)αwithU= 0.12 (thick line) and\n(1/T1TK2)0\nαwithU= 0 (thin line) for α=A(red dashed\nline),B(green solid line), and Csites (blue dotted line),\nrespectively. (Color Online)important role for the anomalous T-dependences of KB,\nsince|χs\nBC|atq=0is strongly enhanced by U, owing to\nproduct of negative χ0\nBCwith positive diagonal terms in\nthe RPA process such as\n[ˆχs]BC= [(1−ˆχ0ˆU)−1ˆχ0]BC\n=χ0\nBC+Uχ0\nBBχ0\nBC+Uχ0\nBCχ0\nCC+···.(3.2)\nNegative off-diagonal susceptibilities, χs\nBCandχs\nAB,\nand positive off-diagonal susceptibility, χs\nAC, atq=0in-\ndicate the ferrimagnetic spin fluctuation where the spin\non B site tends to be opposite to that of the other sites\nas shown in Fig 5(c). The Knight shift is a sum of the\ndiagonal and off-diagonal spin susceptibilities, where the\nformer are dominant at high temperatures since the spin\nfluctuation exhibits local character. The latter become\nimportant with decreasing temperature, resulting in the\nanomalous T-dependence of KBwhich is convex down-\nward at low temperatures.\nThe negative value of χ0\nBCatq=0originats from the\nformfactorfortheinterbandfluctuation, F(2)\nBC(k,0), since\nthe form factor for intraband fluctuation, F(1)\nBC(k,0), is\npositive for any k. The momentum kdependence of\nF(2)\nBC(k,0) in the Brillouin zone is shown in Figs 6(a) and\n6(b). Although F(2)\nBC(k,0) can take positive or negative\nvalues, the numerical result show F(2)\nBC(k,0) is negative\nor zero for any k. It is not a self-evident result owing\nto the phase structure of the wave function as discussed\nlater. The white curved lines in Fig. 6(b) correspond to\nthe zeroline for B and C sites, where F(2)\nBC(k,0) = 0.\nIn order to show the reason of F(2)\nBC(k,0)≤0, we\ninvestigate the phase structure of the wave function.\nProperties of wave functions in the vicinity of the Dirac\npoints have been investigated by analying in terms of the\nLuttinger-Kohn representation28. The analysis of wave\nfunctions in the present paper, on the other hand, is\nbased on the Bloch representation. Figure 7 shows nu-\nmerical results of wave functions for B and C sites and\nfor conduction and valence bands, dB1(k0′)∗,dB2(k0′),\ndC1(k0′) anddC2(k0′)∗, on the Gauss plane, here|k0′−\nk0|= 0.05π, i.e., the momentum k 0′circles the Dirac\npointk0. In order to avoid arbitrary phase factor of the\nwave functions for each band γand each momentum k 0′,\nwe choose dAγ(k0′) is real and positive. Such a represen-\ntation for wave functions was used in ref. 32 to examine\nthe rotation of the base, dα,1anddα,2, around k0. The\ninset shows θ-dependences of absolute values of the wave\nfunctions for the conduction band. The similar result\nwas shown in 28 although there is slight difference in the\nchoice of the horizontal axis. Based on this numerical\nresults, the wave functions in the vicinity of the Dirac\npoints, are approximately given by\ndB1(θ) =eiθ1|dB1|[ei(θ+θB1)−ei(θBZ+θB1)],\ndC1(θ) =eiθ1|dC1|[ei(θ+θC1)−ei(θCZ+θC1)],\ndB2(θ) =eiθ2|dB2|[ei(θ+θB2)−ei(θCZ+θB2)],6\ndC2(θ) =eiθ1|dC2|[ei(θ+θC2)−ei(θBZ+θC2)],(3.3)\nwhereθis the angle of the vector k 0′−k0measured from\nkx-axis. The values of wave functions at θ= 0 are deter-\nmined by θB1,θC1,θB2andθC2. The angleofthe zeroline\nfor B (C) site in the conduction band is θBZ(θCZ), where\n|dB1(θ)|= 0 (|dC1(θ)|=0) as shown in the inset of Fig.\n7, while the zeroline for B (C) site in the valence band\ncorresponds to θCZ(θBZ). The arbitrary phases of the\nwave functions for each band and each momentum, θ1(θ)\nandθ2(θ), disappear in the form factor. Using above\nwave functions with the parameters given by the numer-\nical calculation, we obtain F(2)\nBC(k0,0)≤0 at arbitrary\nθ.\nIfθCZ=θBZ+π(closetothe presentnumericalresult),\nF(2)\nBC(k0,0) =−8cosθ0sin2(θ−θBZ) (3.4)\nwithθ0=−θB1+θC1−θC2+θB2. Weobtain F(2)\nBC(k0,0)≤\n0 in the wide region of parameter, −π/2≤θ0≤π/2.\nThus the condition of F(2)\nBC(k0,0)≤0 is robust when the\nzerolines of B and C sites extend in opposite directions\neach other. When θCZ=θBZ, on the other hand,\nF(2)\nBC(k0,0) = 8cos θ0(1−cos(θ−θBZ))2.(3.5)\nWe obtain F(2)\nBC(k0,0)≤0 withθ0in the present numer-\nical result. Thus the angle between the zerolines of B\nand C sites is an important factor for the ferrimagnetic\nfluctuation in the Dirac electrons of α-(BEDT-TTF) 2I3.\nIV. SUMMARY AND DISCUSSION\nIn summary, we examined the wave function and the\nspin fluctuation in α-(BEDT-TTF) 2I3, using a tight-\nbinding model and the on-site Coulomb interaction\ntreated within the random phase approximation. The\neffect of electron correlation on Kαand (1/T1T)αwith\nA, A′, B and C sites for T >50Kwasinvestigatedpaying\nattention to the inequivalence of these sites (A = A′∝negationslash=\nB∝negationslash= C).\nWe found that zerolines, where the wave function is\nzero for the components of BorCsite. They give the\nvanishing of two Van Hove singularities near the energy\nof the Dirac points only for B-site component of DOS.\nExistence (absence) of the Van Hove singularities plays\nessential role for the T-dependences of Kα.Forthe\nhigh temperature region of 100 < T <300K, with de-\ncreasing T,KBdecreases monotonously while KAand\nKCexhibit a maximum. In the region for 50 < T <\n100K,KBis convex downward, and (1 /T1TK2)Bin-\ncreases leading to an inequality of the Korringa ratio,\n(1/T1TK2)B>(1/T1TK2)A>(1/T1TK2)C, was ob-\ntained. These results are consistent with those of exper-\niment for T >50K25–27.It is found that the anomalous T-dependence of KB\nis ascribed to the ferrimagnetic spin fluctuation which\nisenhanced by the on-site Coulomb interaction. Such\na fluctuation describes a spin on B site being opposite\nto the other spins on A, A′and C sites. The ferri-\nmagnetic spin fluctuation originates from the interband\nfluctuation mainly between B and C sites. The inter-\nband fluctuation relates to the zerolines bounded by two\nDirac points in the Brillouin zone. Such zeroline does\nnot exist in the wave function of graphene, since carbon\natoms in two sublattice are equivalent owing to the in-\nversion symmetry, which corresponds to A and A′sites\ninα-(BEDT-TTF) 2I3. Thus the present results reveal\nthat the inequivalence of BEDT-TTF sites play impor-\ntant roles for observables in NMR as an inner degree of\nfreedom of Dirac electron in molecular conductor with\nthe short range Coulomb interaction.\nThere remain problems to be clarified. In the low\ntemperature region for T <50,Kαfor all compo-\nnents are convex downward with decreasing T25–27, and\n(1/T1T)αexhibits complex T-dependence for very low\ntemperatures27,31in the presence of magnetic field per-\npendicular to the conducting plane. Those behavior can\nnotbeexplainedwithin therandomphaseapproximation\non the on-site Coulomb interaction. It indicates an im-\nportance of a higher order correction such as self-energy\ncorrectionwiththelongrangeCoulombinteraction, since\nat low temperatures, low energy phenomena in the vicin-\nity of Dirac point is dominant and then scale of length\nis much longer than lattice constant. Finally, we note\nthatthe valley splitting owing to the pseudo-spin XY\nferromagnetism in N= 0 Landau states24may also play\nsignificant roles at very low temperatures in the presence\nof magnetic field perpendicular to the conducting plane.\nAcknowledgments\nThe authors are thankful to K. Ishikawa, M. Hirata,\nK. Miyagawa and K. Kanoda for fruitful discussions.\nY.S. is indebted to the Daiko Foundation for financial\naid in the present work. This work was financially sup-\nported in part by Grant-in-Aid for Special Coordination\nFunds for PromotingScience and Technology(SCF), Sci-\nentific Research on Innovative Areas 20110002, and was\nalso financially supported by a Grant-in-Aid for Special\nCoordination Funds for Promoting Science and Tech-\nnology (SCF) from the Ministry of Education, Culture,\nSports, Science and Technology in Japan, and Scientific\nResearch 19740205, 22540366, 23540403 and 24244053\nfrom the Ministry of Education, Culture, Sports, Science\nand Technology in Japan.7\n1N. Tajima and K. Kajita K, Sci. Tech. Adv. Mater. 10,\n024308 (2009).\n2H. Seo, C. Hotta, and H. Fukuyama H, Chem. Rev. 104,\n5005 (2004).\n3A. Kobayashi S. Katayama, and Y. Suzumura, Sci. Tech.\nAdv. Mater. 10, 024309 (2009).\n4H. Kino and H. Fukuyama, J. Phys. Soc. Jpn. 64, 4523\n(1995).\n5H. Seo, J. Phys. Soc. Jpn. 69, 805 (2000).\n6C. Hotta, J. Phys. Soc. Jpn. 72, 840 (2003).\n7T. Takahashi, Synth. Met. 133-134 , 26 (2003).\n8N. Tajima, A. Ebina-Tajima, M. Tamura, Y. Nishio, K.\nKajita, J. Phys. Soc. Jpn. 71, 1832 (2002).\n9A. Kobayashi, S. Katayama, and Y. Suzumura, J. Phys.\nSoc. Jpn. 74, 2897 (2005).\n10K. Kajita, T. Ojiro, H. Fujii, N. Nishio, H. Kobayashi, A.\nKobayashi, R. Kato, J. Phys. Soc. Jpn. 61, 23 (1992).\n11A. Kobayashi, S. Katayama, K. Noguchi, and Y. Suzu-\nmura, J. Phys. Soc. Jpn. 73, 3135 (2004).\n12S. Katayama, A. Kobayashi, and Y. Suzumura, J. Phys.\nSoc. Jpn. 75, 054705 (2006).\n13R. KondoS. Kagoshima, and J. Harada, Rev. Sci. Instrum.\n76, 093902 (2005).\n14H. Kino and T. Miyazaki, J. Phys. Soc. Jpn. 75, 034704\n(2006).\n15S. Ishibashi, T. Tamura, M. Kohyama, and K. Terakura,\nJ. Phys. Soc. Jpn. 75, 015005 (2006).\n16A. Kobayashi, S. Katayama, Y. Suzumura, and H.\nFukuyama, J. Phys. Soc. Jpn. 76, 034711 (2007).\n17M.O. Goerbig, J.-N. Fuchs, G. Montambaux, and F.Pi´ echon, Phys. Rev. B 78, 045415 (2008).\n18A. Kobayashi, Y. Suzumura, and H. Fukuyama, J. Phys.\nSoc. Jpn. 77, 064718 (2008).\n19N.Tajima, S.Sugawara, R.Kato, Y.Nishio, andK.Kajita,\nPhys. Rev. Lett. 102, 176403 (2009).\n20K. Morinari, T. Himura, and T. Tohyama, J. Phys. Soc.\nJpn.78, 023704 (2009).\n21A. Kobayashi, Y. Suzumura, F. Pi´ echon, and G. Montam-\nbaux, Phys. Rev. B 84, 075450 (2011).\n22T. Nishine, A. Kobayashi, and Y. Suzumura, J. Phys. Soc.\nJpn.79, 114715 (2010).\n23T. Nishine, A. Kobayashi, and Y. Suzumura, J. Phys. Soc.\nJpn.80, 114713 (2011).\n24A. Kobayashi, Y. Suzumura, H. Fukuyama, and M. O.\nGoerbig, J. Phys. Soc. Jpn. 78, 114711 (2009).\n25Y. Takano, K. Hiraki, Y. Takada, H. M. Yamamoto, and\nT. Takahashi, J. Phys. Soc. Jpn. 79, 104704 (2010).\n26K. Miyagawa, M. Hirayama, M. Tamura, and K. Kanoda,\nJ. Phys. Soc. Jpn. 79, 063703 (2010).\n27M. Hirata, Ph.D. thesis, Univ. of Tokyo, (2012)\n28S. Katayama, A. Kobayashi, and Y. Suzumura, Eur. Phys.\nJ. B67, 139 (2009).\n29T. Mori, A. Kobayashi, Y. Sasaki, H. Kobayashi, G. Saito,\nand H.Inokuchi, Chem. Lett. 957 (1984).\n30T. Mori, H. Mori, and S. Tanaka, Bull. Chem. Soc. Jpn.\n72, 179 (1999).\n31Y. Shimizu, A. Kobayashi, M. Ito, H. M. Yamamoto, Y.\nTakano, T. Takahashi, private communication.\n32S. Katayama, A. Kobayashi, Y. Suzumura , private com-\nmunication.8\n0123χαβs\nqχBB0χBBsχCC0χCCs\nχAA0χAAs\n(0,0) (π,0) (π,π) (0,0)(a)\n−0.500.5χαβs\nq(b)\n(0,0) (0,0) (π,0) (π,π)χBC0\nχBCsχABsχAB0χACs\nχAC0\nFIG. 5: (a) Momentum dependences of the diagonal compo-\nnents of the spin susceptibilities χs\nααwithU= 0.12 (thick\nline) and the bare susceptibilities χ0\nαβ(thin line) for α=A\n(red dashed line), B(green solid line), and Csites (blue dot-\nted line), where T= 0.02 andω= 0. (b) Momentum depen-\ndences of the off-diagonal components of the spin susceptibi l-\nitiesχs\nαβU= 0.12 (thick line) and the bare susceptibilities\nχ0\nαβ(thin line) for ( α,β) = (A,B) (gold dashed line), ( B,C)\n(wine red solid line), and ( A,C) sites (purple dotted line),\nwhereT= 0.02 andω= 0. (c) Schematic figure for pattern\nof the ferrimagnetic spin fluctuation. (Color Online)9\n($%\nC\u000b\n(−π,−π)\n(π,−π)(π,π)\n㫂㫏 㫂㫐\n\u0014\u000b\n\nD\u000b\nMZM[\n(−π,−π) (π,−π)(−π,π)\nFIG. 6: (a) Momentum kdependence of the form factor\nF(2)\nBC(k,0)for the interband fluctuation in the Brillouin zone.\nThe contour plot corresponding to (a) is shown in (b) (Color\nOnline)10\n−1 0 1−101\n0 2 4 600.20.40.6\nθ [rad]|dB1||dA1|\n|dC1|\nReImdB1 (k0’)*\ndB2 (k0’)\ndC1 (k0’)\ndC2 (k0’)*\nFIG. 7: Numerical results of Wave functions normalized by\ndAγ/|dAγ1|(γ=1,2) on the Gauss plane, for dB1(k0′)∗(the\ngreen thick solid circle), dB2(k0′) (the green thin solid circle),\ndC1(k0′) (the red thick dashed circle) and dC2(k0′)∗(the red\nthin dashed circle), where |k0′−k0|= 0.05πand the angle\nof k0′−k0vector from kx-axis,θ, moves from zero to 2 π.\nThe directions where the values of wave functions move with\nincreasing θare represented by the arrows on the circles, and\nthe values of wave functions at θ= 0 are indicated by the\nfilled or open small circles on the large circles. The inset\nshowsθ-dependences of absolute values of the wave functions\nfor the conduction band, |dA1(k0′)|(the blue dotted line),\n|dB1(k0′)|(the green solid line) and |dC1(k0′)|(the red dashed\nline. (Color Online)" }, { "title": "1702.06038v1.Parametric_pumping_of_spin_waves_by_acoustic_waves.pdf", "content": " 1 \n \n \nParametric pumping of spin waves by acoustic waves \n \nPratim Chowdhury, Albrecht Jander and Pallavi Dhagat \n \nSchool of Electrical Engineering and Computer Science , Oregon State University, Corvallis , USA \n \n \nThe linear and nonlinear interactions between spin waves (magnons) and acoustic waves \n(phonons) in magnetostrictive materials provide an exciting opportunity for realiz ing novel \nmicrowave signal processing devices1–3 and spintronic circuits4,5. Here we demonstrate the \nparametric pumping of spin waves by acoustic waves, the possibility of which has long been \ntheoretically anticipated6,7 but never exp erimentally realized . Spin waves propagating in a thin film \nof yttrium iron garnet (YIG), a magnetostrictive ferrimagnet with low spin and acoustic wave \ndamping , are pumped using an acoustic resonator driven at frequencies near twice the spin wave \nfrequency. The observation of a counter -propagating idler wave and a distinct pump threshold that \nincreases quadratically with frequency non-degeneracy are evidence of a nonlinear parametric \npumping process consistent with classical theory . This demonstration of acoustic parametric \npumping lays the groundwork for developing new spintronic and microwave signal processing \ndevices based on amplification and manipulation of sp in waves by efficient, spatially localized \nacoustic transducers . \nThe interaction between acoustic waves and spin waves includes both linear and nonlinear , parametric \neffects. The l inear coupling betwee n spin waves and acoustic waves, first contemplated t heoretically by Kittel6, has \nbeen shown to radiate acoustic waves from resonant ly excited ferromagnetic precession8 and, conversely , excite \nferromagnetic resonance9 and spin waves4,10,11 in ferromagnetic films upon application of acoustic waves. In the \nnonlinear coupling regime , parametric excitation of acoustic modes by spin waves , as observed in YIG spheres12 has \nbeen explained by theor y developed by Comstock13,14. The converse effect , the parametric pumping of spin waves \nby coherent acoustic waves, however, has not previously been experimentally demonstrated. \nThe parametric excitation of acoustic waves by spin waves is an important consideration in the design of \nferrite -based microwave devices. In most cases , to avoid loss of energy to the acoustic sy stem, the parametric \npumping threshold must not be exceeded15. In some devices such as frequency selective limiters1, however, the \nlosses are the basis of device function . The converse pumping of spin waves by acoustic waves could be similarly 2 \nYIG ZnO exploited for technological applications in signal processing , including in spin wave amplifiers, correlators and \nfrequency selective limiters of acoustic signals. In contrast to established methods of parametric pumping of spin \nwaves by electromagnetic wa ves16–18, acoustic pumping with piezoelectric transducers promises higher efficiency, \nlocalization and ease of integration with micro - and nano -scale circuits. Beyond novel signal processing \napplications, the recent discovery of spin -calori c effects19 and acoustically driven spin currents20,21 provides impetus \nto the study of magnon-phonon interactions to explain the fundamental processes underlying these phenomena. \nParametric pumping involves the nonlinear interaction between three waves, the signal wave a t frequency 𝑓𝑠, \nthe pump at frequency 𝑓𝑝 and the idler wave at frequency 𝑓𝑖. Energy conservation dictates that the three frequencies \nsatisfy the relation \n 𝑓𝑝=𝑓𝑠+ 𝑓𝑖. (1) \nIn the present experiments, the p ump is a standing acoustic wave that interacts with signal and idler spin wave s in a \nmagnetostrictive YIG film. \nIn the degenerate case where 𝑓𝑝 is equal to 2𝑓𝑠, the idler frequency is identical to the signal frequency , \nmaking it difficult to distinguish the idler from the inevitable electromagnetic feedthrough of the signal wave \nexcitation . As a result, although previous experiments13 showed modulation of spin wave transmission under the \ninfluence of an acoustic pump, they did no t convincingly demonstrate the parametric interaction . Here we observe \nnon-degenerate pumping, where the presence of the frequency -shifted, counter -propagating idler as well as a distinct \nthreshold for its appearance provide clear evidence of a nonlinear parametric pumping process. \nThe device used in our experiments, shown in Figure 1, consists of a thin film piezoelectric transducer \nfabricated on one side of a 0.5 mm thick gadolinium galliu m garnet (GGG) substrate with a ~12 µm thick epitaxia l \nYIG film on the opposite side . The transducer , excites l ongitudinal acoustic waves , which resonate in the acoustic \ncavity between the top and bottom free surfaces . The pump frequency is tuned to one of the high-order cavity \nresonance around 3 GHz to obtain la rge-amplitude standing acoustic waves in the YIG. The amplitude of the \nacoustic vibration is controlled by varying the power of the microwave signal applied to the transducer . (See \nMethods for details on device fabrication and calibration.) \nTwo microstrip antennas are used for excitation and detection of spin waves in the YIG film, which form s a \n1.3 mm wide spin wave waveguide spanning the 8 mm distance between the antennas and passing directly beneath \nthe acoustic transducer. A static magnetic bias field , HBIAS, is applied in the film plane , parallel to the waveguide , 3 \nsupporting the propagation of backward volume magnetostatic spin waves between the antennas . Since the pump is \northogonal to the spin waves (see Figure 1(a)), conservation of moment um requires that the idler spin wave \npropagate counter to the signal wave. \nWe first examine the propagation of the signal wave through the YIG waveguide , under the influence of the \nacoustic pump, using a vector network analyzer as illustrated in Fig ure 2. The pump frequency, 𝑓𝑝, is 3022.2 MHz , \ncorresponding to one of the acoustic cavity resonances. The signal spin waves are generated with 1 W applied to \nthe excitation antenna. The bias field is s et to 15.3 mT, the condition at which the transmission of spin waves at 𝑓𝑠=\n𝑓𝑝/2=1511.1 MHz is maximized in the absence o f the acoustic pump (see right -most trace in Figure 2). \nAs the power applied to the a coustic transducer is gradually increased, there is no discerni ble effect on signal \nwave transmission until a threshold of about 100 mW is reached . Beyond this threshold, up to 340 mW, the intensity \nof the transmitted spin wave increases with the acoustic pump power . \nWe postulate that the accompanying shift in the spin wave spectrum to lower frequencies is associated with a \nreduction in magnetization due to the pumping of spin waves from the signal wave into modes that do not couple to \nthe receiving antenna. At pump power levels beyond 340 mW, this background of spin waves causes excessive \nscattering of the signal wave, resulting in the reduction in transmitted power as well . \nNext we observe the counter propagating idler spin wave using a circulator at the excitation antenna as \nillustrated in Figure 3(a). The signal wave is excited using a microwave signal generator swept over a frequency \nrange from 𝑓𝑠=1503 MHz to 1519 MHz. Waves returning to the same antenna are routed to a spectrum analyzer \nthrough the circulator. The acoustic pump power is kept at a constant 340 mW. \nThe detected spectrum is plotted as a function of the signal frequency in Fig ure 3(b). The main diagonal is \nthe signal frequency, appearing here due to unavoidable reflections from the antenna and electromagnetic \nfeedthrough past the circulator. The parametrically pumped counter -propagating idler wave returning to the \ntransmitting antenna is clearly visible as off -diagonals. The plot is an overlay of the spectra observed for three pump \nfrequencies of 3015.5 MHz, 3022.2 MHz, and 3028.9 MHz (corresponding to adjacent resonant modes of the \nacoustic cavity) . In each case, the frequency relation of equation (1) is maintained . We note that these spectra are not \nvisible when the acoustic cavity is driven off -resonance, eliminating the possibility of electromagnetic interference \ncoupled with a non-linearity in the electronic system being the source of the observed frequencies. \nFinally, we examine quantitatively the threshold conditions for parametric pumping for the dege nerate as 4 \nwell as non -degenerate cases . For these experiments, the signal frequency was kept constant at 𝑓𝑠=1511.1 MHz \nwhile the pump frequency was shifted to different acoustic cavity resonances (𝑓𝑝=3008.8 , 3015.5 , 3022.2 , 3028.9 \nand 3035.6 MHz). The observed intensity of the counter -propagating idler wave as a function of the acoustic pump \npower is plotted in Fig ure 4. A clear threshold is seen in each case. The threshold increases the further the pump \nfrequency deviates from degeneracy ( ∆𝑓=𝑓𝑠−𝑓𝑝/2=50 kHz is as close as we can get to the degenerate case \nwhile still being able to distinguish the idler from the signal frequency). As seen in the inset to Fig ure 4, the intensity \n(represented here in amplitude squared, as measured by laser vibrometry22) of acoustic waves required to obtain \nparametric pumping increases quadratically with this frequency offset. We note that the parametric conversion is \nquite significant, with the intensity of the idler wave reaching nearly 6% of the transmitted wave intensity seen in \nFigure 2, assuming that the propagation and transducer losses are similar in both cases. \nA classical theory for parametric pumping of spin waves was derived by Schlömann , et a l.23 In the most \ngeneral form, equating the energy pumped into the wave wi th the damping losses leads to a pumping threshold given \nby \n 𝜂𝑘2=(𝑉𝑘ℎ𝑝)2−(2𝜋Δ𝑓)2, (2) \nwhere ℎ𝑝 is the threshold amplitude of a microwave magnetic pumping field , Vk is a coupling factor that depends on \nthe geometry of the device and Δ𝑓 is the offset in pumping freq uency from the degenerate case. The spin wave \nrelaxation rate, 𝜂𝑘, is related to the spin wave linewidth , Δ𝐻𝑘, by 𝜂𝑘=𝛾𝜇0Δ𝐻𝑘/2. Fitting the parabolic dependence \non Δ𝑓 to the experimentally determined thresholds (see red trace in inset of Fig ure 4), we obtain a spin wave \nlinewidth, Δ𝐻𝑘=85 A/m ( ~1 Oe), which is typical of the YIG films used. \nIn the context of our acoustically pumped device, ℎ𝑝 represents an effective magnetic field resulting from the \nmagneto elastic coupling in the ferromagnetic film. Expanding on the theory of Schlömann23, Keshtgar et al. recently \nderived an expression for the coupling of a longitudinal acoustic pump to backward vol ume magnetostatic waves24, \nwhich, (after accounting for typographical errors ) relates the pumping term in equation (2) to the amplitude, 𝑅, of \nthe acoustic wave as: \n 𝑉𝑘ℎ𝑝=𝛾𝐵1\n𝑀𝑠2𝜋𝑓𝑝\n𝑐𝑅. (3) \nHere 𝛾 is the gyromagnetic ratio, 𝐵1 the magnetoelastic coefficient , 𝑀𝑠 the saturation magnetization and 𝑐 the \nlongitudinal acoustic wave velocity of the magnetic film. Using equation (3) in equation (2), the threshold 5 \namplitude , 𝑅𝑐, for acoustic pumping in the degenerate case (Δ𝑓=0) is \n 𝑅𝑐=𝜇0Δ𝐻𝑘\n2𝑀𝑠\n𝐵1𝑐\n2𝜋𝑓𝑝. (3) \nFor YIG, we use 𝛾=2𝜋×28 GHz/T, 𝐵1=3.5×105 J/m3, 𝑀𝑠=1.4×105A/m and 𝑐=7.2 km/s, giving a \ntheoretical threshold acoustic amplitude of 𝑅𝑐=8.1 pm at 𝑓𝑝=3022 .2 MHz. The corresponding experimentally \ndetermined threshold of 39 pm is somewhat higher, but on the order of the predi cted value. A more comprehensive \ntheoretical model, which takes into account the finite extent of the pump region as well as the non -uniform \ndistribution of acoustic strain through the thickness of the film will be needed to resolve this discrepancy. \nNonetheless, t hese experiments demonstrate that parametric pumping of spin waves by acoustic waves is possible \nand provide insight into nonlinear phonon -magnon coupling in magnetostrictive materials . Localized and efficient \npiezoelectric transducers may th us, in the future, be used to generate, modulate and amplify spin wave signals via \nacoustic pumping in nonlinear microwave signal processing devices and magnonic logic circuits. \nMethods \nDevice fabrication \nThe ~12 m thick epitaxial YIG film was grown on a 0.5 mm thick single -crystal GGG substrate by liquid \nphase epitaxy. Using a wafer saw, t he substrate was subsequently cut into a 1.5 mm wide strip to form the spin wave \nwaveguide. The acoustic transducer was fabricated on the YIG/GGG strip by sputter deposi tion and shadow \nmasking. The active transducer area of approximately 1.3 mm square is defined by the overlap of 180 nm thick Al \nelectrodes sandwiching the 800 nm thick piezoelectric ZnO. Cu microstrip antennas (both 25 m wide) were \npatterned at the ends o f two coplanar waveguides on a printed circuit board . The device was taped to this board with \nthe YIG film facing down. The acoustic transducer was connected to a third coplanar waveguide by wire bonding. \nCalibration \nThe thickness -mode resonances of the acoustic cavity were determined using a network analyzer to display \nthe absorption spectrum ( S11) of the acoustic transducer as shown in Fig ure M1. The acoustic resonances are spaced \napproximately 6.7 MHz apart , limitin g the pump frequency to these discrete values. The amplitude of the acoustic \nvibration , as controlled by the applied microwave signal power , was calibrated using a heterodyne laser \nvibrometer22. At resonance, the combined effect of the transducer efficienc y and the quality factor of the cavity \nresult in standing acoustic waves having an amplitude of 3.3 pm/√𝑚𝑊. 6 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1 | Schematic and photograph of the experimental device . (a) The device is comprised of an \nacoustic transducer and a YIG spin-wave waveguide on opposite surfaces of a GGG substrate. Microstrip \nantennas are used to excite and detect spin waves (represented by the wavy blue arrow) in the waveguide. \nThe acoustic transducer consists of a piezoelectric ZnO film sandwiched between Al electrodes. \nLongitudinal acoustic waves generated by the transducer resonate in t he device creating standing waves , \nas illustrated in red. (b) A photograph of the device. The scale bar is 2 mm. \n \na \nb \nMicrostrip \nantenna \nGGG \nYIG \nSpin wave \nsignal , fs \nHBIAS \nAcoustic p ump, fp \nAcoustic \ntransducer \nYIG \nGGG \nMicrostrip antennas 7 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2 | Spin wave transmission spectra. The transmission spectrum of the signal spin wave s through \nthe YIG waveguide , as measured by a vector network analyzer (VNA), is shown for various levels of \npower applied to the acoustic transducer. The schematic, inset in the top left, shows the experimental \nsetup. The signal spin waves are gen erated with a microwave power of 1 µW applied to the excitation \nantenna under a bias field of 15.3 mT . \n \nVNA \n15.3 mT 8 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3 | Counter -propagating idler spin waves. (a) Sc hematic of experimental setup for observing \ncounter -propagating idler spin waves. A spectrum analyzer (SA) connected to a circulator is used to \nmeasure the frequency spectrum of wave s returning to the excitation antenna. (b) Spectra of waves \nreturning to t he excitation antenna versus signal wave frequency. The strong main diagonal is primarily \nfeedthrough of the excitation signal. The off -diagonals show the counter -propagating idler waves that are \nparametrically excited from the signal wave at different aco ustic pump frequencies. The power applied to \nthe acoustic transducer is 340 mW. The signal spin waves are generated under a bias field of 15.3 mT and \n1 µW applied to the excitation antenna . \n \n \n \na \n b \nSA \n15.3 mT \nfs \nfp 9 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4 | Acoustic parametric pumping of spin waves. Parametrically generated idler wave intensity \nas a func tion of acoustic pump power, plotted for different conditions of frequency non -degeneracy . The \nsignal spin waves are excited with 1 µW applied to the excitati on antenna . The inset shows the threshold \nacoustic intensity (in units of amplitude squared) versus frequency offset. 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Instability of spin waves and magnetostatic modes i n a microwave \nmagnetic field applied parallel to the dc field. J. Appl. Phys. 32, 1006–1014 (1961). \n24. Keshtgar, H., Zareyan, M. & Bauer, G. E. W. Acoustic parametric pumping of spin waves. Solid \nState Commun. 198, 30–34 (2014). 13 \nAcknowledgement s \nThis work was supported in part by the National Science Foundation (Award No. 1414416 ). \n \nAuthor contributions \nP.C. fabricated the devices, performed the measurements and prepared the figures in this manuscript. A.J. \nand P.D. supervised the work, devised th e experiments, interpreted the results and wrote the manuscript. \n \n1 14 \n " }] \ No newline at end of file